Remedial plan
Grade/Cluster
12 ES
Subject
Section
Mathematics
Date
Question 1:
Find the derivative of each of the following functions:
1)

f ( x)  ( x  1)
2

 2 3
f ( x )  ( x 2  1) 6
f ( x )  6( x 2  1) 5 ( 2 x )
f ( x )  12 x ( x 2  1) 5
2) f ( x)  4 x  3x  10
2
f ( x) 
8x  3
2 4 x 2  3x  10
2
2
5
f
(
x
)

x
(
x

1
)(
x
)
3)
f ( x)  ( x 4  x 2 )( x5 )
f ( x)  x 9  x 7
f ( x)  9 x8  7 x 6
3
5
f
(
x
)

(
x

x

1
)
4)
f ( x)  5( x 3  x  1) 4 (3x 2  1)
5) f ( x)  x
3
4x  1
f ( x)  (3 x 2 )( 4 x  1) 
f ( x)  3 x 2 4 x  1 
4
2 4x 1
2x3
4x 1
(x3 )
8x
f
(
x
)

6)
( x 3  1) 2
f ( x ) 
8( x 3  1) 2  2( x 3  1)(3 x 2 )
( x 3  1) 4
8( x 3  1) 2  6 x 2 ( x 3  1)
f ( x ) 
( x 3  1) 4
8( x 3  1)  6 x 2
f ( x ) 
( x 3  1) 3
8x3  8  6 x 2
f ( x ) 
( x 3  1) 3
7) f ( x)  (
8)
x 2  4  3x 2 )
3
2
1
2
3
f ( x ) 
(
2
x  4  3x ) (
3
f ( x ) 
(
2
x  4  3x ) (
2
2
2x
x 4
2
2
2
2
1
2
x
x 4
2
 6 x)
 6 x)
f ( x)  x 3 e 2 x
f ( x)  (3x 2 )(e 2 x )  (2e 2 x )( x 3 )
f ( x)  e 2 x (3x 2  2 x 3 )
9) f ( x)  5
f ( x )  (
ln x
1
)(ln 5)(5ln x )
x
10)
f ( x )  e 2 x (ln xex  1)
f
f
f
f
(1)( e x )  (e x )( x )
( x )  ( 2e )(ln xe  1)  (
)( e 2 x )
x
xe
(e x )(1  x )
2x
x
( x )  ( 2e )(ln xe  1)  (
)( e 2 x )
x
xe
(1  x)
( x )  ( 2e 2 x )(ln xex  1)  (
)( e 2 x )
x
1 x

( x )  (e 2 x ) 2(ln xex  1) 
x 


2x
x
11)
f ( x )  log 6 (1  x  x 2 )
1  2x
1
)(
)
2
1 x  x
ln 6
1  2x
f ( x ) 
(1  x  x 2 )(ln 6)
f ( x )  (
12)
f ( x )  cos(3 x  5)
f ( x )  3 sin( 3 x  5)
13)
f ( x )  sin(ln 3 x 4 )
12 x 3
4
f ( x ) 
cos(ln
3
x
)
4
3x
4
f ( x )  cos(ln 3 x 4 )
x
14)
f ( x)  esin 2 x
f ( x)  (2 cos 2 x)esin 2 x
15)
f ( x )  sin(ln(cos x 3 ))
 3 x 2 sin x 3
f ( x)  (
) cos(ln(cos x 3 ))
3
cos x
16)
f ( x )  ln(sec x  tan x )
sec x tan x  sec 2 x
f ( x ) 
sec x  tan x
sec x (tan x  sec x )
f ( x ) 
sec x  tan x
f ( x )  sec x
17)
f ( x )  x sin x
ln f ( x )  ln x sin x
ln f ( x )  sin x ln x
f ( x )
1
 (cos x )(ln x )  ( )(sin x )
f ( x)
x
sin x
f ( x )  x sin x (cos x ln x 
)
x
18)
f ( x)  e 2 x cos 4 x
f ( x)  (2e 2 x )(cos 4 x)  (4 sin 4 x)(e 2 x )
f ( x)  e 2 x (2 cos 4 x  4 sin 4 x)
19)
f ( x) 
ln x 2
x
2x
)
2
ln x 2
x
(
)( x)  (1)(
)
2
x
f ( x)  2 ln x
x2
(
1
f ( x) 
ln x
2

