MATH1012 Lecture/Tutorial Notes
Instructional Assistant: Mr Zeta CHAN
Web Page: http://ihome.ust.hk/~mazeta
Office: Room 3012 (through Math Support Center (Room 3010)) (via lift 2/lift 3)
E-mail Address: [email protected]
Office Hours: refer to the web page
Table of Contents
Part I: Pre-calculus
Binomial Theorem (out of the syllabus) ………………………………………………………P. 2
Radian Measure ……………………………………………………………………………P. 4
Cotangent, Secant and Cosecant ……………………………………………………………P. 7
Absolute Value
……………………………………………………………………………P. 10
Functions ………………………………………………………………………………………P. 12
Injective Functions ……………………………………………………………………………P. 17
Odd and Even Functions ………………………………………………………………………P. 18
Composite Functions ………………………………………………………………………P. 19
Transformations of Graphs …………………………………………………………………P. 20
Inverse Functions ……………………………………………………………………………P. 21
Exponential and Logarithmic Functions ………………………………………………………P. 23
Trigonometric Functions and Their Inverses …………………………………………………P. 25
Part II: Limit
Limits and Continuity at a Point ……………………………………………………………P. 28
Limits about Infinities, Vertical and Horizontal Asymptotes …………………………………P. 41
Continuity on an Interval
…………………………………………………………………P. 53
Part III: Differentiation
Differentiation Techniques
…………………………………………………………………P. 55
Differentiability
……………………………………………………………………………P. 75
Equations of Tangent Lines …………………………………………………………………P. 77
Extrema and Curve Sketching
……………………………………………………………P. 79
Related Rates …………………………………………………………………………………P. 82
Optimization …………………………………………………………………………………P. 86
Mean Value Theorem ………………………………………………………………………P. 90
L’Hospital’s Rule ……………………………………………………………………………P. 91
Part IV: Integration
Indefinite Integral ……………………………………………………………………………P. 95
Summation …………………………………………………………………………………P. 107
Definite Integral ……………………………………………………………………………P. 108
Fundamental Theorem of Calculus ……………………………………………………………P. 113
P. 1
Binomial Theorem (out of the syllabus)

Pascal’s Triangle:
n=0
n=1
n=2
n=3
n=4
n=5



1
1
1
1
1
1
2
3
4
1
3
6
1
4
1
1
5
10 10 5
1
n
In a row, the (k + 1)th number is   .
k 
Other terms can be obtained by adding the two nearest terms in the row above it.
It is symmetric about the central axis.

 n  n  n  n 1
 n n
n
Binomial Theorem: (a  b)   a   a b     b
 0
1
 n
1.
Expand (x + 2)5.
2.
Expand (2x + y)4.
3.
Expand (x – a)3.
4.
Expand (2x – 3y)6.
5.
Expand (3x – 1)5.
6.
Expand (1 + at)6.
7.
Expand (2 + x2)5.
8.
Expand (m2 + n)6.
9.
Expand (ab + 5y)4.
10. Expand (2t 
1 5
) .
t2
11. (a) Expand (1 + ax)4.
(b) If the coefficient of x in the expansion of (1 + ax)4 is 8, find the constant a.
P. 2
1
1
12. Simplify ( x 2  ) 4  ( x 2  ) 4 .
x
x
13. (a) Simplify (a + b)5 + (a – b)5.
(b) Hence, express ( 3  1)5  ( 3  1)5 in the simplest surd form.
14. Expand [1 + (x + x2)]3.
15. Expand (1 + x + 2x2)3.
16. Expand (1 – 2x + x2)3.
17. Expand (1 + x + x2)6 in ascending powers of x as far as the term in x3.
P. 3


Radian Measure
π radians = π rad = 180°= a straight angle
When angles are measured in radian, the unit may be omitted.


= a straight angle divided by n
n

Length of arc = s = 2r 

Area of sector = A = r 2 
1.
Complete the following table.
Degree
measure

 r
2

1
 r 2
2 2
10° 15° 30°

5
Radian
measure
2.
60° 90°
225°
7
6

4
300°
3
2
750°
3π
Complete the following table.
0

6

4
sin
2
2
cos
2
2

3

2
1
3
tan


and ∠B = . Find ∠C in radians.
4
6
3.
In ∆ABC, ∠A =
4.
The radius of a circle is 25cm. Find the length of the arc and the area of the sector if the central
angle is
5.
5
.
18
The radius of a circle with centre O is 4cm. The angle subtended at the centre by an arc AB is

. Find the length of the arc AB and the area of the sector AOB.
3
6.
The area of a sector is 12cm2 and the length of the arc is 4cm. Find the radius of the circle and
its central angle.
P. 4
7.
The area of a sector is 48cm2 and its perimeter is 32cm. Find the radius of the circle.
8.
The radius of the Earth at the equator is 6378km. If two places at the equator differ by

180
in
longitude, find the distance apart in km.
9.
Points A, B, C and D divide the circumference of a circle in the ratio 1 : 2 : 3 : 4. Find the
angles of the cyclic quadrilateral ABCD in radians.
10. Assume that the Earth revolves around the Sun in a circular orbit with uniform speed and it
takes 365 days for a complete revolution. It is given that the distance between the Sun and the
Earth is 1.50×108km. If the Earth revolved for 7 days, find
(a) the angle subtended at the centre,
(b) the area of the sector formed.
11. ABCD is a square of side 4cm. Two arcs BD and AC with centres C and D respectively are
drawn. Find the area bounded by the arcs and the side BC.
12. On a plane, two wheels of radii 70cm and 10cm have their centres 120cm apart. Find the
length of the belt which passes tightly around the wheels.
13. A wire 20cm long is bent into a sector of radius rcm and arc length scm.
(a) Express s in terms of r.
(b) Find the maximum area and the corresponding radius.
(c) Find the central angle for the maximum area.
14. A right circular cone is 10cm in diameter and 12cm high. An ant, starting at a point A on the
circumference of the base, moves around the lateral face for a loop and returns to A. Find the
length of the shortest path of the ant.
15. Evaluate cos
2
.
3
16. Evaluate tan
5
.
3
17. Evaluate sin(
4
).
3
P. 5
18. Evaluate cos(
7
).
6
19. Evaluate tan
9
.
4
20. Evaluate sin
7
.
3
21. Evaluate sin
23
.
4
22. Evaluate sin
50
.
3
23. Evaluate sin
5
7
5
7
.
cos
 cos sin
6
4
6
4
24. Simplify cos(–π – θ).
25. Simplify sin( 
3
).
2
26. Simplify tan( 
3
).
2

27. Simplify sin(  A) cos( A  ) .
2
sin(
28. Simplify

2
  ) cos(

2
cos(   )
)

sin(   ) cos(

2
sin(   )
P. 6
)
.
Cotangent, Secant and Cosecant
1
1
1
, sec 
and csc 
tan
cos 
sin 

New trigonometric ratios: cot  

2
2
2
2
2
2
Pythagorean Identities: sin   cos   1 , tan   1  sec  and cot   1  csc 
7
Evaluate sec(  ) .
6
1.
2.
Evaluate cot(
17
).
3
3.
Evaluate csc( 
20
).
3
4.
Evaluate cot(
49
).
4
5.
Evaluate cos 2
6.
Simplify cot(–2π + θ).
7.
Simplify sec(θ – π).
8.
Simplify csc( 
9.
Simplify sec(
5
11

csc
 cos( ) .
4
6
3
3
).
2
3
  ) tan(   ) cos( ) .
2
10. Simplify tan( A 
3

) csc(  A) .
2
2
11. Simplify sin(  A) cos(  A) cot(
3

 A)  sin(  A) cos(2  A) .
2
2
3
)
sin(2   )
cos( )
2


12. Simplify
.


cot(   )
cos(   ) sin(   )
2
2
tan(
P. 7
3
)
tan( A   )
2  sin( A  2 ) .
13. Simplify

3

cos( A)
csc(  A) cot(  A)
2
2
tan( A 
3
3
 A) cot(  A)
2
2
14. Simplify
.
3

sec( 2  A) cos(  A) cot(  A)
2
2
sin(  A) csc(
3

3
 B) sin(  B) cot(B  )
2
2
2 .
3
sin( B) cos(  B) cot(  B)
2
sin(
15. Simplify
16. Simplify
sec(  )  cos(   )
.
csc(   )  sin(2   )
17. If 2tan2θ + 3sec2θ = 18 and
18. If sin   
19. If sin   
22. If sin  
2
    , find the value of cos θ.
4
3
and    
, find the values of cos θ and cot θ.
2
5
1
2
20. If cot θ = –1 and
21. If cos   

and

2
3
   2 , find the values of sec θ and cot θ.
2
    , find the values of sin θ and cos θ.
24
and sin θ < 0, find the values of sec θ and cot θ.
25
4
and cos α < 0, find the value of tan α – cos α.
5
23. If ∠A is in the second quadrant and sin A 
12
, find the value of 2cot A – 5csc A.
13
P. 8
24. Given that cos  
sin   cos 
1
3
and
.
   2 , find the value of
2
2
tan   sec 
25. Given that cos   
1

3
and cot β = 1, where
, find the value of
    and    
2
2
2
2 sin   sec 
.
2 cos   csc 
26. Suppose that tan A   3 and tan B 
3
, where

2
 A   and   B 
3
. Find the
2
cot A cot B  1
csc A  sec B
and
 tan A cot B .
cot B  cot A
sin A  cos B
values of
27. Given that sin   
and tan(
1
4

and θ is in the 4th quadrant, find the values of cos θ, cot θ, cos(   )
5
2
3
) .
2
28. Given that sec  
p
, where p and q are positive, and csc α < 0, find the values of tan α and
q
sin α.
29. If sin  
t 1

and
    , find the values of cos α and cot α.
t 1
2
30. If tan  
m2  n2
and m ≠ n, find the values of cos θ and csc θ.
2mn
P. 9
Absolute Value

 x if x  0
Definition: x  
 x if x  0

x2  x

ab  a b

a |a|

, where b ≠ 0
b |b|

x p  x , whenever xp is defined
p
1.
Solve the equation |x| = 7.
2.
Solve the equation |3x – 5| = 31.
3.
Solve the equation |x| + 9 = 8.
4.
Solve the equation |21 – (x – 3)| = 6.
5.
Solve the equation |x2 – 5x| = 6.
6.
Solve the equation |x2 – 3x – 4| = 6.
7.
Solve the equation |x2 + x – 13| = 7.
8.
Solve the inequality |x| < 2.
9.
Solve the inequality |x – 5| > 3.
10. Solve the inequality |x – 6| < 9.
11. Solve the inequality |7 + x| > 0.
12. Solve the inequality |2x + 7| ≥ 13.
13. Solve the inequality |5 – 3x| ≤ 10.
14. Solve the inequality 2 + |x| ≥ 8.
P. 10
15. Solve the inequality |x2 – 3x – 4| > 6.
16. Solve the inequality |x2 + 2x – 16| < 8.
17. Solve the inequality |x – 2|(x – 4) > 0.
18. Solve the inequality (3x – 1)|x – 2| < 2.
19. Solve the inequality (x – 3)|x + 1| < –3.
20. Solve the inequality
4x  1
 2.
3x  5
21. Solve the inequality |x – 1| < 2x.
22. Solve the inequality |x – 2| + |x + 3| ≥ 6.
23. Solve the inequality |x – 2| + |x + 3| < 4.
24. Consider the function f ( x)  10  x  3 | x  2 | .
(a) Find the domain of f.
(b) Solve the equation f (x) = 2.
P. 11


Functions
If a relation maps every x to a unique y, then the relation is said to be a function.
To determine whether a relation is a function, we use the vertical line test: If there is a vertical
line intersecting the graph more than once, then it is not a function. Otherwise, it is a function.

The domain of a function is the largest set of real numbers which can be mapped by the
function.




The range of a function is the largest set of real numbers to which the function maps.
The open interval (a, b) is the set {x: a < x < b}, where a can be –∞ and b can be ∞.
The open-closed interval (a, b] is the set {x: a < x ≤ b}, where a can be –∞.
The closed-open interval [a, b) is the set {x: a ≤ x < b}, where b can be ∞.



The closed interval [a, b] is the set {x: a ≤ x ≤ b}.
The union of two sets A and B which contains all elements in A and B is A  B .
{a, b} is the set containing a and b.


R = (–∞, ∞) is the set of all real numbers.
R\{a} = (, a)  (a, ) is the set of all real numbers except a.

The independent variable in the formula representation of a function is dummy, i.e. “f (x) =
x2” can be changed to “f (y) = y2”.
Is y a function of x for the following graph?
1.
y
4
2.
x
O
4
Is y a function of x for the following graph?
y
4
3.
x
O
4
Is y a function of x for the following graph?
y
4
O
P. 12
x
4
4.
Is y a function of x for the following graph?
y
4
5.
x
O
4
Is y a function of x for the following graph?
y
4
6.
x
O
4
Is y a function of x for the following graph?
y
4
7.
x
O
4
Is y a function of x for the following graph?
y
4
8.
x
O
4
Is y a function of x for the following graph?
y
4
O
P. 13
x
4
9.
Is y a function of x for the following graph?
y
4
x
O
4
10. Is y a function of x for the following graph?
y
4
x
O
4
11. Is y a function of x for the following graph?
y
4
O
12. Is y a function of x for the following graph?
13. Is y a function of x for the following graph?
P. 14
x
4
14. Is y a function of x for the following graph?
15. Find the domain and range of the function f (x) = 3x4 – 10.
16. Find the domain of the function g ( x)  ( x 2  4) x  5 .
17. Find the domain and range of the function h(u)  3 u  1 .
18. Find the domain and range of the function F (w)  4 2  w .
19. Find the domain and range of the function f ( x)  4  x 2 .
20. Find the domain and range of the function f ( x)  ( x  1)( x  2) .
21. Find the domain and range of the function f (x) = (9 – x2)3/2.
22. Find the domain and range of the function g (t ) 
1
.
1 t 2
23. Find the domain and range of the function g ( y ) 
24. If f (2x) = x + 5, find the value of f (6).
x
25. Given that g ( )  2 x 2  4 x  3 .
2
(a) Find the value of g(5).
(b) Find g(x).
P. 15
y 1
.
( y  2)( y  3)
x
26. (a) Given that g ( )  4 x  16 . Find g(x).
2
x
(b) Do the functions g(x) and g ( ) have the same domain?
2
27. If g(x + 1) = 2x2 + 4x + 2, find g(x).
28. Let g(2x – 1) = 6x – 7. Find g(x).
P. 16

Injective Functions
For a function, if no two distinct x’s are mapped to the same y, then the function is said to be
1.
injective/one-to-one.
To determine whether a function is injective/one-to-one, we use the horizontal line test: If
there is a horizontal line intersecting the graph more than once, then it is not
injective/one-to-one. Otherwise, it is injective/one-to-one.
Is the function f (x) = 3x + 4 injective/one-to-one?
2.
Is the function f (x) = |2x + 1| injective/one-to-one?
3.
Is the function f ( x) 
4.
Is the function f (x) = –(6 – x)2 injective/one-to-one?
5.
Is the function f ( x) 
6.
Is the function f (x) = x2 – 2x + 8 injective/one-to-one?

