Mathematics IM
Worked Examples
CALCULUS: THE DERIVATIVE
Produced by the Maths Learning Centre,
The University of Adelaide.
May 4, 2013
The questions on this page have worked solutions and links to videos on the following
pages. Click on the link with each question to go straight to the relevant page. You will
need to have the question handy to refer to while watching the videos.
Questions
1. See Page 4 for worked solutions.
Evaluate the following limits (if possible):
(b)
(a)
lim (x3 + 2) sin(π/2 − x)
x→0
√
√
x− a
(d)
(c)
lim
x→a
x−a
1
2
lim
− 2
x→0 x
x
2
x − 3x − 4
lim
x→ 2
x+2
2. See Page 6 for worked solutions.
(a) Explain why you can’t use the Limit Laws to evaluate lim
x→π/2
1
− tan x .
cos x
(b) Find the limit in (a) by bringing the expression to a common denominator and
multiplying top and bottom by (1 + sin x).
3. See Page 8 for worked solutions.
Find, if possible,
(a)
3x2 − 4x
x→∞ x3 − 2
lim
(b)
3x3 − 4x
x→∞ x3 − 2
lim
(c)
3x4 − 4x
x→∞ x3 − 2
lim
(d)
4. See Page 9 for worked solutions.
(
Determine if lim g(x) and g 0 (2) exist where g(x) =
x→2
2x + 3 if x > 2
2x2 − 1 if x ≤ 2
5. See Page 11 for worked solutions.
Find the derivative of the function y = 2x3 +
12
when x = 2.
x2
6. See Page 12
√ for worked solutions.
If f (x) = x2 + 1, find f 0 (x) and f 00 (x).
7. See Page 13 for worked solutions.
√
Differentiate (a) (x3 + 4x) 2x + 4
(b)
x4 − 2x
x−3
(c)
x2 + 2x
√
x x
Simplify your answer in each case.
8. See Page 15 for worked solutions.
p
Differentiate: (i) sin (3x2 )
(ii)
cos (2x)
(iii) x3 cos (1/x) .
What are the domains of the functions in (ii) and (iii)?
1
lim sin
x→0
1
x
9. See Page 17 for worked solutions.
dy
dy
Let y = cos 2x − 2 sin x. Find
and hence find all solutions of the equation
=0
dx
dx
with 0 ≤ x ≤ 2π.
2
10. See Page 18 for worked solutions. p
The mass of a special wire is x(1 + (x)) kg, where x is measured in metres from one
end of the wire. Note that the mass is not a linear function of x. Find the rate of
change of the mass with respect to the distance from the end of the wire when x = 1
and when x = 4.
11. See Page 19 for worked solutions.
Find the equation of the tangent line to the curve y = e2x cos πx at the point where
x = 0.
12. See Page 20 for worked solutions.
Which of the following functions are 1-1?
(a)
f (x) = x3 + 2
(b)
g(k) = k 2 − 7k + 10
(c)
h(x) =
x−1
x+2
13. See Page 22 for worked solutions.
Find the inverse of the function g(x) = x3 + 2, and plot both g and g −1 on the same
axes.
14. See Page 23 for worked solutions.
Sketch the graphs of y = 2x and its inverse function y = log2 x. Find the equation of
the tangent to the curve y = log2 x at the point where x = 1.
15. See Page 24 for worked solutions.
The magnitude of an earthquake is measured on the logarithmic scale known as the
Richter scale. If an earthquake has intensity I, its magnitude M (I) is defined as
M (I) = log10 ( II0 ), where I0 is a minimum level of intensity used for comparisons.
(a) The famous San Francisco earthquake of 1906 measured 8.25 on the Richter scale
and the 1989 earthquake in Newcastle, Australia, measured 5.50 on the Richter
scale. How many times more intense was the San Francisco earthquake than the
one in Newcastle?
(b) The largest earthquake ever measured occured in Chile in 1960, and was about
18 times more intense than the San Francisco earthquake of 1906. What did the
earthquake in Chile measure on the Richter scale?