x2
20)
f ( x)  4 tan x
4 sec 2 x
f ( x) 
2 4 tan x
2 sec 2 x
f ( x) 
4 tan x
ln x 2
x
21)
f ( x )  ln( x  5 x )
4
2
3
2
3
ln( x 4  5 x 2 )
2
3 4 x 3  10 x
f ( x )  ( 4
)
2 x  5x 2
3( 2 x )( 2 x 2  5)
f ( x ) 
2( x 2 )( x 2  5)
f ( x) 
f ( x ) 
3( 2 x 2  5)
x ( x 2  5)
22)
f ( x)  3  10
 x2
f ( x)  3(2 x)(ln 10)(10  x )
2
f ( x)  6 x(ln 10)(10  x )
2
23)
f ( x)  3(sin 2 x  cos 2 x)10
f ( x)  3(1)10
f ( x)  0
24)
f ( x)  cos 4 x  3 sin

12
f ( x)  4 cos3 x(  sin x)
f ( x)  4 cos3 x sin x
Question 2:
The distance of a moving particle from the origin at time t
S (t )  sin t  2 cos t Find the velocity at time
t 

4
S (t )  cos t  2 sin t



S ( )  cos( )  2 sin( ) 
4
4
4
 2
2
Question 3:
3


h
(
x
)

f
(
x
)
Given that
and f (3)  1 and f (3)  2
Find h (3)
2
h( x)  3 f ( x) ( f ( x))
2
h(3)  3 f (3) ( f (3))
h(3)  3(1) 2 (2)  6
Question 4:
2
f
(
x
)

x
Let
and g ( x)  3x  6 Find  f ( g ( x))
 f ( g ( x)) 
(3x  6)2  3x  6
Question 5:
Find the equation of tangent line to the graph of equation
f ( x)  x 4  5x 3  2 At x = 2
f ( x)  4 x 3  15 x 2
m  4(2) 3  15(2) 2  28
A (2, -24)
y  m( x  x1 )  y1
y  28( x  2)  24
y  28 x  56  24
y  28 x  32
Question 6:
Find all points on the graph of f ( x)  x
tangent line to the graph is horizontal
3
 5x 2  2 x  1
where the
f ( x)  3x 2  10 x  2
3x 2  10 x  2  0
5  19
5  19 3 5  19 2
5  19
 y(
)  5(
)  2(
)  1  11
3
3
3
3
5  19
5  19 3 5  19 2
5  19
x2 
 y(
)  5(
)  2(
) 1  1
3
3
3
3
5  19
A(
,11)
3
5  19
B(
,1)
3
x1 
Question 7:
If the cost function in dollars for q units is
C (q )  100q  100
Demand function for q units
p  D( q )  100 
50
ln q
a) Find the marginal revenue
R  qp
50
)
ln q
50 q
R  100 q 
ln q
R  q (100 
1
(50 q )
q
R   100 
(ln q ) 2
50 ln q  50
R   100 
(ln q ) 2
50 ln q 
b) Find the marginal cost
C ( q )  100q  100
C ( q )  100
c) Find the marginal profit from one more unit when 8 units
are sold
P  R C
P ( q )  100 q 
50 q
 (100 q  100)
ln q
50 q
 100
ln q
1
50 ln q 
(50 q )
q
P ( q ) 
(ln q ) 2
50 ln q  50
P ( q ) 
(ln q ) 2
50 ln 8  50
P (8) 
 $12.48
(ln 8) 2
P(q) 
Question 8:
Assume that the total revenue received from the sale of x
items is given by
R( x)  30 ln( 2 x  1)
While the total cost to produce x items is
C ( x) 
x
2
Find the number of items that should be manufactures so
that profit is maximum
P ( x )  R ( x )  C ( x )  30 ln( 2 x  1) 
P ( x )  30(
x
2
2
1
)
0
2x  1
2
60
1