1
injective/one-to-one?
x5
1
injective/one-to-one?
x2
P. 17




1.
Odd and Even Functions
If f (–x) = f (x) for all x in its domain, then f is said to be even.
The graph of an even function is symmetric about the vertical axis.
If f (–x) = – f (x) for all x in its domain, then f is said to be odd.
The graph of an odd function has a rotational symmetry of order 2 about the origin.
Determine whether each of the following functions is odd, even or none of them.
(a) f (x) = x2 + x4 + x6 + x8
(b) f (x) = x3 + x5 + x7
(c) f (x) = 3x2 + 5x4 + 7x6
(d) f (x) = 2x3 + 4x5 + 6x7
(e) f (x) = 7
(f) f (x) = x5 – x3 – 2
1
x
(g)
f ( x) 
(h)
f ( x)  3 x
(i)
f ( x)  4 x
(j)
f (x) = x3/5
(k)
(l)
(m)
(n)
(o)
(p)
f (x) = 2|x|
f (x) = x|x|
f (x) = log|x|
f (x) = sin x
f (x) = cos x
f (x) = tan x
P. 18

Composite Functions
Given two functions f and g, the composite function f  g is defined by
( f  g )( x)  f ( g ( x)) .
The domain of f  g is the set of all x in the domain of g such that g(x) is in the domain of f.
1.
Let f (x) = x2 – 4, g(x) = x3 and F ( x) 

1
. Find the following functions and their domains.
x3
(a) F(g(y))
(b) g(F( f (x)))
2.
(c)
f ( x  4)
(d)
F(
3x  1
)
x
Let f (x) = |x|, g(x) = x2 – 4, F ( x)  x and G ( x) 
1
. Find the following functions and
x2
their domains.
(a) f  G
(b) G  G
(c) f  g  G
(d) G  g  f
(e) F  g  g
3.
Let g(x) = x2 + 3. Solve f for each of the following equations.
(a) ( f  g )( x)  x 2
1
x 3
( f  g )( x)  x 4  6 x 2  9
(b) ( f  g )( x) 
2
(c)
(d) ( f  g )( x)  x 4  6 x 2  20
(e) ( g  f )( x)  x 4  3
(f) ( g  f )( x)  x 2 / 3  3
4.
Find a polynomial f that satisfies ( f (x))2 = 9x2 – 12x + 4.
5.
Find a polynomial f that satisfies ( f (x))2 = x4 – 12x2 + 36.
6.
Find a polynomial f that satisfies f ( f (x)) = x4 – 12x2 + 30.
7.
Find a polynomial f that satisfies f ( f (x)) = 9x – 8.
P. 19
Transformations of Graphs
y = f (x)
Translation
Vertical Transformation
k units upward
Enlargement/Reduction k times
Horizontal Transformation
y = f (x) + k
k units leftward
y = f (x + k)
y = k f (x)
1
times
k
y = f (k x)
Whole Graph Reflection
about the x-axis
y = – f (x)
about the y-axis
y = f (–x)
Half Graph Reflection
reflect upward
y = | f (x)|
reflect leftward
y = f (|x|)

1.
DO NOT follow the order you learnt in primary school for successive horizontal
transformations.
Describe the transformation from y = f (x) to y = f (x – 1) + 2.
2.
Describe the transformation from y = f (x) to y = f (x – 2) + 1.
3.
Describe the transformation from y = f (x) to y = 2f (x + 3).
4.
Describe the transformation from y = f (x) to y = 3f (x – 1) – 5.
5.
Describe the transformation from y = f (x) to y = –3f (x – 2) + 4.
6.
Describe the transformation from y = f (x) to y = f (2x – 4).
7.
Describe the transformation from y = f (x) to y = 2f (2x – 1).
8.
Describe the transformation from y = f (x) to y = f (3x – 6) + 1.
9.
Describe the transformation from y = f (x) to y  6 f (
x2
)  1.
3
10. Describe the transformation from y = f (x) to y = – f (2x + 1) + 13.
P. 20





1.
Inverse Functions
Let f be an injective/one-to-one function with domain D and range R. Its inverse function f –1
is defined with domain R and range D by f –1(y) = x, where f (x) = y, for all y in R.
domain of f –1 = range of f
range of f –1 = domain of f
f –1( f (x)) = x for all x in D.
f ( f –1(y)) = y for all y in R.
The graph of y = f –1(x) can be obtained by reflecting the graph of y = f (x) about the line y = x.
Find the inverse function and its domain and range for the function f (x) = 2x.
2.
Find the inverse function and its domain and range for the function f (x) = 3x + 5.
3.
Find the inverse function and its domain and range for the function f ( x) 
4.
Find the inverse function and its domain and range for the function f (x) = 6 – 4x.
5.
Find the inverse function and its domain and range for the function f (x) = x2 + 4, where the
domain of f is [0, ∞).
6.
Find the inverse function and its domain and range for the function f (x) = 3x3.
7.
Find the inverse function and its domain and range for the function f (x) = x4 + 4, where the
domain of f is (–∞, 0].
8.
Find the inverse function and its domain and range for the function f ( x)  x  2 .
9.
Find the inverse function and its domain and range for the function f ( x)  3  x .

x
 1.
4
10. Find the inverse function and its domain and range for the function f ( x) 
1
.
x5
11. Find the inverse function and its domain and range for the function f ( x) 
2x
.
x2
12. Find the inverse function and its domain and range for the function f ( x) 
2
, where the
x 1
domain of f is [0, ∞).
P. 21
2
13. Find the inverse function and its domain and range for the function f (x) = x2 – 2x + 6, where
the domain of f is [1, ∞).
14. Find the inverse function and its domain and range for the function f (x) = –x2 – 4x – 3, where
the domain of f is (–∞, –2].
15. Find the inverse function and its domain and range for the function f ( x) 
domain of f is [0, ∞)\{3}.
P. 22
6
, where the
x 9
2


Exponential and Logarithmic Functions
The domain and range of the exponential function f (x) = bx, where b > 0 and b ≠ 1, are R and
(0, ∞) respectively.



The domain and range of the logarithmic function g(x) = logbx, where b > 0 and b ≠ 1, are (0,
∞) and R respectively.
logbx is the answer of the question “What power of b is x?”.
The logarithmic function g(x) = logbx is the inverse of the exponential function f (x) = bx.
logbbx = x for any real number x.


blogb x  x for any x > 0.
The graph of y = g(x) = logbx is the reflection image of the graph of y = f (x) = bx about the line
y = x.
n
1
2
3
4
5
10
100
1000
1
(1  ) n
n
2
2.25
2.3704
2.4414
2.4883
2.5937
2.7048
2.7169

1
1
It can be proved that (1  ) n increases as n increases and (1  ) n is always less than 3.
n
n

1
The constant e ≈ 2.71828182845904 is the number that (1  ) n approaches as n increases.
n

e is an irrational number, like π.





The natural exponential function F(x) = ex is the exponential function with base e.
The natural logarithm function G(x) = ln x is the logarithmic function with base e, i.e. ln x =
logex.
ln x is the answer of the question “What power of e is x?”.
The natural logarithm function G(x) = ln x is the inverse of the natural exponential function
F(x) = ex.
ln ex = x for any real number x.
eln x = x for any x > 0.
The graph of y = G(x) = ln x is the reflection image of the graph of y = F(x) = ex about the line
1.
y = x.
Solve the equation ln x = 3.
2.
Solve the equation ln x = –1.
3.
Solve the equation ln x 


1
.
3
P. 23
4.
Let ln x = 0.36, ln y = 0.56 and ln z = 0.83. Evaluate the following expressions.
x
(a) ln
y
(b) ln x2
(c) ln xz
(d) ln
(e)
ln 3
(f)
ln
xy
z
x
z
e2 x5/ 2
y
5.
100 grams of a particular radioactive substance decays according to the function m(t) =
100e–t/650, where t > 0 measures time in years. When does the mass reach 50 grams?
6.
The population P of a small town is growing according to the function P(t) = 100et/50, where t
measures the number of years after 2010. How long does it take the population to double?
7.
Convert 2x to base e.
8.
Convert log2(x2 + 1) to base e.
9.
Convert 10 ln 10 to base e.
1
P. 24
Trigonometric Functions and Their Inverses
Domain:
Range:
sin
cos
tan
R
R
[–1, 1]
[–1, 1]
R\{
cot
(2n  1)
}
2
R
R\{nπ}
R
sec
(2n  1)
}
2
(,1]  [1, )
R\{
csc
R\{nπ}
(,1]  [1, )

In order to define the inverses of the trigonometric functions, their domains have to be
restricted so that they become injective/one-to-one.

The inverse of sine is the inverse sine function or the arcsine function, denoted by sin–1 or
arcsin.
The inverse of cosine is the inverse cosine function or the arccosine function, denoted by





cos–1 or arccos.
The inverse of tangent is the inverse tangent function or the arctangent function, denoted by
tan–1 or arctan.
The inverse of cotangent is the inverse cotangent function or the arccotangent function,
denoted by cot–1 or arccot.
The inverse of secant is the inverse secant function or the arcsecant function, denoted by
sec–1 or arcsec.
The inverse of cosecant is the inverse cosecant function or the arccosecant function, denoted
by csc–1 or arccsc.
sin
Domain:
Range:
cos
tan
[ , ]
2 2
[0, π]
( , )
2 2
(0, π)
[–1, 1]
[–1, 1]
R
R
 
sin
Domain:
Range:
–1
cos
[–1, 1]
 
[ , ]
2 2
1.
Evaluate sin–11.
2.
Evaluate tan–11.
3.
Evaluate cos–1(–1).
4.
Evaluate sin–1(–1).
5.
Evaluate csc–1(–1).
6.
Evaluate sin 1
–1
cot
 
–1
cot–1
tan
[–1, 1]
R
[0, π]
( , )
2 2
R
 
(0, π)
3
.
2
P. 25
sec
csc

3
[0, )  [ , )
2
2
(,1]  [1, )

3
(0, ]  ( , ]
2
2
(,1]  [1, )
sec–1
(,1]  [1, )
csc–1
(,1]  [1, )

3
[0, )  [ , )
2
2

3
(0, ]  ( , ]
2
2
2
).
2
7.
Evaluate cos 1 (
8.
Evaluate cos–12.
9.
1
Evaluate cos 1 ( ) .
2
10. Evaluate tan 1 3 .
11. Evaluate cot 1 (
1
3
).
12. Evaluate sec–12.
13. Evaluate cos(cos–1(–1)).
14. Evaluate cos 1 (cos

7
).
6
In this case, cos–1 is not the inverse of cos.

15. Evaluate tan 1 (tan ) .
4
16. Evaluate tan(tan–11).
17. Evaluate tan 1 (tan
3
).
4
18. Evaluate csc–1(sec 2).
19. Simplify tan(cos–1x).
20. Simplify cot(tan–12x).
x
21. Simplify cos(sin 1 ( )) .
3
22. Simplify cos(tan 1 (
x
9  x2
)) .
P. 26
23. Given that sin2θ = 2sinθcosθ, simplify sin(2cos–1x).
24. Given that cos2θ = cos2θ – sin2θ = 1 – 2sin2θ = 2cos2θ – 1, simplify cos(2sin–1x).
P. 27
Limits and Continuity at a Point

As x approaches 3, f (x) approaches 4. lim f ( x)  4 .


The y-coordinate of the hole with two tails is the limit. (not a precise statement)
The y-coordinate of the dot is the function value.

lim f ( x)  3 . f (2) = 5. lim f ( x)  f (2) . f (x) is discontinuous at x = 2.
x 3
x 2
x 2
P. 28

lim f ( x)  2 . f (1) = 2. lim f ( x)  f (1) . f (x) is continuous at x = 1.
x 1
x 1

Let lim f ( x) and f (a) both exist. f (x) is continuous at x = a if and only if lim f ( x)  f (a) .

Left hand limit (the y-coordinate of the hole with a left tail): As x approaches 2 from the left,
x a
x a
f (x) approaches 3. lim f ( x)  3 .
x 2

Right hand limit (the y-coordinate of the hole with a right tail): As x approaches 2 from the
right, f (x) approaches 3. lim f ( x)  3 .
x 2

lim f ( x) and lim f ( x) both exist and lim f ( x)  lim f ( x)  L if and only if lim f ( x)
x a 
x a
x a
exists and lim f ( x)  L .
x a
P. 29
x a
x a

Let lim f ( x) and f (a) both exist. f (x) is continuous from the left/left-continuous at x = a
x a
if and only if lim f ( x)  f (a) .
x a

Let lim f ( x) and f (a) both exist. f (x) is continuous from the right/right-continuous at x
x a
= a if and only if lim f ( x)  f (a) .
x a

Methods for evaluating limits:

Substitute (continuity assumed)

Reduce

Multiply by

the conjugate
the conjugate
Reduce by the highest power of x in the denominator (for lim f ( x) / lim f ( x) only)
x 

Formulas, e.g. lim


Change of variable
L’Hospital’s Rule

sin 
 0

x 
1
f (  )  f (a)
 1 , lim (1  ) x  e and lim
 f (a)
x 
 a
 a
x
Squeeze Theorem/Sandwich Theorem: If f (x) ≤ g(x) ≤ h(x) on an open interval
containing a, except possibly at x = a and, lim f ( x)  lim h( x)  L , then lim g ( x) exists
x a
and lim g ( x)  L .
x a
1.
Let f ( x)  sin
1
.
x
(a) Find the value of f (
1
) , where n is an integer.
n
(b) Find the value of f (
2
) , where n is an integer.
(2n  1)
(c) Comment on lim f ( x) .
x 0
2.
Evaluate lim
3.
Evaluate lim
x 1
8x  1
.
x3
4 x  4 x
.
x  0 4 x  4 x
P. 30
x a
x a
4.
1 1

Evaluate lim r 3 .
r 3 r  3
5.
1
1

Evaluate lim 5  h 5 .
h 0
h
6.
1
1

Evaluate lim 2  sin  2 .
 0
sin 
7.
Evaluate lim (
8.
Evaluate lim
9.
Evaluate lim
x 2
x 0
1
2
 2
).
x  2 x  2x
(1  2 x)(1  3x)(1  4 x)  1
.
x
(1  x) 4  (1  4 x)
.
x 0
x  x4
( x  a) 50  x  a
.
x a
xa
10. Evaluate lim
11. Evaluate lim
x 4
x 2  16
.
4 x
e2 x  1
.
x 0 e x  1
12. Evaluate lim
(2 x  1) 2  9
.
x 1
x 1
13. Evaluate lim
14. Evaluate lim
x c
x 2  2cx  c 2
.
xc
x3  8
.
x2 x  2
15. Evaluate lim
P. 31
( y  2)3
.
y  2 y 4  16
16. Evaluate lim
17. Evaluate lim
x  1
x 1
.
x  2x  3
2
cos 2 x  3 cos x  2
.
x 
cos x  1
18. Evaluate lim
19. Evaluate lim
x
3
2
sin 2 x  6 sin x  5
.
sin 2 x  1
x2  5x  6
.
x 3 x 2  2 x  15
20. Evaluate lim
x 2  3x  2
.
x 1 x 2  5 x  6
21. Evaluate lim
x2  1
.
x 1 3 x 2  x  2
22. Evaluate lim
3t 2  7t  2
.
t 2
2t
23. Evaluate lim
w 2  5kw  4k 2
.
w  k
w 2  kw
24. Evaluate lim
25. Evaluate lim 4
x  3
x2  5x  6
.
x 2  x  12
x3  x 2  2 x  2
.
x 1
x2  x  2
26. Evaluate lim
27. Evaluate lim
x 9
28. Evaluate lim
x
2
x 3
.
x9
sin x  1
sin x  1
.
P. 32
29. Evaluate lim
16  h  4
.
h
30. Evaluate lim
2 x 2
.
x2
h 0
x2
31. Evaluate lim
x 1
x 1
4x  5  3
.
32. Evaluate lim
x2  5  3
.
x2  4
33. Evaluate lim
x
.
x2 2
x2
x 0
34. Evaluate lim
x 4
35. Evaluate lim
x 0
36. Evaluate lim
x a
37. Evaluate lim
x 0
38. Evaluate lim
t a
3( x  4) x  5
3 x 5
.
1 x  1 x
.
x
x2  a2
x a
, where a > 0.
x
cx  1  1
.
3t  1  3a  1
.
ta
a  a2  x2
39. Evaluate lim
, where a > 0.
x 0
x2
40. Evaluate lim
x2
x  2  3x  2
.
5x  1  4 x  1
P. 33
x7 1
.
x  1 x  1
a2 – b2 = (a – b)(a + b)
a3 – b3 = (a – b)(a2 + ab + b2)
an – bn = (a – b)(an – 1 + an – 2b + … + bn – 1)
a2 + b2 cannot be factorized.
a3 + b3 = (a + b)(a2 – ab + b2)
an + bn = (a + b)(an – 1 – an – 2b + … + bn – 1) for odd n
41. Evaluate lim






xn  an
, where n is a positive integer.
xa
42. Evaluate lim
x a
x 2
.
x  16
4
43. Evaluate lim
x 16
3
1 h 1
.
h
3
x 1
.
x 1
44. Evaluate lim
h 0
45. Evaluate lim
x 1
46. Find the constants b and c such that lim
x 2
x 2  bx  c
 6.
x2
x
.
x  0 cos x
47. Evaluate lim
48. Evaluate lim
x 0
sin 4 x
.
x
sin
49. Evaluate lim
x 0
x
x
5.
sin 5 x
.
x  0 sin 3 x
50. Evaluate lim
51. Evaluate lim
x 0
sin ax
, where b ≠ 0.
bx
P. 34
tan 5 x
.
x 0
x
52. Evaluate lim
53. Evaluate lim
x 0
tan 7 x
.
sin x
sin 3x
.
x  0 tan 4 x
54. Evaluate lim
55. Evaluate lim
 0
tan 9
.
tan 6
sin x 2
.
x 0
x
56. Evaluate lim
57. Evaluate lim
sin 2 8
2
 0
.
sin x  x
.
x 0
3x
58. Evaluate lim
59. Evaluate lim
x 0
6 x  sin 2 x
.
2 x  3 sin 4 x
sin( x  2)
.
x2
x2  4
60. Evaluate lim
61. Evaluate lim
sin( x  3)
.
x 2  8 x  15
62. Evaluate lim
sin 4 x
.
x2 2
x  3
x 0
1  cos 2 x
.
x 0
2x2
63. Evaluate lim
64. Evaluate lim
 0
cos 2   1