3
1. Click here to go to question list.
Evaluate the following limits (if possible):
1
2
(b)
lim
(a)
lim (x3 + 2) sin(π/2 − x)
− 2
x→0 x
x→0
x
√
√
x− a
x2 − 3x − 4
(c)
lim
(d)
lim
x→a
x→ 2
x−a
x+2
Click here to see video of this example on YouTube.
4
5
2. Click here to go to question list.
(a) Explain why you can’t use the Limit Laws to evaluate lim
x→π/2
1
− tan x .
cos x
(b) Find the limit in (a) by bringing the expression to a common denominator and
multiplying top and bottom by (1 + sin x).
Click here to see video of this example on YouTube.
6
7
3. Click here to go to question list.
Find, if possible,
3x2 − 4x
3x3 − 4x
3x4 − 4x
(b)
lim
(c)
lim
x→∞ x3 − 2
x→∞ x3 − 2
x→∞ x3 − 2
Click here to see video of this example on YouTube.
(a)
lim
8
(d)
lim sin
x→0
1
x
4. Click here to go to question list.
(
Determine if lim g(x) and g 0 (2) exist where g(x) =
x→2
Click here to see video of this example on YouTube.
9
2x + 3 if x > 2
2x2 − 1 if x ≤ 2
10
5. Click here to go to question list.
12
when x = 2.
x2
Click here to see video of this example on YouTube.
Find the derivative of the function y = 2x3 +
11
6. Click here√to go to question list.
If f (x) = x2 + 1, find f 0 (x) and f 00 (x).
Click here to see video of this example on YouTube.
12
7. Click here to go to question list.
√
Differentiate (a) (x3 + 4x) 2x + 4
(b)
x4 − 2x
x−3
Simplify your answer in each case.
Click here to see video of this example on YouTube.
13
(c)
x2 + 2x
√
x x
14
8. Click here to go to question list.
p
Differentiate: (i) sin (3x2 )
(ii)
cos (2x)
(iii) x3 cos (1/x) .
What are the domains of the functions in (ii) and (iii)?
Click here to see video of this example on YouTube.
15
16
9. Click here to go to question list.
dy
dy
Let y = cos 2x − 2 sin x. Find
and hence find all solutions of the equation
=0
dx
dx
with 0 ≤ x ≤ 2π.
Click here to see video of this example on YouTube.
17
10. Click here to go to question list. p
The mass of a special wire is x(1 + (x)) kg, where x is measured in metres from one
end of the wire. Note that the mass is not a linear function of x. Find the rate of
change of the mass with respect to the distance from the end of the wire when x = 1
and when x = 4.
Click here to see video of this example on YouTube.
18
11. Click here to go to question list.
Find the equation of the tangent line to the curve y = e2x cos πx at the point where
x = 0.
Click here to see video of this example on YouTube.
19
12. Click here to go to question list.
Which of the following functions are 1-1?
(a)
f (x) = x3 + 2
(b)
g(k) = k 2 − 7k + 10
Click here to see video of this example on YouTube.
20
(c)
h(x) =
x−1
x+2
21
13. Click here to go to question list.
Find the inverse of the function g(x) = x3 + 2, and plot both g and g −1 on the same
axes.
Click here to see video of this example on YouTube.
22
14. Click here to go to question list.
Sketch the graphs of y = 2x and its inverse function y = log2 x. Find the equation of
the tangent to the curve y = log2 x at the point where x = 1.
Click here to see video of this example on YouTube.
23
15. Click here to go to question list.
The magnitude of an earthquake is measured on the logarithmic scale known as the
Richter scale. If an earthquake has intensity I, its magnitude M (I) is defined as
M (I) = log10 ( II0 ), where I0 is a minimum level of intensity used for comparisons.
(a) The famous San Francisco earthquake of 1906 measured 8.25 on the Richter scale
and the 1989 earthquake in Newcastle, Australia, measured 5.50 on the Richter
scale. How many times more intense was the San Francisco earthquake than the
one in Newcastle?
(b) The largest earthquake ever measured occured in Chile in 1960, and was about
18 times more intense than the San Francisco earthquake of 1906. What did the
earthquake in Chile measure on the Richter scale?
Click here to see video of this example on YouTube.
24
25