2x  1
2
2 x  1  120
2 x  119
x
119
2
Question 9:
The sales of a new personal computer (in thousands) is given
by
S (t )  100  90e 0.3t
Where t represent time in years
Find the rate of change of sales at each time
a) After 1 year
b ) after 5 years
S (t )  ( 0.3)( 90)e 0.3t
S (t )  27e 0.3t
S (1)  20
S (5)  6
c) What is happening to the rate of change of sales as time
Goes on? Rate is decreasing
d) Does the rate of change of sales ever equal zero
The rate of change of sales never equals zero, it gets closer
and closer to zero as t increases
Question 10:
x
Find the slope of the tangent line to the curve xe at x = 1
f ( x)  xe x
f ( x)  (1)(e x )  (e x )( x)
f ( x)  e x (1  x)
f (1)  e(1  1)  2e
Question 11:
Find the rate of change of the calcium level in the blood
1
1
stream is given by: C (t )  2 (2t  1) 2 with respect to time
after 4 days
3
1 1
C (t ) 
(
)( 2t  1) 2 ( 2)
2 2
3
1
C (t ) 
( 2t  1) 2
2
3
1
1
C ( 4) 
(9) 2 
2
54
Question 12:
The figure is a graph of a
graph at x=2. Find
f ( 2) 
function f and a tangent line to the
f (2) 
1 5
 2
20
Question 13:
Find a relative maximum about the
graph
Relative maximum at x = 3
Question 14:
Find the horizontal asymptote of the given function
2 x 2  5x  1
f ( x) 
x2  2
Y= 2 is the horizontal asymptote
Question 15:
x2
Let f ( x)  5 x  9 and g ( x)  5  1
Compute ( f g ( x))
x2
( f g ( x))  5(  1)  9   x 2  4
5