.
P. 35
1  cos 
.
  0 sin 
65. Evaluate lim
66. Evaluate lim
 0
1  cos 
2
.
cos x  1
.
x  0 sin 2 x
67. Evaluate lim
68. Evaluate lim
x 0
1  cos 2 x
.
5 x 2  x3
1  cos mx
.
x 0
x2
69. Evaluate lim
70. Evaluate lim
 0
sec  1

.
tan x  sin x
.
x 0
sin 3 x
71. Evaluate lim
72. Given that cos2θ = cos2θ – sin2θ = 1 – 2sin2θ = 2cos2θ – 1, sin 2
cos 2

2


2

1  cos 
and
2
1  cos 
1  2 cos x  cos 2 x
, evaluate lim
.
x

0
2
x2
x y
x y
73. Given that sin x  sin y  2 sin
, evaluate lim
cos
x 0
2
2
sin(3x 


)  sin( x  )
3
3 .
x
74. Given that cos x  cos y  2 sin
yx
yx
cos 5 x  cos x
sin
, evaluate lim
.
x 0
2
2
x2
75. Given that sin x  sin y  2 cos
x y
x y
sin( x  h)  sin( x  h)
sin
, evaluate lim
.
h

0
2
2
h
sin x
.
x    x
76. Evaluate lim
P. 36
77. Evaluate lim
x

2
cos x
x

.
2
78. Evaluate lim (  2 x) tan x .
x

2
79. Evaluate lim [(2 x   )2 (tan2 x  1)] .
x

2

80. Evaluate lim tan 2 x tan(  x) .

4
x
4
81. Evaluate lim ( x  3) csc x .
x 3
82. Evaluate lim (1  x) tan
x 1
x
2
.
 x 2  1 if x  1
83. Let f ( x)  
. Evaluate lim f ( x) , lim f ( x) and lim f ( x) .
x 1
x 1
x 1
 x  1 if x  1
0


84. Let f ( x)   25  x 2
 3x

if x  5
if  5  x  5 .
if x  5
(a) Evaluate lim f ( x) , lim f ( x) and lim f ( x) .
x 5
x 5
x 5
(b) Evaluate lim f ( x) , lim f ( x) and lim f ( x) .
x 5
x 5
x5
x3
.
2 x
(a) Find the domain of f.
85. Let f ( x) 
(b) Evaluate lim f ( x) .
x 3
P. 37
86. Evaluate lim e x .
x 0
87. Evaluate lim | x |  x .
x 1
88. Evaluate lim
1 x
.
1 x
89. Evaluate lim
1 x
.
x 1
x 0
x 1
| x4|
.
x4 4  x
90. Evaluate lim
91. Let f ( x) 
| x 1|
( x  1) . Evaluate lim f ( x) , lim f ( x) and lim f ( x) .
x  1
x 0
x1
x 1
3x  k if x  2
92. Let f ( x)  
. Find the value of k such that lim f ( x) exists.
x2
 x  2 if x  2
 x 2  5 x if x  1
93. Let f ( x)   3
. Find the value of k such that lim f ( x) exists.
x 1
 kx  7 if x  1
94. Let g(x) = f (1 – x), lim f ( x)  6 and lim f ( x)  4 . Evaluate lim g ( x) and lim g ( x) .
x 1
x 1
x 0
 x  1 if x  0

if x  0 at x = 0.
95. Determine the continuity of f ( x)   0
 x  1 if x  0

 x2 1

96. Determine the continuity of f ( x)   x  1 if x  1 at x = 1.

if x  1
 3
 x2  x

97. Determine the continuity of f ( x)   x  1

 2
if x  1 at x = –1.
if x  1
P. 38
x 0
 x 2  4x  3

if x  3 at x = 3.
98. Determine the continuity of f ( x)   x  3

2
if x  3

 sin x

99. Determine the continuity of f ( x)   x

 0
if x  0
at x = 0.
if x  0
 x5  x3  3x

100. Determine the continuity of f ( x)   sin x

0

if x  0 at x = 0.
if x  0
 cos x  1
if x  0
2

101. Determine the continuity of f ( x)   x
at x = 0.
1
 
if x  0
2

if x  2
1  x

102. Let f ( x)    1 if  2  x  2 .
x  1
if x  2

(a) Evaluate lim f ( x) and lim f ( x) .
x  2
x 2
(b) Find the value(s) of x at which f (x) is discontinuous.
if x  4
 x(1  x)

 3x
if  4  x  5 .
103. Let f ( x)  
 x 2  3x  20
if x  5

(a) Evaluate lim f ( x) and lim f ( x) .
x  4
x5
(b) Find the value(s) of x at which f (x) is discontinuous.
 x 2  3x  2

if x  1 . Find the value of k such that f (x) is continuous at x = –1.
104. Let f ( x)   x  1

k
if x  1

3 sin x


105. Let f ( x)   x

 k
if x  0
if x  0
. Find the value of k such that f (x) is continuous at x = 0.
P. 39
1  cos x


106. Let f ( x)   2 x

 k
if x  0
if x  0
. Find the value of k such that f (x) is continuous at x = 0.
 x 2  x if x  1

107. Let f ( x)   k
if x  1 .
3x  5 if x  1

(a) Find the value of k such that f (x) is left-continuous at x = 1.
(b) Find the value of k such that f (x) is right-continuous at x = 1.
(c) Is there a value of k such that f (x) is continuous at x = 1?
108. Let 1 ≤ g(x) ≤ sin2x + 1 on an open interval containing 0, except possibly at x = 0. Find
lim g ( x) .
x0
109. Show that lim x 2 sin
x 0
1
 0.
x
110. (a) Disprove that “  x  x sin
(b) Show that lim x sin
x 0
111. Show that lim sin x sin
x 0
1
 x for all x ≠ 0”.
x
1
 0.
x
1
 0.
x
P. 40
Limits about Infinities, Vertical and Horizontal Asymptotes

As x approaches a, f (x) can be made as positive large as we want. lim f ( x)   .

As x approaches a, f (x) can be made as negative large as we want. lim f ( x)   .

As x approaches a from the left, f (x) can be made as positive large as we want. lim f ( x)   .

As x approaches a from the left, f (x) can be made as negative large as we want.
x a
x a
x a
lim f ( x)   .
x a 
P. 41

As x approaches a from the right, f (x) can be made as positive large as we want.
lim f ( x)   .
x a 

As x approaches a from the right, f (x) can be made as negative large as we want.
lim f ( x)   .
x a 


lim g ( x)   and lim g ( x)   . lim g ( x) doesn’t exist.
x 3 
x 3
x3
If a point moves along a curve y = f (x) and the point approaches a straight line, then the
straight line is said to be an asymptote of the graph of y = f (x).

If lim f ( x)   or lim f ( x)   , then the line x = a is a vertical asymptote of the
x a
x a
graph of y = f (x).
P. 42

As x increases without bound, f (x) approaches L. lim f ( x)  L .

As x decreases without bound, f (x) approaches M. lim f ( x)  M .

As x increases without bound, f (x) can be made as positive large as we want. lim f ( x)   .

As x decreases without bound, f (x) can be made as negative large as we want.
x 
x  
x 
lim f ( x)   .
x  

Any infinite limit doesn’t exist. lim f ( x)   and lim g ( x)   do not exist, where a can
x a
x a
be replaced by a–, a+, ∞ or –∞.

1
1
a
a
lim (1  ) x  e , lim (1  ) x  e , lim (1  ) x  ea and lim (1  ) x  ea
x 
x  
x 
x  
x
x
x
x
P. 43

Squeeze Theorem/Sandwich Theorem for Limits at Positive Infinity: If f (x) ≤ g(x) ≤ h(x)
for all x > m for some m and lim f ( x)  lim h( x)  L , then lim g ( x) exists and
x 
x 
x 
lim g ( x)  L .
x 

Squeeze Theorem/Sandwich Theorem for Limits at Negative Infinity: If f (x) ≤ g(x) ≤ h(x)
for all x < M for some M and lim f ( x)  lim h( x)  L , then lim g ( x) exists and
x  
x  
x  
lim g ( x)  L .
x  

If lim f ( x)  b or lim f ( x)  b , then the line y = b is a horizontal asymptote of the graph
x 
x  
of y = f (x).
1.
Evaluate lim
2
2
2
, lim
and lim
.
3
3
x 3 ( x  3) 3
x 3 ( x  3)
( x  3)
2.
Evaluate lim
x5
x5
x5
, lim
and lim
.
2
2
x

4
x 4 ( x  4)
( x  4)
( x  4) 2
3.
Evaluate lim
x2
x2
x2
, lim
and lim
.
3
3
x 1 ( x  1) 3
x 1 ( x  1)
( x  1)
4.
Evaluate lim
5.
x2  4x  3
x2  4x  3
x2  4x  3
Evaluate lim
, lim
and lim
.
x  2 ( x  2) 2
x 2
x 2
( x  2) 2
( x  2) 2
6.
(a) Evaluate lim
x 3
x 4
x 1
x  2
x4
x4
x4
, lim
and lim
.
x  2 x ( x  2)
x( x  2) x  2 x( x  2)
x  2
x3  5 x 2  6 x
x3  5 x 2  6 x
x3  5 x 2  6 x
lim
lim
,
and
.
x  2
x  2 
x4  4x2
x4  4x2
x4  4x2
x3  5 x 2  6 x
.
x 2
x4  4x2
(b) Evaluate lim
7.
Evaluate lim
8.
Evaluate lim
x 1
x2  5x  6
.
x 1
z 5
.
z  4 ( z  10 z  24) 2
2
P. 44
9.
Evaluate lim csc .
 0
10. Evaluate lim csc x .
x 0
11. Evaluate lim


tan
.
3
2
12. Evaluate lim (10 cot x) .
x 0
13. Evaluate lim
ex
.
1  ex
14. Evaluate lim
x
.
ln x
15. Evaluate lim
x
.
ln x
16. Evaluate lim
1  sin x
.
cos x
x 0
x 1
x 0
x
17. Let f ( x) 

2
x 2  7 x  12
.
xk
(a) Find the value(s) of k such that lim f ( x) exists.
x k
(b) Find the value(s) of k such that lim f ( x)   .
x k
(c) Find the value(s) of k such that lim f ( x)   .
x k
2


if x  2
18. Determine the continuity of f ( x)   x  2
at x = 2.

2
if
x

2

P. 45
19. For each of the following functions f, find all vertical asymptotes of the graph of y = f (x).
(a)
f ( x) 
x 1
x  4x2  4x
(b)
f ( x) 
x5
x 2  25
(c)
f ( x) 
x7
x  49 x 2
(d)
f ( x) 
x 2  3x  2
x10  x9
(e)
f ( x) 
x3  10 x 2  16 x
x2  8x
(f)
f ( x) 
x 2  9 x  14
x2  5x  6
(g)
f ( x)  sec
(h)
f ( x)  tan
(i)
3
4
x
2
, where |x| < 2
x
10
f (x) = 2 – ln x2
(j)
f ( x) 
ex
( x  1)3
(k)
f ( x) 
cos x
x2  2x
(l)
f ( x) 
1
(m) f ( x) 
x sec x

x  sin x
20. Evaluate lim tan 1 x .
x  
1
1
21. Evaluate lim (1  )(3  2 )2 .
x 
x
x
tan 1 x
.
x 
x
22. Evaluate lim
23. Evaluate lim (5 
x  
100 sin 4 x3

).
x
x2
P. 46
x
.
x  x  1
24. Evaluate lim
x2  4
.
x   x 2  4
25. Evaluate lim
2 x 2  3x  4
.
x 
5x2  1
26. Evaluate lim
2 x 2  3x
.
x  5  x 2
27. Evaluate lim
( x  1)( x  2)( x  3)
.
x 
5 x3
28. Evaluate lim
(3x  1)(2 x  3)
.
x   (5 x  3)(4 x  5)
29. Evaluate lim
x 2  3x  1
.
x   ( x  1) 2
30. Evaluate lim
x2
.
x  3x  8
31. Evaluate lim
x3  8
.
x  x 2  2 x  7
32. Evaluate lim
5x2  6 x  2
.
x 
10 x3  7
33. Evaluate lim
5x  7
.
x  2 x 
x
34. Evaluate lim
35. Evaluate lim (
x 
36. Evaluate lim
x 
3x
2x

).
x 1 x 1
4x2  x
.
x2  9
P. 47
37. Evaluate lim
x 
x x x
.
x 1
x  cos x
.
x 
x 1
38. Evaluate lim
2 x  2 x
.
x   2 x  2 x
39. Evaluate lim
t2  4
40. Evaluate lim
.
t  t  4
41. Evaluate lim
4x2  1  1
.
x
42. Evaluate lim
y2  4
.
y4
x  
y  
43. Evaluate lim ( x  m  x ) .
x 
44. Evaluate lim ( x 2  2 x  x) .
x 
45. Evaluate lim ( x 4  1  x 2 ) .
x 
46. Evaluate lim ( x 2  1  x) .
x  
47. Evaluate lim ( x  1  x ) x  2 .
x 
48. Evaluate lim ( 4 x 2  2 x  1  2 x  1) .
x 
49. Evaluate lim x( x 2  2 x  2 x 2  x  x) .
x 
P. 48
50. Evaluate lim (3 x3  x  3 x3  3 ) .
x 
51. Evaluate lim
x 0
1
.
3  41 / x
1
52. Evaluate lim (1  )  x .
x 
x
2
53. Evaluate lim (1  )  x .
x 
x
54. Evaluate lim (1 
x 
3 x
) .
x2
55. Evaluate lim x[ln(1  x)  ln x] .
x 

If f (y) is continuous at y  lim g ( x) , then lim f ( g ( x))  f (lim g ( x)) .
x a
x a
ln(1  3x)
.
x 0
x
56. Evaluate lim
57. Evaluate lim
x 0
ln( a  x)  ln a
, where a > 0.
x
58. Evaluate lim (
x 
x 1 x
) .
x 1
59. Evaluate lim ( x  1) ln(
x 
x 1
).
x 1
60. Evaluate lim (
x2  2x x
) .
x2  1
61. Evaluate lim (
x2  2x  3 x
) .
x 2  3x  28
x 
x 
62. Evaluate lim (1  tan x)cot x .
x 
P. 49
x a
2
63. Evaluate lim (cos x)cot x .
x 0
64. Show that lim
x 
sin x
0.
x
65. Show that lim
 
cos 
2
 0.
cos x 5
66. Show that lim
 0.
x 
x
cos x
 0.
x  e 3 x
67. Show that lim
68. For each of the following functions f, find all horizontal asymptotes of the graph of y = f (x).
(a) f (x) = 2x
(b) f (x) = –3e–x
(c)
f ( x) 
50
e2 x
(d) f (x) = sin x
(e) f (x) = 1 – ln x
(f) f (x) = |ln x|
(g)
f ( x) 
2e x  3e2 x
e 2 x  e3 x
(h)
f ( x) 
3e x  e x
e x  e x
(i)
f ( x) 
(j)
f ( x)  4 x(3x  9 x 2  1)
(k)
f ( x) 
(l)
f ( x) 
3
x6  8
4 x 2  3x 4  1
4 x3
2 x3  9 x 6  15 x 4
4 x3  1
2 x3  16 x 6  1
P. 50
69. For each of the following functions f, find all vertical and horizontal asymptotes of the graph
of y = f (x).
(a) f (x) = e1/x
(b)
f ( x) 
cos x  2 x
x
(c)
f ( x) 
1
x 9
(d)
f ( x) 
3x
4  x2
(e)
f ( x) 
2x2
1  x2
(f)
f ( x) 
x 1
x  x2  6x
(g)
f ( x) 
3x  1
x 1
(h)
f ( x) 
x2  1
x 1
(i)
f ( x) 
x3  x 2  1
x 2  3x  4
(j)
f ( x)  x 
(k)
f ( x) 
(l)
x2  9
f ( x) 
x( x  3)
(m) f ( x) 
2
3
1
x 1
1
2

x  3 x 1
x2  4x  3
x 1
(n)
f ( x) 
2 x3  10 x 2  12 x
x3  2 x 2
(o)
f ( x) 
3x 4  3x3  36 x 2
x 4  25 x 2  144
(p)
f ( x) 
2e x  10e x
e x  e x
(q)
f ( x) 
(r)
f ( x) 
(s)
f ( x) 
2e x  5e3 x
e 2 x  e3 x
x  100
x 2  100
9x2  4
x2
P. 51
(t)
16 x 4  64 x 2  x 2
f ( x) 
2x2  4
(u)
f ( x)  16 x 2 (4 x 2  16 x 4  1)
(v)
f ( x) 
x2  2x  6  3
x 1
(w) f ( x)  | x |  | x  1 |
(x)
f ( x) 
| 1  x2 |
x( x  1)
(y)
f ( x) 
x 1
x 1
2/3
P. 52

Continuity on an Interval
Continuity in an open interval: f is said to be continuous in (a, b) if and only if f is
continuous at every point in (a, b).

Continuity on an open-closed interval: f is said to be continuous on (a, b] if and only if f is
continuous in (a, b) and left-continuous at b.

Continuity on a closed-open interval: f is said to be continuous on [a, b) if and only if f is
continuous in (a, b) and right-continuous at a.

Continuity on a closed interval: f is said to be continuous on [a, b] if and only if f is
continuous in (a, b), right-continuous at a and left-continuous at b.

Intermediate Value Theorem: If f is continuous on a closed interval [a, b] and N is a number
between f (a) and f (b) exclusive, then there exists c in (a, b) such that f (c) = N.

Bolzano Theorem: If f is continuous on a closed interval [a, b] and f (a) f (b) < 0, then there
exists c in (a, b) such that f (c) = 0.
1.
Determine the interval(s) on which the function f ( x) 
3x 2  6 x  7
is continuous.
x2  x  1
2.
Determine the interval(s) on which the function f ( x) 
x5  6 x  17
is continuous.
x2  9
3.
Determine the interval(s) on which the function f ( x) 
x2
is continuous.
x2  4
4.
Determine the interval(s) on which the function f ( x)  3 x 2  2 x  3 is continuous.
5.
Determine the interval(s) on which the function f (x) = (2x – 3)2/3 is continuous.
6.
Determine the interval(s) on which the function f ( x)  x 4  1 is continuous.
7.
Determine the interval(s) on which the function f (x) = (x2 – 1)3/2 is continuous.
8.
Determine the interval(s) on which the function f (x) = (x – 1)3/4 is continuous.
9.
Determine the interval(s) on which the function f ( x) 
1
is continuous.
x 4
 x 2  3x if x  1
10. Determine the interval(s) on which the function f ( x)  
is continuous.
if x  1
 2x
P. 53
 x3  4 x  1 if x  0
11. Determine the interval(s) on which the function f ( x)  
is continuous.
3
if x  0
 2x
12. Show that the equation x3 – 5x2 + 2x = –1 has a real root in the interval (–1, 5).
13. Show that the equation
x 4  25x3  10  5 has a real root in the interval (0, 1).
14. Show that the equation 2x3 + x – 2 = 0 has a real root in the interval (–1, 1).
15. Show that the equation  x5  4 x 2  2 x  5  0 has a real root in the interval (0, 3).
16. Show that the equation x + ex = 0 has a real root in the interval (–1, 0).
17. Show that the equation xln x – 1 = 0 has a real root in the interval (1, e).
18. Show that the equation xex = 2 has at least one real root in [0, 1].
19. Let f (x) = x3 – 8x + 5.
(a) Find the values of f (–4), f (0), f (1) and f (3).
(b) Show that the equation f (x) = 0 has exactly three real roots.
20. Show that the equation x3 + 10x2 – 100x + 50 = 0 has three real roots in the interval (–20, 10).
21. Show that the equation 70x3 – 87x2 + 32x – 3 = 0 has three real roots in the interval (0, 1).
22. Let f be a function continuous on [0, 1] and 0 ≤ f (x) ≤ 1 for all x on [0, 1]. Show that the
equation f (x) = x has at least one real root on the interval [0, 1].
23. Suppose you begin a 2-hour hike to the mountaintop at 7 a.m. on day 1 and start a 2-hour hike
back to the foot of the mountain at 7 a.m. on day 2. Assume the mountaintop is 3 miles from
the foot of the mountain. Let f (t) and g(t) be your distances in miles from the foot of the
mountain t hours after 7 a.m. on day 1 and on day 2 respectively.
(a) Find the values of f (0), f (2), g(0) and g(2).
(b) Let h(t) = f (t) – g(t). Find the values of h(0) and h(2).
(c) Show that there exists some point along the trail that you pass at exactly the same time on
both days.
P. 54

Differentiation Techniques
√ is an operator. When it is applied to 9, the result is 3.

d
is a differential operator. When it is applied to a function, the result is another function.
dx

Constant Rule:

Power Rule:

Constant Multiple Rule:

Sum and Difference Rule:
1.
Find the derivative of each of the following functions.
(a) y = x5
(b) f (t) = t11
(c) f (x) = 5
(d) g(x) = e3
(e) h(t) = t
(f) f (v) = v100
2.
Find the derivative of each of the following functions.
(a) f (x) = 5x3
(b)
g ( w) 
d
(k )  0
dx
d p
( x )  px p 1
dx
d
(kf ( x))  kf ( x)
dx
d
( f ( x)  g ( x))  f ( x)  g ( x)
dx
5 12
w
6
(c) p(x) = 8x
(d)
g (t )  6 t
(e) g(t) = 100t2
(f)
3.
f ( s) 
s
4
Find the derivative of each of the following functions.
(a) f (x) = 3x4 + 7x
(b) g(x) = 6x5 – x
(c) f (x) = 10x4 – 32x + e2
(d)
f (t )  6 t  4t 3  9
(e) g(w) = 2w3 + 3w2 + 10w
(f)
1
s(t )  4 t  t 4  t  1
4
P. 55
4.
Find the derivative of each of the following functions.
(a) f (x) = (2x + 1)(3x2 + 2)
(b) g(r) = (5r3 + 3r + 1)(r2 + 3)
(c) h(x) = (x2 + 1)2
(d) h( x)  x ( x  1)
5.
Find the derivative of each of the following functions.
(a)
f ( w) 
(b)
y
(c)
g ( x) 
w3  w
w
12s 3  8s 2  12s
4s
x2  1
x 1
x3  6 x 2  8 x
x2  2x
xa
, where a is a nonnegative constant
x a
(d) h( x) 
6.
(e)
y
(f)
y
x 2  2ax  a 2
, where a is a constant
xa
The following table shows the derivatives of f (x) and g(x) at x = 1, 2, 3, 4 and 5.
x
1
2
3
4
5
f '(x)
3
5
2
1
4
g'(x)
2
4
3
1
5
Evaluate the following expressions.
d
( f ( x)  g ( x))
(a)
dx
x 1
(b)
d
(1.5 f ( x))
dx
x2
(c)
d
(2 x  3g ( x))
dx
x4

Exponential Rule:

Product Rule:

d x
(e )  e x
dx
d
( f ( x) g ( x))  f ( x) g ( x)  g ( x) f ( x)
dx
d f ( x)
g ( x) f ( x)  f ( x) g ( x)
(
)
Quotient Rule:
dx g ( x)
( g ( x))2
P. 56
7.
8.
Find the derivative of each of the following functions.
(a)
(b)
(c)
(d)
g(x) = 6x – 2xex
f (t) = t5et
g(w) = ew(5w2 + 3w + 1)
g(w) = ew(w3 – 1)
(e)
s(t )  4et t
Find the derivative of each of the following functions.
(a)
f ( x) 
x
x 1
(b)
f ( x) 
x3  4 x 2  x
x2
(c)
f ( x) 
ex
ex  1
2e x  1
2e x  1
(e) y = (3t – 1)(2t – 2)–1
9.
(d)
f ( x) 
(f)
h( w) 
w2  1
w2  1
(g)
g ( x) 
ex
x2  1
(h)
y  (2 x  1)(4 x  1)1
Find the derivative of each of the following functions.
(a) f (x) = 3x–9
4
(b) y  3
p
6
t7
(c)
g (t )  3t 2 
(d)
y
(e)
g (t ) 
t 3  3t 2  t
t3
(f)
p( x) 
4 x 3  3x  1
2 x5
w 4  5 w2  w
w2
P. 57
10. Find the derivative of each of the following functions.
(a)
g ( x) 
(b) h( x) 
( x  1)e x
x2
( x  1)(2 x 2  1)
x3  1
h( x ) 
xe x
x 1
(d) h( x) 
x 1
x 2e x
(c)
11. Find the derivative of each of the following functions.
4  x2
x2
(a)
f ( x) 
(b)
f ( x)  4 x 2 
2x
5x  1
2r  r
r 1
7
(d) h(x) = (5x + 5x)(6x3 + 3x2 + 3)
(c)
h( r ) 
12. The following table shows the values of f (x), f '(x), g(x) and g'(x) at x = 1, 2, 3, 4 and 5.
x
1
2
3
4
5
f (x)
5
4
3
2
1
f '(x)
3
5
2
1
4
g(x)
4
2
5
3
1
g'(x)
2
4
3
1
5
Evaluate the following expressions.
d
( f ( x) g ( x))
(a)
dx
x 1
(b)
d f ( x)
(
)
dx g ( x) x  2
(c)
d
( xf ( x))
dx
x 3
(d)
d f ( x)
(
)
dx x  2 x  4
(e)
d xf ( x)
(
)
dx g ( x) x  4
(f)
d f ( x) g ( x)
(
)
dx
x
x4
P. 58
d
d
(sin x)  cos x
(cos x)   sin x
dx
dx
d
d
(tan x)  sec2 x
(cot x)   csc 2 x
 Trigonometric Rules:
dx
dx
d
d
(sec x)  sec x tan x
(csc x)   csc x cot x
dx
dx
13. Find the derivative of each of the following functions.
(a) y = sin x + cos x
(b) y = 5x2 + cos x
(c) y = x sin x
(d)
y
cos x
sin x  1
(e)
y
1  sin x
1  sin x
(f)
y = sin x cos x
( x 2  1) sin x
sin x  1
(h) y = cos2x
(g)
y
(i)
y
x sin x
1  cos x
14. Find the derivative of each of the following functions.
(a) y = tan x + cot x
(b) y = sec x + csc x
(c) y = sec x tan x
(d)
y
tan w
1  tan w
(e)
y
cot x
1  csc x
(f)
y
tan t
1  sec t
(g)
y
1
sec z csc z
(h) y = csc2θ – 1
P. 59
15. Find the derivative of each of the following functions.
(a)
y
sin x
1  cos x
(b) y = x cos x sin x


(c)
y
1
2  sin x
(d)
y
2 cos x
1  sin x
(e)
y
x cos x
1  x3
(f)
y
1  cos x
1  cos x
The derivative of a function f (β) at β = a, i.e. f '(a), represents the slope of tangent to the curve
y = f (β) at β = a.
f (a  x)  f (a)
f (  )  f (a)
First Principles at x = a: f (a)  lim
or f (a)  lim
x  0
 a
x
 a
16. For each of the following questions, find the slope of tangent to the curve y = f (x) at x = a
from first principles/by definition.
(a) f (x) = x2 – 5; a = 3
(b) f (x) = –3x2 – 5x + 1; a = 1
(c) f (x) = –5x + 1; a = 1
(d) f (x) = 5; a = 1
(e)
f ( x) 
1
; a = –1
x
(f)
f ( x) 
4
; a = –1
x2
P. 60
17. For each of the following questions, find the slope of tangent to the curve y = f (x) at x = a
from first principles/by definition.
(a) f (x) = 2x + 1; a = 0
(b) f (x) = 3x2 – 4x; a = 1
(c) f (x) = x2 – 4; a = 2
(d)
f ( x) 
1
;a=1
x
(e) f (x) = x3; a = 1
(f)
f ( x) 
1
;a=0
2x  1
(g)
f ( x) 
1
; a = –1
3  2x
(h)
f ( x)  x  1 ; a = 2
(i)
f ( x)  x  3 ; a = 1
(j)
f ( x) 
x
; a = –2
x 1
18. For each of the following questions, find the slope of tangent to the curve y = f (x) at x = a
from first principles/by definition.
(a) f (x) = 8x; a = –3
(b) f (x) = x2; a = 3
(c) f (x) = 4x2 + 2x; a = –2
(d) f (x) = 2x3; a = 10
1
1
(e) f ( x) 
; a
4
x
1
;a=1
x2
(f)
f ( x) 
(g)
f ( x)  2 x  1 ; a = 4
(h)
f ( x)  3x ; a = 12
(i)
f ( x) 
1
;a=5
x5
(j)
f ( x) 
1
;a=2
3x  1
P. 61
19. For each of the following questions, find the slope of tangent to the corresponding graph at a
from first principles/by definition.
(a)
y
1
;a=1
1 t
(b) y = t – t2; a = 2
(c) c  2 s  1 ; a = 25
(d) A = πr2; a = 3


The derivative f '(x) is the function which maps x to the slope of the tangent to curve y = f (β) at
β = x.
f ( x  x)  f ( x)
f (  )  f ( x)
First Principles: f ( x)  lim
or f ( x)  lim
x  0
 x
x
 x
20. Find the derivative of each of the following functions from first principles/by definition.
(a)
f ( x)  3 x  1
(b)
f ( x)  x  2
(c)
f ( x) 
2
3x  1
(d)
f ( x) 
1
x
21. Find the derivative of each of the following functions from first principles/by definition.
(a) f (x) = mx + b
(b) f (x) = ax2 + bx + c
(c)
f ( x)  ax  b
P. 62
22. Evaluate each of the following limits by expressing it as a derivative of some function at some
point.
(a)
lim
h 0
2h  2
h
(2  h) 4  16
h 0
h
(b) lim
(c)
3x 2  4 x  7
x 1
x 1
lim
(d) lim
h 0
9h  9
h
(1  h)8  (1  h)3  2
h 0
h
(e)
lim
(f)
lim
x100  1
x 1 x  1
sin(