( f g ( x) )  2 x
Question 16:
A piece of carton 12 cm by 10 cm is to be used to make an open box for
doughnuts. Squares of equal sides x are cut out of each corner then the
sides are folded to make the box
1. Prove that the volume of the open box is V(x) = 4x3– 44x2+ 120x.
V ( x)  x(10  2 x)(12  2 x)
V  x(120  20 x  24 x  4 x 2 )
V  120 x  44 x 2  4 x 3
2. When does the volume reach its maximum?
What is the maximum volume?
V ( x )  12 x 2  88 x  120  0
x1  5.5
x 2  1.8
max volume  V (1.8)  4(1.8) 3  44(1.8) 2  120(1.8)  97
3. If the volume of a doughnut is 5cm3, how many doughnuts
will you be able to fit in the box?
4 x 3  44 x 2  120 x  5
4 x 3  44 x 2  120 x  5  0
x1  0.04
x2  6
x3  5
4. What is the rate of change of the volume at the maximum?
V (1.8)  12(1.8) 2  88(1.8)  120  0.48
5. Check the concavity of the volume function?
V ( x )  24 x  88  0
x
88
 3 .7
24
-
3.7
+
f (x) concave down on  ,3.7
f(x) concave up on
3.7,
Question 17:
f ( x)  ln x
Compute
f ( x), f ( x), f ( x), f ( 4) ( x), f (5) ( x)
f ( n ) ( x)
n is a positive integer
1
 x 1
x
f ( x)   x  2
f ( x) 
f ( x)  2 x 3
f
( 4)
( x)  6 x  4
f
(5)
( x)  24 x 5
f
(n)
( x)  (1)( 2)( 3)( 4)      ( n) x  n
Question 18
f ( x)  e x
Compute
f ( x), f ( x), f ( x), f ( 4) ( x), f (5) ( x)
f ( n ) ( x)
n is a positive integer
f ( x)  e x
f ( x)  e x
f ( x)  e x
f ( 4) ( x)  e x
f (5) ( x)  e x
f ( n ) ( x)  e x
Question 19
f ( x)  sin x
Compute
f ( x), f ( x), f ( x), f ( 4) ( x), f (5) ( x)
f ( n ) ( x)
n is a positive integer
f ( x)  cos x
f ( x)   sin x
f ( x)   cos x
f ( 4 ) ( x)  sin x
f ( 5) ( x)  cos x
f ( n ) ( x)   sin x      neven
f ( n ) ( x)   cos x      nodd
Question 20
The percent of concentration of a certain drug in the bloodstream x hours after
the drug is administrated is given by:
K ( x) 
3x
x2  4
1. Find the time at which concentration is maximum
3( x 2  4)  2 x(3x)
K ( x) 
( x 2  4) 2
3x 2  12  6 x 2
K ( x) 
( x 2  4) 2
 3x 2  12
K ( x)  2
0
2
( x  4)
 3x 2  12  0
x2
2. Find the maximum concentration
K (2) 
3(2)
 0.75
2
(2)  4
Question 21
When an object is dropped straight down, the distance (in feet) that travels in t
seconds is given by
s(t )  16t 2
1. Find the velocity after 3 seconds , 5 seconds , 8 seconds
V (t )  s (t )  32t
V (3)  32(3)  96 feet / s
V (5)  32(5)  160 feet / s
V (8)  32(8)  256 feet / s
2. Find the acceleration
a(t )  s(t )  32 feet / s 2
Question 22
A ball is fired straight up; its position equation is given by:
s(t )  16t 2  140t  37
1. Find the maximum height of the ball
v(t )  32t  140  0
t  4.375s
s(4.375)  16(4.375) 2  140(4.375)  37  343.25
2. Find the time and the velocity when the ball hits the ground
s(t )  16t 2  140t  37  0
t  9s
v(9)  32(9)  140  148
Question 23
A car rolls down a hill; its distance from its starting point is given by
s(t )  1.5t 2  4t
1. How far will the car move in 10 seconds
s(10)  1.5(10) 2  4(10)  190
2. What is the velocity at 5 seconds , at 10 seconds
v(t )  3.5t  4
v(5)  3.5(5)  4  21.5
v(10)  3.5(10)  4  39
3. How can you tell that the car will not stop
The velocity is not becoming zero
4. What is the acceleration at 5 seconds , at 10 seconds
a (t )  3.5
a (5)  3.5
a (10)  3.5
5. What is happening o the velocity and acceleration as t increases
Velocity is increasing
Question 24
The total profit (in thousands of dollars) from the sale of x units of a certain
prescription drug is given by
P( x)  ln(  x 3  3x 2  72 x  1)
0  x  10
1. Find the number of units that should be sold in order to maximize the total
profit
 3 x 2  6 x  72
P ( x) 
0
3
2
 x  3 x  72 x  1
 3 x 2  6 x  72  0
x6
2. What is the maximum profit
P(6)  ln( (6) 3  3(6) 2  72(6)  1)  5.78
Question 25
In the summer the activity level of a certain type of lizard varies according
to the time of day
A biologist has determined that the activity level is given by the function
a(t )  0.008t 3  0.288t 2  2.304t  7
Where t is the number of hours after 12 noon
When is the activity level highest, when it is lowest?
a (t )  0.008(3t 2 )  0.288(2t )  2.304  0
0.024t 2  0.576t  2.304  0
t1  19
t2  5
5
19
Activity level highest at t = 5
Activity level lowest at t = 19
Question 26
The percent of concentration of a certain drug in the bloodstream x hours after
the drug is administrated is given by:
4x
K ( x)  2
3x  27
1. On what time intervals is the concentration of the drug increasing
2. On what intervals it is decreasing
4(3x 2  27)  6 x(4 x)
K ( x) 
(3x 2  27) 2
12 x 2  108  24 x 2
K ( x) 
0
2
2
(3x  27)
 12 x 2  108  0
x  3
-3
K (x) increase on: (- 3, 3)
K(x) decrease on (- ,3)U (3, )
Question 27
3
Find
1. The critical points
A (- 2 3 , - 64), B (0, 80), C (2
3 , -64)
2. The intervals where the function is increasing or decreasing
F(x) decrease on: (  , -2 3 ) U (0, 2 3 , -64)
F (x) increase on: (- 2 3 0) U (2 3 ,  )
3. The values of the relative extrema
Relative minimum of -64 at x = - 2 3
Relative minimum of -64 at x = 2 3
Relative maximum of 80 at x = 0
4. The intervals where the function is concave up or down
F(x) concave up on: (  , -2) U (2,  )
F (x) concave down on: (- 2, 2)
Question 28
Find all the relative extrema as well as where the function is increasing or
decreasing, critical points
2 x2
f ( x)  xe
f ( x)  (1)(e 2 x )  (2 xe2 x )( x)
2
f ( x)  e
2 x 2
2
(1  2 x 2 )  0
1  2x2  0
x  0.7
A(0.7,3.2)
B(0.7,3.2)
-0.7
0.7
f(x) increase on: (- 0.7, 0.7)
f(x) decrease on (- ,0.7)U (0.7, )
relative minimum of -3.2 at x = - 0.7
relative maximum of 3.2 at x = 0.7
Question 29
A small company manufactures and sells bicycles
The production manager has determined that the cost and demand
functions for q ( q  0 ) bicycles per week are
1 3
C (q)  10  5q  q
60
p  D(q)  90  q
1. Find the maximum weekly revenue
R  qp  q (90  q)  90q  q 2
R  90  2q  0
q  45
R (45)  90(45)  (45) 2  2025
2. Find the maximum weekly profit
P  R  C  (90q  q 2 )  (10  5q 
P  R  C   (90  2q)  (5 
1 3
q)
60
1 2
q)
20
1 2
q  2q  85  0
20
q  26
P 
P(26)  645.2
3. Find the price the company should charge to realize maximum profit
P(26 ) = 90-26=64
Question 30 (Exam 2010-2011)
Let
f ( x) 
x4
 x3  1
4
1. State the domain of the function and find the yintercept.
Domain : all real numbers ................(1mark)
Y-intercept is 1............(1mark)
2. Find the critical points of the function.
(4 marks)
f ( x)  x 3  3 x 2 ......................................(1 mark )
critical points  f ( x)  0....................(0.5 marks )
x3  3x 2  0