(g) lim
6
h 0
h
cos(

(h) lim
6
 h) 
h 0
(i)
lim
x

4
3
2
h
cot x  1
x
tan(
(j)
1
2
 h) 
lim
h 0

4
5
1
 h) 
6
3
h
23. (a) Prove the power rule for positive integer power from first principles/by definition, i.e.
d n
 n  xn
( x )  lim
 nx n 1 .
 x   x
dx
(b) Prove the power rule for negative integer power from first principles/by definition, i.e.
d n
 n  xn
( x )  lim
 nx  n 1 .


x
dx
 x
(c) Prove the power rule for rational power from first principles/by definition, i.e.
m
m

m
d n
 n  x n m n 1
d n
 n x
and
( x )  lim
 x
( x )  lim
 x   x
 x
dx
n
dx
 x
m
m
m
P. 63

m
n
m

m  n 1
.
x
n
24. (a) Prove the sine rule from first principles/by definition, i.e.
d
sin   sin x
(sin x)  lim
 cos x .
 x
dx
 x
(b) Prove the cosine rule from first principles/by definition, i.e.
d
cos   cos x
(cos x)  lim
  sin x .
 x
dx
 x
dy dy du

dx du dx

Chain Rule:

Square Root Rule:
d
dx
f ( x) 
f ( x)
2 f ( x)
25. Find the derivative of each of the following functions.
(a) y = (3x + 7)10
(b) y = (5x2 + 11x)20
(c) y = sin5x
(d) y = cos x5
(e) y = e5x – 7
(f)
y  7x 1
(g)
y  x2  1
(h)
ye
(i)
y = tan5x2
(j)
y  sin
x
x
4
(k) y = sec ex
(l)
y  e x
2
26. Find the derivative of each of the following functions.
(a) y = (3x2 + 7x)10
(b) y = (x2 + 2x + 7)8
(c)
y  10 x  1
(d)
y  x2  9
(e) y = 5(7x3 + 1)–3
(f) y = cos(5t + 1)
(g) y = sec(3x + 1)
(h) y = csc ex
P. 64
(i)
y = tan ex
(j) y = etan t
(k) y = sin(4x3 + 3x + 1)
(l) y = csc(t2 + t)
(m) y  sin(2 x )
(n) y = cos4θ + sin4θ
(o) y = (sec x + tan x)5
(p) y = sin(4cos z)
27. Find the derivative of each of the following functions.
(a) f (x) = xe7x
(b) g(t) = 2tet/2
(c) f (x) = 15e3x
(d) y = 3x2 – 2x + e–2x
x
e3 x
(e)
g ( x) 
(f)
f (x) = (1 – 2x)e–x
2e x  3e x
3
(h) A = 2500e0.075t
(g)
y
28. Find the derivative of each of the following functions.
(a)
y  1 cot 2 x
(b)
y  (3x  4)2  3x
(c) f (x) = sin(sin(ex))
(d) f (x) = sin2(e3x + 1)
(e) f (x) = sin5(cos3x)
(f) f (x) = cos4(7x3)
(g)
y  tan(e
3x
)
(h) y = (1 – e–0.05x)–1
(i)
y x x
(j)
y x x x
(k) h(x) = f (g(x2))
(l) h(x) = [ f (g(xm))]n
P. 65
29. Find the derivative of each of the following functions.
(a)
y(
x 5
)
x 1
(b)
y(
ex 8
)
x 1
(c)
y  ex
2
1
sin x3
(d) y = tan(xex)
(e) y = θ2sec5θ
(f)
y(
3x 5
)
4x  2
(g) y = [(x + 2)(x2 + 1)]4
(h) y = e2x(2x – 7)5
(i)
y  x 4  cos 2 x
te t
t 1
(k) y = (p + π)2sinp2
(l) y = (z + 4)3tan z
(j)
y
(m) f (x) = xe–x
(n)
(o)
(p)
(q)
(r)
(s)
f ( x)  e  x x
y = e–xsin x
y = sin x + 4e0.5x
y = e6xsin x
y = e5xcsc x
f ( z )  z 2 (e3 z  4) 
2z
z 1
2
30. Let f '(0) = 3, f '(1) = 5 and g(x) = f (sin x). Evaluate the following expressions.
(a) g'(0)
(b)

g ( )
2
(c) g'(π)
31. Let f (0) = –3, f (1) = 3, f '(0) = 3, f '(1) = 5 and g(x) = sin(π f (x)). Evaluate the following
expressions.
(a) g'(0)
(b) g'(1)
P. 66
32. The following table shows the values of f (x), f '(x), g(x) and g'(x) at x = 1, 2, 3, 4 and 5.
x
1
2
3
4
5
f (x)
0
3
5
1
0
f '(x)
5
2
–5
–8
–10
g(x)
4
5
1
3
2
g'(x)
2
10
20
15
20
Let h(x) = f (g(x)) and p(x) = g(f (x)). Evaluate the following expressions.
(a) h'(3)
(b) h'(2)
(c) p'(4)
(d) p'(2)
(e) h'(5)
33. The following table shows the values of f '(x), g(x) and g'(x) at x = 1, 2, 3, 4 and 5.
x
1
2
3
4
5
f '(x)
–6
–3
8
7
2
g(x)
4
1
5
2
3
g'(x)
9
7
3
–1
–5
Let h(x) = f (g(x)) and k(x) = g(g(x)). Evaluate the following expressions.
(a) h'(1)
(b) h'(2)
(c) h'(3)
(d) k'(3)
(e) k'(1)
(f) k'(5)
34. Evaluate each of the following limits by expressing it as a derivative of some function at some
point.
(a)
1
1

lim x  1 3
x2
x2
( x 2  3)5  1
x2
x2
(b) lim
(c)
lim
x 0
4  sin x  2
x
sin(
(d) lim
h 0
(e)

2
 h)  sin
2
2
4
h
1
1

5
10
3[(1  h)  7]
3(8)10
lim
h 0
h
P. 67

Implicit Differentiation: If y is defined as an implicit function of x, i.e. the relation is
represented by an equation involving x and y, then we can differentiate with respect to x on
both sides to obtain
dy
.
dx
35. For each of the following questions, find
(a)
(b)
(c)
(d)
dy
.
dx
x4 + y4 = 2
x = ey
y2 = 4x
y2 + 3x = 2
(e) sin y = 5x4 – 5
(f)
x 2 y 0
(g) cos y = x
(h) tan xy = x + y
36. For each of the following questions, find
dy
.
dx
(a) sin xy = x + y
(b) exy = 2y
(c) x + y = cos y
(d)
x  2y  y
(e) cos y2 + x = ey
x 1
(f) y 
y 1
(g)
x3 
x y
x y
(h) (xy + 1)3 = x – y2 + 8
(i) 6x3 + 7y3 = 13xy
(j) sin x cos y = sin x + cos y
(k)
x4  y 2  5x  2 y3
(l)
x  y 2  sin y
P. 68
37. Find the derivative of each of the following functions.
(a) y = x5/4
(b)
y  3 x2  x  1
(c) y = (5x + 1)2/3
(d)
y  e x x3
(f)
2x
4x  3
y = x(x + 1)1/3
(g)
y  3 (1  x1 / 3 )2
(h)
y
(e)
y4
5
x
xx
38. For each of the following questions, find
dy
.
dx
x  3 y4  2
(a)
3
(b)
(c)
(d)
(e)
(f)
x2/3 + y2/3 = 2
xy1/3 + y = 10
(x + y)2/3 = y
xy + x3/2y–1/2 = 2
xy5/2 + x3/2y = 12
39. For each of the following questions, find
dy
.
dx
(a) y3 = ax2
(b) x + y3 – xy = 1
x 2 (4  x)
4 x
4
(d) x = 2(x2 – y2)
(e) y2(x + 2) = x2(6 – x)
(c)
y2 
(f)
5 x  10 y  sin x
(g) (x2 + y2)(x2 + y2 + x) = 8xy2
(h)
3x7  y 2  sin 2 y  100 xy
P. 69
40. For each of the following questions, find
(a)
(b)
(c)
(d)
(e)
(f)
x + y2 = 1
2x2 + y2 = 4
x + y = sin y
x4 + y4 = 64
e2y + x = y
sin x + x2y = 10
(g)
y  xy  1
d2y
.
dx 2
d
1
d
1
(ln x)  , x  0 and
(ln | x |) 
dx
x
dx
x

Logarithmic Rule:

Exponential Rule with an Arbitrary Base:
d x
(b )  b x ln b
dx

Logarithmic Differentiation: Sometimes, it is more convenient to find the derivative of y = f
(x) by taking natural log and then differentiate with respect to x on both sides.
41. Find the derivative of each of the following functions.
(a) y = ln7x
(b) f (x) = x2ln x
(c) y = ln x2
(d) y = ln2x8
(e) f (x) = ln|sin x|
ln x 2
x
(f)
y
(g)
y  ln
x 1
x 1
(h) y = exln x
(i) y = (x2 + 1)ln x
(j) y = ln|x2 – 1|
(k) y = ln(ln x)
(l) f (x) = ln(cos2x)
(m) y 
ln x
ln x  1
(n) y = ln(ex + e–x)
42. Find the derivative of each of the following functions.
(a) y = 8x
(b) y = 53t
(c) y = 5(4x)
(d) y = 4–xsin x
(e) y = x3(3x)
P. 70
(f)
P
40
1  2 t
(g) A = 250(1.0454t)
(h) y = ln 10x
43. Find the derivative of each of the following functions.
(a) f (x) = xe
(b) f (x) = 2x
(c)
(d)
(e)
(f)
(g)
(h)
(i)
f ( x)  2 x
2
g(y) = ey(ye)
s(t) = cos 2t
r = e2θ
y = ln(x3 + 1)π
f (x) = (2x – 3)x3/2
y = tan x0.74
2x
2x  1
(k) f (x) = (2x + 1)π
(j)
(l)
f ( x) 
f (x) = xπ + πx
44. Find the derivative of each of the following functions.
(a) y = 4log3(x2 – 1)
(b) y = log10x
(c) y = cos x ln(cos2x)
(d) y = log8|tan x|
1
(e) y 
log 4 x
(f) y = log2(log2x)
45. Find the derivative of each of the following functions.
(a) f (x) = ln(3x + 1)4
2x
(b) f ( x)  ln 2
( x  1)3
(c)
f ( x)  ln 10 x
(d)
f ( x)  log 2
(e)
(2 x  1)( x  2)3
f ( x)  ln
(1  4 x) 2
(f)
8
x 1
f (x) = ln(sec4x tan2x)
P. 71
46. Use logarithmic differentiation to evaluate
dy
.
dx
(a) y = xcos x
(b) y = xln x
(c)
yx
x
(d) y = (x2 + 1)x
(e) y = (sin x)ln x
(f) y = (tan x)x – 1
(g)
( x  1)10
y
(2 x  4)8
(h)
tan10 x
y
(5 x  3)6
(i)
( x  1)3 / 2 ( x  4)5 / 2
y
(5 x  3) 2 / 3
x8 cos 3 x
(j) y 
x 1
(k) y = (sin x)tan x
(l)
1
y  (1  ) 2 x
x
(m) y = x10x
(n) y = (2x)2x
(o) y = (1 + x2)sin x
(p)
y  xx
(q)
y  (ln x) x
10
2
47. Evaluate each of the following limits by expressing it as a derivative of some function at some
point.
(a)
ln x  1
x e x  e
lim
ln(e8  h)  8
h 0
h
(b) lim
(c)
(3  h)3 h  27
h 0
h
lim
5 x  25
x2 x  2
(d) lim
P. 72

Inverse Function Rule:
dx
1

dy
dy
dx
48. For each of the following questions, find the derivative of the inverse of the given function at
the specified point on the graph of the inverse function.
(a) f (x) = 3x + 4; (16, 4)
(b)
f ( x) 
1
x  8 ; (10, 4)
2
(c) f (x) = –5x + 4; (–1, 1)
(d) f (x) = x2 + 1, where x ≥ 0; (5, 2)

)
4
(f) f (x) = x2 – 2x – 3, where x ≤ 1; (12, –3)
(e) f (x) = tan x; (1,
49. Find ( f –1)'(3) if f (x) = x3 + x + 1.

d
1
(sin 1 x) 
dx
1  x2
d
1
(tan 1 x) 
Inverse Trigonometric Rules:
dx
1  x2
d
1
(sec1 x) 
dx
x x2  1
50. Find the derivative of each of the following functions.
(a) f (x) = sin–1(2x)
(b) f (x) = x sin–1x
(c) f (w) = cos(sin–1(2w))
(d) f (x) = sin–1(ln x)
(e) f (x) = sin–1(e–2x)
(f)
f (x) = sin–1(esin x)
P. 73
d
1
(cos 1 x)  
dx
1  x2
d
1
(cot 1 x)  
dx
1  x2
d
1
(csc1 x)  
dx
x x2  1
51. Find the derivative of each of the following functions.
(a) f (x) = tan–1(10x)
(b)
f ( x)  x cot 1
x
3
(c) f (y) = tan–1(2y2 – 4)
1
z
(d)
g ( z )  tan 1
(e)
f ( z )  cot 1 z
(f)
f ( x)  sec1 x
(g)
f ( x)  cos 1
1
x
(h) f (t) = (cos–1t)2
(i) f (u) = csc–1(2u + 1)
(j) f (t) = ln(tan–1t)
1
(k) f ( y )  cot 1 2
y 1
(l)
f (w) = sin(sec–1(2w))
(m)
(n)
(o)
(p)
(q)
f (x) = sec–1(ln x)
f (x) = tan–1(e4x)
f (x) = csc–1(tan ex)
f (x) = sin(tan–1(ln x))
f (s) = cot–1(es)
1
f ( x) 
1
tan ( x 2  4)
(r)
P. 74
Differentiability
f (  )  f (a)
exists, then f (x) is said to be differentiable at the point x = a.
 a

If f (a)  lim

f (a)  lim
f (  )  f (a)
is said to be the left hand derivative of f (x) at x = a.
 a

f (a)  lim
f (  )  f (a)
is said to be the right hand derivative of f (x) at x = a.
 a



1.
2.
 a
 a
 a
f (x) is not differentiable at x = a if one of the following conditions holds.
 f (x) is discontinuous at x = a.
 f (x) has a corner at x = a.
 f (x) has a vertical tangent line at x = a.
If f (x) is differentiable at x = a, then f (x) is continuous at x = a.
If f is differentiable at every point on an open interval (a, b), then f is said to be differentiable
on the open interval (a, b).
Determine which of the following functions are differentiable at x = 0.
(a) f (x) = x2/3
(b)
1

 x sin
f ( x)  
x

 0
(c)
 x 2  x if x  0
f ( x)  
if x  0
 x
(d)
2 x if x  0
f ( x)  
 0 if x  0
if x  0
if x  0
For each of the following functions, find f '(x) and locate the points at which f (x) is not
differentiable.
if x  3
 4x
(a) f ( x)   2
2 x  6 if x  3
(b)
 x2
f ( x)  
2
( x  2)
if x  2
if x  2
3

3.
(c) f (x) = |x + 8|
Hint: Find the differentiation rule for g(x) = |x| first.
(d) f (x) = |x2 – 1|
(e) f (x) = |x – a| + |x – b| + |x – c|, where a < b < c
Let f (x) = x|x|.
(a) Find f '(x) for x ≠ 0.
(b) Show that f '(0) exists.
(c) Show that f '(x) is continuous at x = 0.
P. 75
4.
1

 x3 sin
Let f ( x)  
x

 0
if x  0
if x  0
.
(a) Find f '(x) for x ≠ 0.
(b) Show that f '(0) exists.
(c) Is f '(x) continuous at x = 0? Explain.
5.
 x3
if x  1
Let f ( x)  
. Find the values of a and b so that f is differentiable at x = 1.
ax  b if x  1
6.
1

 x 2 cos
Let f ( x)  
x

ax

b

if x  0
if x  0
be differentiable at x = 0. Find the values of a and b.
7.
Prove the following statements.
(a) The derivative of a differentiable even function is an odd function.
(b) The derivative of a differentiable odd function is an even function.
8.
Let f and g be two differentiable functions such that g(x + y) = g(x)f (y) + f (x)g(y) for all real
numbers x and y, f (0) = 1, f '(0) = 0, g(0) = 0 and g'(0) = 1. Show that g'(x) = f (x) for all real
numbers x.
9.
Let f be a differentiable function such that f (x + y) = f (x)f (y) for all real numbers x and y, and
f (x) = 1 + xg(x), where lim g ( x)  1 . Show that f '(x) = f (x) for all real numbers x.
x 0
10. Let f be a differentiable function such that f (x + y) = f (x) + f (y) + 2xy(x + y) for all real
numbers x and y, and f '(0) = 1. Show that f '(x) = 1 + 2x2 for all real numbers x.
11. Let f be a differentiable function such that | f (x) – f (a)| ≤ (x – a)2. Show that f '(x) = 0 for all
real numbers x.
P. 76

1.
Equations of Tangent Lines
By substituting (a, f (a)) into the equation y = f '(a)x + c, we can find the equation of the line
tangent to the curve y = f (x) at x = a.
For each of the following questions, find the equation of the tangent to the curve at the given
point.
(a) y = (x – 2)(x + 4); (–1, –9)
(b)
y
7x
; (0, 0)
x  4x  1
(c)
y
1
1
; (1, )
x 4
5
(d)
y  x ; (4, 2)
2
2

, 3)
4
(f) y = sin x + cos x; (π, –1)
(g) x + 3y – y2 = 0; (4, –1)
(e) y = 3tan x; (
(h)
y

2
sin( x  y ) ; (0,

)
2
2.
Find the equations of the tangents to the curve x2/3 + y2/3 = a2/3 at the points of intersection with
the line y = x.
3.
Show that the equation of the tangent to the ellipse
x2 y 2

 1 at the point (x1, y1) is
a 2 b2
x1 x y1 y
 2  1.
a2
b
4.
Find the coordinates of the point(s) at which the tangent to the curve y = x3 – 3x + 1 is/are
horizontal.
5.
Show that the tangents to the curve y = x3 – 9x2 + 30x + 1 cannot be parallel to the x-axis.
6.
1
3
Find the coordinates of the points on the curve y  x3  x 2  3x  5 where the inclination
3
2
of the tangent is
7.

.
4
Find the x-coordinates of the points at which the tangent to the curve y = x3 + 3x2 – 4x + 5 are
parallel to the line 4x + y = 1.
P. 77
8.
Find the equation(s) of the tangent(s) to the curve y 
4x
which is/are parallel to the line
x 1
4x + y = 5.
9.
Find the coordinates of the point(s) at which the tangent to the curve y = x3/2 – x1/2 is/are
perpendicular to the line x + y = 0.
10. Show that the tangents to the curves 3y2 – 8x – 18y + 48 = 0 and x2 + y2 = 25 at the point (3, 4)
are perpendicular to each other.
P. 78





Extrema and Curve Sketching
If for all x1, x2 on an interval I, f (x1) < f (x2) whenever x1 < x2, then f is said to be increasing on
the interval I.
If for all x1, x2 on an interval I, f (x1) > f (x2) whenever x1 < x2, then f is said to be decreasing
on the interval I.
If f '(x) > 0 for all x on an interval I, then f is increasing on the interval I.
If f '(x) < 0 for all x on an interval I, then f is decreasing on the interval I.
If f (a) ≥ f (x) for all x on some open interval containing a, then f (a) is said to be a
local/relative maximum value of f and (a, f (a)) is said to be a local/relative maximum point
of f.

If f (a) ≤ f (x) for all x on some open interval containing a, then f (a) is said to be a
local/relative minimum value of f and (a, f (a)) is said to be a local/relative minimum point
of f.


A local/relative maximum value or a local/relative minimum value is said to be a local/relative
extreme value. A local/relative maximum point or a local/relative minimum point is said to be
a local/relative extreme point.
Let a be in the domain of f. If f '(a) = 0 or f '(a) doesn't exist, then a is said to be a critical
value/number of f and (a, f (a)) is said to be a critical point of f.
To test whether a critical point of a continuous function is a local/relative maximum point, a

local/relative minimum point or otherwise, we use the first derivative test.
Let I be an interval containing a. If f (a) ≥ f (x) for all x on I, then f (a) is said to be a

global/absolute maximum value of f on the interval I and (a, f (a)) is said to be a
global/absolute maximum point of f on the interval I.
Let I be an interval containing a. If f (a) ≤ f (x) for all x on I, then f (a) is said to be a

global/absolute minimum value of f on the interval I and (a, f (a)) is said to be a
global/absolute minimum point of f on the interval I.
A global/absolute maximum value or a global/absolute minimum value is said to be a

global/absolute extreme value. A global/absolute maximum point or a global/absolute

1.
minimum point is said to be a global/absolute extreme point.
To find the global/absolute extreme points of a continuous function on a closed interval, we
compare the function values at the critical points and the endpoints.
For each of the following functions, find the increasing intervals, the decreasing intervals, the
maximum points and the minimum points.
(a) y = x3 – 3x + 1
(b) y = 2x3 – 9x2 + 27
(c) y = (x + 1)3
(d) y = x(x – 5)2
(e) y = x6 – 3x2
(f)
y  x x3
P. 79
2.
Find the maximum and minimum values of the following functions.
(a)
(b)
(c)
(d)
3.
y = sin x + cos x
y = x + tan x
y = sin2x – 2sin x + 2
y = 2sin2x + cos2x
Find the critical values/numbers of the following functions.
(a)
f ( x)  3 x 2  x  2
(b)
f ( x)  (2 x  5) x 2  4
(c)
f ( x)  x 2 3 2 x  5
(d)
f ( x) 
2x  3
x2  9
(e) f (x) = |x – 3| + |2x – 9|
4.
Find the maximum and minimum points of the following functions.
(a) f (x) = cos2x + sin x for –π ≤ x ≤ π
(b) f (x) = (x + 2)3(3x – 1)4
(c)
f ( x)  3 8  x 3
(d) f (x) = (4 – x)1/3
5.
(e)
f ( x) 
x
1  x2
(f)
f ( x) 
1  sin x
for –π ≤ x ≤ π
1  sin x
1

If f ( x)  k sin x  sin 3x has a maximum value or a minimum value at x  , find the
3
3
value of k and determine whether it is a maximum value or a minimum value.
6.
Let f (x) = x3 + 3kx + 5.
(a) Show that if k > 0, then f (x) has no maximum points or minimum points.
(b) Show that if k < 0, then f (x) has a maximum point and a minimum point.
7.
Let f ( x)  x 2 
k
.
x
(a) Find the value of k so that f (x) may have a minimum at x = –3.
(b) Show that f (x) cannot have a maximum point for any value of k.
P. 80
8.
Sketch the graphs of the following functions.
(a)
(b)
(c)
(d)
(e)
(f)
(g)
(h)
y = 2x3 + 3x2 – 12x
y = x3 + x2 – 5x
y = x4 – 2x3 + 1
y = 8x2 – x4
y = 2x3 – 3x2 – 12x + 9
y = x5 – 15x3
y = x4 – 4x3 + 4x2
y = 2 + 2x2 – x4
(i)
y
x
x 1
2
x 2  3x
x2  3
(k) y2 = x2(x + 1)
(l) y2 = 4x2(1 – x2)
(m) y = 2sin x + sin2x for –π ≤ x ≤ π
(n) y = sin xcos3x for 0 ≤ x ≤ π
9.
(j)
y
(o)
f ( x) 
x2
x2
(p)
f ( x) 
1
1

5
x 1 x  2
(q)
f ( x)  xe  x
2
/2
The graph of y = x3 + px2 + qx + r, where p, q and r are constants, touches the x-axis at x = 1
and crosses the x-axis at x = 4.
(a) Find the values of p, q and r.
(b) Find the maximum and minimum values of y.
(c) Sketch the graph of the function for –1 ≤ x ≤ 4.
10. Let C be the curve y = ax3 + bx2 + cx + d, where a, b, c and d are constants. Its y-intercept is –4.
The tangent to C at x = –3 is horizontal. When x = 1,
d2y
dy
 16 .
 16 and
dx 2
dx
(a) Find the values of a, b, c and d.
(b) Find the coordinates of the maximum and minimum points of C.
(c) Sketch C for –5 ≤ x ≤ 1.
P. 81
Related Rates

Displacement: s

Velocity: v 

Acceleration: a 

dv d 2 s

dt dt 2
Use the following steps to solve the problem.
1. Write down the given rate.
2. Express the relation between 2 variables as an equation.
3. Differentiate on both sides of the equation with respect to time t like implicit
4.
1.
ds
dt
differentiation.
Substitute the given rate and find the required quantity.
A particle moves along the x-axis so that its displacement x 
t3
 2t 2  3t , where t is the
3
elapsed time.
(a) Find its velocity and acceleration when t = 1.
(b) When does the particle change the direction of motion?
2.
The position of an object moving along the x-axis at time t is given by x = 3t2 – 5t – 2. Find
(a) the velocity when t = 1,
(b) the acceleration when t = 2,
(c) the time when the velocity equals 4.
3.
The position of an object moving along the x-axis at time t is given by x = t3 – 6t2 + 9t + 5.
Find
(a) the position(s) of the object when the velocity is zero,
(b) the position(s) of the object when the acceleration is zero.
4.
A particle is in oscillating motion so that its displacement x = 5cos2t. Find the velocity and
acceleration at
(a) x = –5,
(b) x = 0.
5.
A particle is moving along a straight line from a fixed point O. Its displacement scm at time t
seconds after passing O is given by s = t3 – 12t. Find
(a) the acceleration of the particle when it is instantaneous rest,
(b) the velocity of the particle when it is at O again.
P. 82
6.
A particle P travels in a straight line so that its displacement in metres from a fixed point O is
given by s = 15t + 6t2 – t3, where t is the time in seconds after leaving O. Find
(a) the average velocity of P over the first 2 seconds,
(b) the value of s when P is instantaneous rest,
(c) the velocity of P when its acceleration is instantaneously zero.
7.
A sandbag is dropped at a height of 80m. Its distance sm from the ground after t seconds is
given by s = –5t2 + 80. Find the velocity and acceleration of the sandbag when it is 35m above
the ground.
8.
A point mass moves along the x-axis and its displacement is given by x = 2t3 + pt2 + qt,
0 ≤ t ≤ 4, where p and q are constants. When t = 1 and t = 2, the direction of its motion reverse.
(a) Find the values of p and q.
(b) Find the velocity and acceleration of the point mass at t = 4.
9.
The radius of a circular disc is increasing at a constant rate of 0.04cm/s. Find the rate at which
the area is increasing when the radius is 20cm.
10. A snowball is melting and its volume is decreasing at a rate of 12cm3/min. When its radius is
6cm, find
(a) the rate of change of its radius,
(b) the rate of change of its surface area.
11. An inverted conical vessel has a base radius of 12cm and a height of 20cm. Water flows into
the vessel at a rate of 5cm3/s. How fast is the water level rising when the water is 15cm deep?
12. The volume V of certain gas under varying pressure P obeys the law PV = 240. At the instant
when P = 60units, P is increasing at a rate of 5units/s. Find the rate at which the volume is
changing at this instant.
13. A light is hung 6m above a straight horizontal path. A man 2m tall is walking at 2.4m/s away
from the light.
(a) How fast is his shadow lengthening?
(b) At what rate is the tip of his shadow moving?
14. N is a fixed point on the circumference of a circle with centre O and radius 18cm. A variable
point P moves along the circumference at a constant speed of 6cm/s. Find
(a) the rate of increase of ∠PON in rad/s.
(b) the rate of increase of the area of the sector PON.
P. 83
15. A ladder 5m long has one end leaning against a vertical wall. The bottom of the ladder is being
moved towards the wall at 0.3m/s. When the bottom of the ladder is 1.4m from the wall,
(a) how fast is the top of the ladder sliding up the wall?
(b) what is the rate of change of the angle of inclination of the ladder to the horizontal?
16. A rocket is launched vertically and is tracked by a radar station. The station is 5km from the
launch site. When the distance between the rocket and the station is 13km, this distance is
increasing at a rate of 7500km/h. What is the vertical speed of the rocket at that instant?
17. The volume of the water with a depth of hcm in a hemispherical bowl of radius 10cm is given
1
by V  h 2 (30  h) cm3. Water is poured into the bowl at a rate of 2cm3/s.
3
(a) If the radius of the water surface is rcm, express r in terms of h.
(b) When the depth is 4cm, find
(i) the rate of change of the water level,
(ii) the rate of change of the radius of the water surface.
18. R(–1, 0) is a fixed point and P(s, 0) is a variable point on the x-axis. s is increasing at a rate of
5units/s. Q is a point on the curve y = x2 such that PQ is perpendicular to the x-axis.
(a) (i) Express the area A of ∆PQR in terms of s.
(ii) Find the rate at which A is increasing when s = 4.
(b) Find the rate of increase of the length of QR when s = 2.
19. The sides of a reactangle PQRS vary with time such that PQ increases at 0.2m/s while PS
decreases at 0.3m/s. When PQ = 20m and PS = 15m, find
(a) the rate of change of the area of PQRS,
(b) the rate of change of the length of the diagonal PR.
20. Boyle’s law for confined gases states that under constant temperature, PV = C, where P is its
pressure, V is its volume and C is a constant. At time t (≥ 0) minutes, the pressure is
(2t + 20)mmHg. At t = 0, the volume is 60cm3. Find the rate at which the volume is changing
at t = 5.
21. The surface area of a sphere is increasing at a rate of 4cm2s–1. How fast is its volume
increasing when the surface area is 36πcm2?
P. 84
22. An inverted conical tank has a base radius of 4m and a height of 8m. Water flows into the tank
at a rate of 12m3s–1.
(a) Find the rate of the water level rising when the water is 2m deep.
(b) If the volume of the water is equal to one-eighth of the capacity of the tank, find the rate
of the water level rising.
23. A straight line with inclination θ meets the parabola y = x2 at the origin O and A, where

. B is the foot of perpendicular from A to the x-axis. Given that θ is increasing at a
2
rate of 0.01rads–1.
0<θ<
(a) Let the area of ∆OAB be S. Find the rate of change of S when  
(b) Find the rate of change of OA when OA = 2 .
P. 85