x 2 ( x  3)  0............(0.5 marks )
 x  0 or x  3........(1 mark )
the points are (0,1) and (3, 5.75)............(1 mark )
3. Find the open intervals where the function is increasing or decreasing.
(2 marks)
0
3
The critical points determine three intervals: (-∞, 0), (0, 3) and (3, +∞)
f(x) is increasing on (3,+∞)………………………………………………………..(1 mark)
f(x) is decreasing on (-∞,3)……………………….(1 mark)
4. Determine the locations and values of all relative extrema.
Relative minimum of -5.75 at x=3.
(2 marks)
(2 marks)
5. Find the regions where the function is concave upward or concave
downward.
0
(3 marks)
f ( x)  3x 2  6 x......................................(0.5 marks )
inflection points  f ( x)  0....................(0.5 marks )
3x 2  6 x  0
 3x( x  2)  0............(0.5 marks)
 x  0 or x  2........(0.5 marks)
f is concave upwards over the intervals (-, 0) and (2, )......(0.5marks)
f is concave downwards over the interval (0, 2).....(0.5 marks )
6. Find the coordinates of the inflection points.
(1 mark)
Points of inflection are: (0, 1) and (2, -3) ………..(1 mark)
7. Summarize the information above in a table.
x
-∞
0
f'(x)
–
f”(x)
+
f(x)
up
(2 marks)
2
–
3
-
down
+∞
+
+
up
+
up
Question 31
A. Writing in Math
1. Explain in your own words when a point is
considered an inflection point for a given function f.
A function f admits an inflection point at a certain
x-value c if the second derivative vanishes at this
point or doesn’t exist, the function defined at the
this value, and the sign of the second derivative
changes around that point.
2. What is the difference between a relative and an
absolute extremism?
An absolute extremum occur over the whole domain
of the function while the relative extremum occurs
over a particular interval only