3
.
1.
Optimization
Find two positive numbers whose sum is 100 such that their product is as large as possible.
2.
Divide 40 into two parts such that the product of one part with the square of the other is the
greatest.
3.
The perimeter of a rectangle is 72m. What are its dimensions that yield the maximum area?
4.
Given a piece of square metal sheet of side 24cm, an open box can be made by cutting equal
squares of side xcm from the four corners and folding up the sides.
(a) If the volume of the box is Vcm3, express V in terms of x.
(b) Find the value of x that gives a box of largest possible volume.
5.
A rectangular box of height ym has a square base of side xm. The volume of the box is 32m3.
(a) Express y in terms of x.
(b) If the perimeter of a lateral surface is Pm, express P in terms of x.
(c) Find the values of x and y which make P a minimum.
6.
In ∆XYZ, XY = XZ = 13cm and YZ = 10cm. A rectangle PQRS is inscribed in the triangle
with PQ parallel to YZ. Let PQ = xcm and QR = ycm.
(a) Express y in terms of x.
(b) If the area of the rectangle is Acm2, express A in terms of x.
(c) Find the greatest area of the rectangle.
7.
A lidless rectangular box with a square base of side xcm has a volume of 500cm3.
(a) If the height of the box is hcm, express h in terms of x.
(b) If the total surface area of the box is Scm2, express S in terms of x.
(c) Find the dimensions of the box that minimize the amount of materials used to make the
box.
8.
The metal used to make the top and bottom of a cylindrical can costs $0.04/cm2, while the
metal used to make the lateral surface costs $0.02/cm2. The capacity of the can is 108πcm3.
(a) If the base radius of the can is rcm, find the cost $C in making the can in terms of r.
(b) What should r be to minimize the cost of making the can?
9.
Find the largest possible area of the rectangle inscribed in the circle x2 + y2 = 4.
10. Find the dimensions of the rectangle of the greatest area that is inscribed in a semicircle of
radius 2 2 .
P. 86
11. (a) A rectangle PQRS has its lower base on the x-axis and upper vertices on the curve
y = 12 – x2. If P(s, t) is in the first quadrant, express the area A of PQRS in terms of s.
(b) Find the largest possible area of the rectangle.
12. Which points on the curve y = x2 are nearest to the point (0, 1)?
13. Find the longest distance from the point (4, 3) to a point on the circle x2 + y2 = 9.
14. A man has a stone wall beside a field. He has 1200m of fencing material and he wishes to
make a rectangular area using the wall as one side. What should the dimensions of the
rectangle be in order to enclose the largest possible area?
15. The total surface area of a cylindrical can is 192πcm2.
(a) If its base radius is rcm and its height is hcm, express h in terms of r.
(b) If the volume of the can is Vcm3, express V in terms of r.
(c) Find the base radius if the volume is at maximum.
16. A cylinder of base radius rcm and height hcm is inscribed in a sphere of radius 3cm.
(a) Express h in terms of r.
(b) Find the largest possible volume of the cylinder.
17. A right cone of height h and base radius r is inscribed in a sphere of constant radius R.
(a) Express r in terms of h.
(b) Find the largest possible volume of the cone.
18. Rectangular posters of area 1728cm2 are to be made with left and right margins 3cm and, top
and bottom margins 4cm.
(a) If the printing area is xcm wide and ycm high, express y in terms of x.
(b) Find the dimensions of the poster if the printing area is at maximum.
19. Each page of a book contains 187.5cm2 printing area having 5cm margins at top and bottom,
and 2.5cm margins at left and right.
(a) If the page is xcm wide, express the height hcm of the page in terms of x.
(b) Find the minimum possible area of such a page.
P. 87
4
20. P(t, ) is a point on the curve xy = 4 in the first quadrant. The tangent to the curve at P
t
intersects the x-axis and y-axis at A and B.
(a) Find the equation of the tangent.
(b) Express AB2 in terms of t.
(c) Find the shortest length of AB.
21. The two sides and the base of a trough are to be made by bending a long rectangular piece of
tin 36cm wide. The cross section of the trough is a trapezium with sides making angles of 120°
with the base.
(a) If a side of the trough is xcm, express the width wcm of the base and the width Wcm of
the top in terms of x.
(b) Express the cross sectional area Acm2 of the trough in terms of x.
(c) What should the length of a side be to maximize the volume of the trough?
22. PQ = 6km, ∠PQR = 90°and QR = 4km. A man walks from P to some point on PQ with a
speed of 8km/h and goes straight to point R with a speed of 3km/h. Find his minimum
travelling time.
23. A funnel of capacity V is in the shape of right cone.
(a) If the base radius of the funnel is r, express the surface area S of the funnel in terms of V
and r.
(b) Find the ratio of the height to the base radius if the least amount of material is to be used
in making the funnel.
24. A window frame has a shape of a rectangle of height h surmounted by a semicircle of radius r.
The perimeter of the frame is constant L.
(a) Express h in terms of L and r.
(b) Find the ratio h : r that will admit the most amount of light.
25. A farmer has 1000m of fences, which he wishes to use to fence a horizontal rectangular field
with three equal parts divided by two vertical fences. Find the dimensions and the area of the
field that has the maximum area.
26. A cylindrical lidless container has a capacity of 24πm3. The cost of the material in making the
bottom and the curved part are $150/m2 and $50/m2 respectively. Find the dimensions and the
cost of the container that has the minimum cost.
27. Find the maximum volume of a right cylinder that can be inscribed in a right cone of height
12cm and base radius 4cm, if the axes of the cylinder and the cone coincide.
P. 88
28. A wire 30m long is cut into two pieces. One of them will be bent into a circle and the other
will be bent into an equilateral triangle.
(a) Where should the wire be cut so that the sum of the areas of the circle and the triangle is
maximized?
(b) Where should the wire be cut so that the sum of the areas of the circle and the triangle is
minimized?
29. Show that the greatest isosceles triangle inscribed in a circle with a fixed radius is equilateral.
30. A billboard 5m tall is located on the top of a building with its lower edge 15m above the eye
level of a viewer. Find the horizontal distance between the billboard and the viewer that
maximizes the angle between the lines of sight of the top and bottom of the billboard.
31. P is a point moving in quadrant I on a circle centered at the origin O of radius r. F is a fixed
point on the positive x-axis such that OF = l and Q is a point on the right of F. Let ∠POF = θ
and ∠PFQ = α.
(a) Show that tan  
r sin 
.
r cos   l
(b) When l = 2r, find the least value of α.
(c) When r = 2l, find the greatest value of α.
32. At some instant, ship A is 60km due north of ship B. A is sailing due east at 20km/h while B is
sailing due north at 15km/h.
(a) Express the distance skm between A and B at the time t hours after that instant in terms of
t.
(b) Find the time and distance between A and B when they are closest.
33. A point mass moves along the x-axis and its displacement is given by x = 2t3 – 9t2 + 12t,
0 ≤ t ≤ 4.
(a) When will the velocity be at maximum?
(b) When will the acceleration be at minimum?
P. 89

Mean Value Theorem
Rolle’s Theorem: If f is continuous in [a, b], differentiable on (a, b) and f (a) = f (b), then there
exists c on (a, b) such that f '(c) = 0.

Mean Value Theorem: If f is continuous in [a, b] and differentiable on (a, b), then there exists
c on (a, b) such that f (c) 
1.
f (b)  f (a)
.
ba
By using Mean Value Theorem, show that
(a) |cos x – cos y| ≤ |x – y|
(b)
sin px
 p
x
(c)
x
 ln(1  x)
1 x
2.
Let f (x) be a continuous function defined on [3, 6]. If f (x) is differentiable on (3, 6) and
| f '(x) – 9| ≤ 3, show that 18 ≤ f (6) – f (3) ≤ 36.
3.
Let P(x) = anxn + an – 1xn – 1 + … + a0 be a polynomial with real coefficients and
an
a
 n 1    a0  0 . By using Mean Value Theorem, show that the equation P(x) = 0 has
n 1
n
at least one real root between 0 and 1.
4.
By using Mean Value Theorem, show that for 0 < x ≤ 1, 1 + x < ex < 1 + ex.
5.
A function f defined on an interval [a, b] is said to be Lipschitz-continuous if there exists a
positive constant k such that | f (x1) – f (x2)| ≤ k|x1 – x2| for all x1, x2 on [a, b]. Show that if f is
differentiable with | f '(x)| ≤ K for all x on [a, b], where K is a positive constant, then f is
Lipschitz-continuous.
6.
Let f be a function defined in (0, ∞). If f '(t) is an increasing function, show that
f (n) + f '(n)(t – n) ≤ f (t) ≤ f (n) + f '(n + 1)(t – n) for all t in (n, n + 1).
7.
Let a > 0. The function f ( x)  ln x  ln
a
is defined on the set of all positive real numbers.
x
By using Mean Value Theorem and the facts that ln 1 = 0 and
positive real numbers a and b, ln
a
 ln a  ln b .
b
P. 90
d
1
(ln x)  , show that for any
dx
x
L’Hospital’s Rule
0 
,
, 0∙∞, ∞ – ∞, 1∞, 00 and ∞0
0 

Indeterminate forms:

L’Hospital’s Rule: If lim f ( x) and lim g ( x) are both 0 or both   and lim
x a
or equals   , then lim
x a
1.
xa
x a
f ( x)
f ( x)
, where a can be replaced by a–, a+, ∞ or –∞.
 lim
x

a

g ( x)
g ( x)
Use L’Hospital’s Rule to evaluate the following limits.
(a)
x3  3x 2  2 x  6
x 3
x2  9
lim
x  tan x
x  0 sin 2 x
(b) lim
(c)
e x  e x
x  0 sin x
lim
tan 1 x
x 0
x
(d) lim
e2 x  1
x  0 ln(1  x )
(e)
lim
(f)
lim
x 2  3x  2
x 
3x3  1
x  sin x
x 0
x3
(g) lim
e x  e x  2 x
x 0
x  sin x
(h) lim
(i)
(j)
24  12 x 2  x 4  24 cos x
x 0
sin 6 x
x tan x
lim
x 0
1  x2  1
lim
ln(e x  x 2 )
x 
x2
(k)
lim
(l)
lim
f ( x)
exists
g ( x)
sin 2 x  x 2
x  0 x 2 sin 2 x
ax 1
, where a > 0
x 0
x
(m) lim
1  ln x  x x
x 1 1  ln x  x
(n) lim
P. 91
2.
Use L’Hospital’s Rule to evaluate the following limits.
(a)
lim sin x ln x
x 0 
(b) lim x csc 2 x
x 0
(c)
lim (  x) tan
(d)
lim x3e x
(e)
x 
x
2
x 
lim x 4 ln x
x 0 
(f)
1
lim x[(1  ) x  e]
x 
x
(g)
1
1
lim x[(1  ) x  e ln(1  ) x ]
x 
x
x
1
(h) lim (  csc x)
x 0 x
(i)
lim (
1
1
 2)
2
sin x x
(j)
lim (
x
1

)
x  1 ln x
(k)
(l)
x 0
x 1
lim (tan 5x  tan x)
x  / 2 
lim (
x 0
1
1
 )
tan x x
P. 92
3.
Use L’Hospital’s Rule to evaluate the following limits.
(a)
lim x1 / x
(b)
3
lim (1  ) x / 2
x 
x
(c)
(d)
x 
lim (tan x)cos x
x  / 2 
lim xsin x
x 0 
x 1/ x
)
sin x
(e)
lim (
(f)
lim (
(g)
x 0
x 
sin 2 x 1 / x 2
)
x
lim x x
x 0 
(h) lim (1  sin x)1 / x
x 0
(i)
(j)
lim (1  sin 2 x)1 / x
x 0
lim (cos x)ln sin x
x  / 2 
(k) lim (
sin 1 x 1 / x 2
)
x
(l)
sin x 1 / x 2
)
x
x 0
lim (
x 0
1
(m) lim x 1 x
x 1
(n)
lim (tan x) tan 2 x
x  / 4
a x  1 1/ x
)
for a > 1 and 0 < a < 1.
a 1
4.
Evaluate lim (
5.
Let f (x) be a continuously differentiable function such that f (0) = 0 and f '(0) ≠ 0. Show that
x 
lim x f ( x )  1 .
x 0 

A function f (x) is said to be continuously differentiable if and only if f (x) is differentiable and
f '(x) is continuous.
P. 93
6.
7.
Evaluate the following limits.
( x 2  1) sin x
x 1
x
ln(1  sin )
2
(a)
lim
(b)
lim
(c)
lim
x  sin x
x   x  sin x
2 x  sin 2 x
x   (2 x  sin x)esin x
Let f be a function such that f (xy) = f (x) + f (y) for all real numbers x and y.
1
(a) Show that for x ≠ 0, f ( x)   f ( ) .
x
(b) Hence, show that if f is continuously differentiable at every x ≠ 0, then f ( x) 
f (1)
for
x
any non-zero real number x.
8.
Let f be a function such that for any real numbers t and x, f ( x)  f (t )  ( x  t ) f (
Suppose f has a continuous third derivative.
(a) Show that for x ≠ t, f (
xt
)
2
4( f ( x)  f (
xt
xt
))
2
.
(b) By finding the value of f '''(x), show that f ''(x) is a constant.
P. 94
xt
).
2
Indefinite Integral

Indefinite Integral: If
d
F ( x)  f ( x) , then
dx
 f ( x)dx  F ( x)  C , where 
is called the
integral sign, f (x) is called the integrand, dx is called the differential, F(x) or F(x) + C is
called the primitive function/antiderivative and C is called the arbitrary constant.
n
 x dx 

Power Rule:
1.
Evaluate
 6 xdx .
2.
Evaluate
 7dx .
3.
Evaluate
 9 x dx .
4.
Evaluate
 x dx .
5.
Evaluate
 4 dx .
6.
Evaluate
12
7.
Evaluate

8.
Evaluate

9.
Evaluate
x
10. Evaluate
x
11. Evaluate
x
x n 1
 C , where n  1
n 1
2
8
1
x dx .
6
x dx .
5
x 4 dx .
dx
3
dx
7
.
.
x dx .
P. 95
3
12. Evaluate
x
x
dx .
2
13. (a) Show by differentiation that
1
 ( x  1) dx  3 ( x  1)
2
3
1
 ( x  1) dx  3 x
2
3
 x 2  x  C1 and
 C2 .
(b) What is the relation between C1 and C2?
14. If

d2y
dy
and y in terms of x.
 12x 2 , find
2
dx
dx
Trigonometric Rules:
dx
15. Evaluate
 cos
16. Evaluate
 sin
17. Evaluate
 cos
18. Evaluate
 sin
2
2
x
2
2
.
x
dx
 cos xdx  sin x  C
 sin xdx   cos x  C
 sec xdx  tan x  C
 csc xdx   cot x  C
 sec x tan xdx  sec x  C  csc x cot xdx   csc x  C
.
sin x
dx .
2
x
cos x
dx .
2
x

Constant Multiple Rule:

Sum and Difference Rule:
 kf ( x)dx  k  f ( x)dx
 ( f ( x)  g ( x))dx   f ( x)dx   g ( x)dx
19. Evaluate
 (2 x
20. Evaluate
 (3  5x
21. Evaluate
 (3x  2)( x  5)dx .
2
 6 x  1)dx .
4
 8 x5 )dx .
P. 96
22. Evaluate
 (2 x  1) dx .
23. Evaluate
( x
24. Evaluate
 (4
25. Evaluate
u
26. Evaluate

27. Evaluate
x4  5x  1
 x6 dx .
28. Evaluate

29. Evaluate
(
30. Evaluate
(
31. Evaluate
 (2 sin x  3cos x)dx .
32. Evaluate
 (1  sin x  cos x)dx .
33. Evaluate
 sin 2 cos 2 d .
34. Evaluate
1  2 cos 3 
 cos 2  d .
2
2
2
3

3
 6)dx .
x2
y
1
2 y
)dy .
(u 5  3u )du .
s (3 s  4 s )ds .
(2 x  1)(3  x)
dx .
x4
1
x  ) 2 dx .
x
2y 1 2
) dy .
3
y2


P. 97
t
dt .
2
35. Evaluate
 4 sin
36. Evaluate
 (sin 2  cos 2 ) dt .
37. Evaluate
 ( cos
38. Evaluate
 (tan x  5) cos xdx .
39. Evaluate
 1  cos 2u .
40. Evaluate
 cos u  sin u du .
41. Evaluate
 ( x  1)( x  2)( x  3)dx .
42. Evaluate

2
t
t
5
2
x

4
)dx .
sin 2 x
du
cos 2u
x 1
dx .
3
x 1
43. (a) Show that
1
1
1

 2 .
2
2
sin  cos  cos  sin 
2
(b) Hence, evaluate
44. Evaluate
2
 sin
2
d
.
 cos 2 
cos 2t
dt .
2
2t
 sin
P. 98
 e dx  e
x
x
C

Exponential Rule:

x
Exponential Rule with an Arbitrary Base:  b dx 

Reciprocal Rule:
bx
C
ln b
1
 x dx  ln | x | C
45. Find all the antiderivatives of the function f (x) = ex.
46. Find all the antiderivatives of the function h(y) = y–1.
1
47. Evaluate
 2 y dy .
48. Evaluate

49. Evaluate
e
50. Evaluate
10t 5  3
 t dt .
t 1
dt .
t
x2
dx .
51. For the function f (u) = 2eu + 3, find the antiderivative F that satisfies the condition F(0) = 8.
52. For the function f ( y ) 
3 y3  5
, find the antiderivative F that satisfies the condition F(1) = 3.
y
53. For the derivative y(t ) 
3
 6 , find the function y that satisfies the condition y(1) = 8.
t
1
54. For the derivative f (t )  , find the function f (t) that satisfies the condition f (1) = 4.
t
55. Evaluate

1 x
dx .
x
P. 99

Integration by Change of Variable:
56. Evaluate
 ( x  3) dx .
57. Evaluate
 ( x  1)
58. Evaluate
 (3x  2) dx .
59. Evaluate
 (9  5x)
60. Evaluate

61. Evaluate

62. Evaluate
x
63. Evaluate
 5x
64. Evaluate

dy du
5
1
6
dx .
7
3/ 4
dx .
1
dt .
4t  7
3
1  8t dt .
x 2  5dx .
6  x5 dx .
4
v
3v 2  1
v2
65. Evaluate

66. Evaluate
 (x
67. Evaluate
 (x
68. Evaluate
 (2  s )
4
1  2v 3
2
2
dv .
dv .
 5x)3 (2 x  5)dx .
x2
dx .
 4 x  1)3
2 5
dy
 du dx dx   du du
6s  s3 ds .
P. 100
69. Evaluate
4  s2
 (s3  12s  3)2 / 7 ds .
70. Evaluate
x
71. Evaluate
t
72. Evaluate
 x( x  1) dx .
73. Evaluate
 x(2  3x) dx .
74. Evaluate
x
75. Evaluate
x
76. Evaluate

77. Evaluate
 (1  x)
1
2
1
3
(1  ) 4 dx .
x
(1 
3
1 7/8
) dt .
t2
7
6
1  x dx .
2
x  2dx .
x
dx .
x3
2x
2/3
dx .
x
x2
78. (a) Show by differentiation that 
dx 
 C1 .
(1  x 2 )2
2(1  x 2 )
(b) Use the substitution u = 1 – x2 to show that
(c) What is the relation between C1 and C2?
79. Evaluate
 (cos 7 x  3sin 2x)dx .
80. Evaluate
 4 sin(1  2x)dx .
P. 101
x
 (1  x )
2 2
dx 
1
 C2 .
2(1  x 2 )
x
dx .
2
81. Evaluate
 5 sec
82. Evaluate
 csc (7  2x)dx .
83. Evaluate
 tan 8x sec8xdx .
84. Evaluate
 cot 4 csc 4 dx .
85. Evaluate
 x sin( x )dx .
86. Evaluate

87. Evaluate
 (1  3cos x)
88. Evaluate
 (1  2 sin x)
89. Evaluate
 sin
90. Evaluate
 cos
91. Evaluate
 sin
92. Evaluate

93. Evaluate
 cos d .
2
2
x
x
2
1
cos 2 x dx .
x
2
cos x
2
3
3
sin xdx .
dx .
t cos tdt .
t sin tdt .
cos t
dt .
5
t
sin t
dt .
3
cos t
3
P. 102
94. Evaluate
 sec
95. Evaluate
 cot
96. Evaluate
 tan sec d .
97. Evaluate
 tan
98. Evaluate
 cot
99. Evaluate
 cot 7u csc
100. Evaluate
 cot
101. Evaluate
sec4 
 tan2  d .
102. Evaluate
cot 3 
 csc d .
103. Evaluate
 sin x sin(cos x)dx .
104. Evaluate
 sec
105. Evaluate

106. Evaluate
 (1  sin
4
tdt .
3
2u csc 2udu .
4
4
2 sec2 2d .
2
x csc4 xdx .
3
2
5
7udu .
x 3x
csc dx .
2
2
x sec2 (tan x)dx .
x
1 x
2
cos 1  x 2 dx .
sin  cos 
d .
2
 )5
P. 103
107. Evaluate

1  cos x
dx .
sin 2 x
108. (a) Evaluate
 sin  cosd
by the substitution u = sin θ.
(b) Evaluate
 sin  cosd
by the substitution v = cos θ.
(c) Show that the results obtained in (a) and (b) are equivalent.
109. Evaluate
 (3x
110. Evaluate
6x2  2
 ( x3  x  6)3 dx .
111. Evaluate
x
112. Evaluate
 tan
113. Evaluate

114. Evaluate
 (6 x  5)
115. Evaluate
 (2 x  1)
116. Evaluate
 x 1 .
117. Evaluate

118. Evaluate
e
2
 2)3 x3  2 x  1dx .
x 4  2dx .
7
3
xsec xdx .
sin x cos x
9 sin 2 x  5 cos 2 x
dx
8
10
dx .
.
dx .
dx
dx
.
1 x
1 3 x
dx .
P. 104
119. Evaluate
 sin(1  4 x)dx .
120. Evaluate
 sec (2x  1)dx .
121. Evaluate
 cos
122. Evaluate

123. Evaluate
 x ln x .
124. Evaluate
 cot kx  csc kx .
125. Evaluate
x
2
5
3x sin 3xdx .
cos x
dx .
x
dx
dx
3
1  3x 2 dx .
e 2 x dx
126. Evaluate

127. Evaluate
sec2 x
 2  3 tan x dx .
128. Evaluate
x
2
129. Evaluate

xdx
.
4x  5
3
ex  1
.
x2
dx .
 4x  5
130. Let f (x) = xn|x|, where n is a positive integer.
(a) Show that f '(0) exists and find its value.
(b) Show that f '(x) = (n + 1)xn – 1|x|.
(c) Hence, find
x
n
| x | dx .
P. 105
131. Evaluate
x
132. Evaluate

3
5  3x dx .
dx
.
x 1
P. 106
Summation

n
 f (i)  f (1)  f (2)    f (n) , where ∑ is called the summation sign and i is called the
i 1
running index.
n

Constant Multiple Rule:
n
 kf (i)  k  f (i)
i 1
i 1
n

i 1


1.
n
n
i 1
i 1
 ( f (i)  g (i))   f (i)   g (i)
Sum and Difference Rule:
The running index is dummy, i.e.
n
n
i 1
k 1
 f (i)   f (k ) .
n
nk
i 1
i 1 k
 f (i)   f (i  k )
Compensation Rule:
Evaluate the following expressions.
5
(a)
 (1  n)
2
n2
9
(b)
 log
k 1
99
(c)
10
1
k
k 1
1
 ( n  n  1)
n 1
2.
What do the following question marks represent?
(a)
n
?
k 1
k 0
 kqk 1   ?
2n
(b)
 (1)
k
kpk  2np 2 n  ?
k 0
(c)
(d)
3.
?
d2 n k
( x )   k (k  1) x k 2 , where n ≥ 2
2 
dx k 0
?
n
?
k 0
?
 (1) k 1 sin(2k  1)   (1) ? sin(2k  1)
(a) Show that (i + 1)3 – i3 = 3i2 + 3i + 1.
n
(b) Hence, show that
i
i 1
n
(c) Show that
i
i 1
3
[
2

1
n(n  1)(2n  1) .
6
n(n  1) 2
] .
2
(d) For each of the following questions, use (b) and (c) to deduce a formula for evaluating the
expression.
(i) 13 + 33 + … + (2n – 1)3
(ii) 12 – 22 + 32 – 42 + …– (2n)2 + (2n + 1)2
P. 107
Definite Integral

n
Let f be defined on [a, b].
 f (a  i 
i 1
ba ba
is said to be the right Riemann sum of f
)
n
n
on [a, b] with n subintervals of equal length. The left Riemann sum and the midpoint
Riemann sum are defined in a similar way.

Let f be defined on [a, b], xi  a  i 

b
a
n
f ( x)dx  lim  f ( xi *)
n 
i 1
ba
for i = 0, 1, …, n and xi – 1 ≤ xi* ≤ xi.
n
ba
is said to be the definite integral of f from a to b, where a is
n
called the lower limit and b is called the upper limit. If this limit exists for all possible
choices of xi*, then f is said to be integrable on [a, b].
P. 108


1.
The definite integral of f from a to b represents the net area, i.e. the sum of the areas of the
parts above the x-axis and below the curve y = f (x) minus the sum of the areas of the parts
below the x-axis and above the curve y = f (x).
If f is continuous on [a, b] or has only a finite number of jump discontinuities, then f is
integrable on [a, b].
(a) Sketch the graph of y = 2x + 1 for 0 ≤ x ≤ 5.

(b) What is the exact value of
4
1
(2 x  1)dx ?
(c) Find the midpoint Riemann sum of

4
1
(2 x  1)dx with 3 subintervals of equal length.
(d) What is the relation between the sum obtained in (c) and
2.

4
1
(2 x  1)dx ?
(a) Sketch the graph of y = x3 for 0 ≤ x ≤ 2.
(b) Find the right Riemann sum of

2
0
x3dx with 5 subintervals of equal length.
(c) Draw the rectangles corresponding to the partition in (b).
(d) State whether the sum obtained in (b) overestimates or underestimates
3.
Evaluate the definite integral

2
0
xdx by definition.
P. 109

2
0
x3dx .
4.
(a) Sketch the graph of y  4  x 2 for –2 ≤ x ≤ 2.
(b) Find the exact value of

2
4  x 2 dx .
0
(c) Find the midpoint Riemann sum of
5.

2
4  x 2 dx with 5 subintervals of equal length.
0
(a) For a function y = f (x), if m ≤ f (x) ≤ M for all x in [a, b], show that
b
m(b  a)   f ( x)dx  M (b  a) .
a
(b) Hence, find a lower bound and an upper bound for
6.
5
 (x
2
(a) Sketch the curves y = x2 and y = x3 on the same graph.
1
(b) Which integral is greater,
 x dx
(c) Which integral is greater,

0
2
1
2
or
x 2 dx or
1
 x dx ?
3
0

2
1
x3dx ?
P. 110
2
 1)dx .

a
b
f ( x)dx   f ( x)dx
f ( x)dx  0

Zero width property:

Reversing property:

Splitting property:

Dummy variable property:

Rectangular property:

Constant Multiple Rule:

Sum and Difference Rule:
7.
What is the value of
8.
If y = f (x) is integrable on [–1, 3], show that
9.
(a) Find
a

a

b
a

b
a
c
b
a
c
f ( x)dx   f ( x)dx   f ( x)dx

b
a

b
a
b
f ( x)dx   f (u )du
a
kdx  k (b  a)

b
a
b
kf ( x)dx  k  f ( x)dx
a
b
b
b
a
a
a

( f ( x)  g ( x))dx   f ( x)dx   g ( x)dx
a
f ( x)dx   f ( x)dx ?
b

0
1
0
1
2
 (2 x  3)dx , given that
0
1

  x sin xdx  2 2 x sin xdx .
2


2
2
1
 x dx  3 .
0
0
x3dx  0 .
2
P. 111
2
3
1
0
3
f ( x)dx   f ( x)dx   f ( x)dx  0 .
 3dx .

11. Show that
b
1
(b) Hence, find
10. Show that
a

12. If
  cos xdx  A , find
2

2


2
0
cos xdx .
13. (a) Simplify (x + 1)3 – (x – 1)3.
(b) Given that
14. (a) Find

2
0
1
1
1
2
0
0
xdx .
(b) Hence, find
1
 x dx  3 , find the value of  ( x  1) dx   ( x  1) dx .

2
2
| x | dx .
P. 112
3
0
3
Fundamental Theorem of Calculus

Fundamental Theorem of Calculus: Let f be continuous on [a, b].
a < x < b and


b
a
d x
f (t )dt  f ( x) if
dx a
f ( x)dx  F (b)  F (a) , where F is any primitive function/antiderivative of f.
Net Change Theorem: The definite integral of a rate of change from a to b is the net change
b
 F ( x)dx  F (b)  F (a) .
from a to b, i.e.
a

Integration by Change of Variable:
1.
Evaluate
2.

2
Evaluate

3
3.
Evaluate

1
4.
Evaluate

8
5.
Evaluate

2
6.
Evaluate
7.
Evaluate
a
x(2 x  5)dx .
1
2
( y 3  4 y  7)dy .
x 1
dx .
3
x
1

dy du
x 4 dx .
0
1
b
(t 

1 2
) dt .
3t
( x  cos x)dx .
2
0

 cot
2
2
xdx .
4
8.
Evaluate


9.
Evaluate

4

6
10. Evaluate
0
(sin x  cos x)2 dx .
3
1
dy
du , where u = g(x)
g ( a ) du
 du dx dx  
| x | dx .
| x  3 | dx .
P. 113
g (b )
11. Evaluate
2
|x
0
1
x | dx .
12. Evaluate

13. Evaluate
 (4 x  3) dx .
14. Evaluate

1
15. Evaluate

4
16. Evaluate

4
17. Evaluate
5
1
1
dt .
 2 (1  2t ) 4
0

x2  4x
03
x3  6 x 2  1
5
4 | x |dx .

23. Evaluate
1
(1  x ) 4 dx .
x
1
19. Evaluate
22. Evaluate
x x 2  9dx .
1

21. Evaluate
3
3
4
18. Evaluate
20. Evaluate
| x 2  1 | dx .
5
1
2
1

0


2
0


6
0


2
0
dx .
( y  1) y  3dy .
y 1  y y dy .
cos 
d .
(2  sin  ) 2
sin 2 cos 3 2d .
cos 3 d .
P. 114
24. Evaluate


0
x2
dx .
2
x sin

25. Evaluate
26. Evaluate
1  tan 2 x
4 (1  tan x)2 dx .
3


3
0
tan3 x
dx .
sec x
27. (a) Show that


2
0

(a cos 2   b sin 2  )d   2 (a sin 2   b cos 2  )d , where a and b are
0
constants.
(b) Hence, evaluate


2
0
(a cos 2   b sin 2  )d .



cos x
sin x
28. (a) Use the substitution x   u to show that 
dx   2
dx .
0
cos x  sin x
cos x  sin x
2
(b) By considering the sum of these integrals, find their common value.
29. (a) Show that




4
0
(b) Deduce that
(c) Hence, evaluate
30. (a) Show that

a
0


0
4
f ( x)dx   4 f (
4
0
2
0
 x)dx .

1  sin 2 x
dx   4 tan 2 xdx .
0
1  sin 2 x


4
0
1  sin 2 x
dx .
1  sin 2 x
a
x m (a  x)n dx   x n (a  x)m dx , where m and n are constants.
(b) Hence, evaluate
0

8
0
x 2 3 8  x dx .
P. 115
31. (a) Show that

a
f ( x)dx  
0
0

(b) Hence, show that
(c) Show also that
(d) Evaluate

 x
4


a
a
f ( x)dx .
a
f ( x)dx  2 f ( x)dx when y = f (x) is an even function.
0
f ( x)dx  0 when y = f (x) is an odd function.
sin xdx .
  3 cos
2

a
a

(e) Evaluate
a
4
d .
2
32. Simplify the following expressions.
(a)
(b)
(c)
d x 2
(t  t  1)dt
dx 3
d x t
e dt
dx 0
d x3 dp
dx 2 p 2
(d)
d 10 dz
dx x 2 z 2  1
(e)
d 1 4
t  1dt
dx x
(f)
d 0 dp
dx x p 2  1
(g)
d x
1  t 2 dt
dx  x
(h)
d e2 x 2
ln t dt
dx e x
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