APPLICATIONS ON DIFFERENTIATIONS
Multiple-choice questions
π
3
1 If y = |cos (x)|, the rate of change of y with respect to x at x = k, 2 < k < 2 , is:
A sin (k)
B sin (k)
C cos (k)
D cos (k)
E k sin (1)
2 The equation of the tangent to the curve with equation y = ex, at the point (–1, e), is given by:
x
A y = e + 1
x 1
B y = e + e + e
C y = 1  ex
1
2
D y = e x + e
E y = ex
3 The minimum value of e –x + 2ex occurs where x =
A 0
B 2 2
1
C – loge 2
2
D 1
E 2.82
x
4 For f: R R, f(x) = ex  e , the coordinates of the turning point of the graph of y = f(x) are:
A (1, 0)

B (1, )
e
2
C (1, )
e
1
D (1, e )
E (1, 0)
5 The equation of the normal of the graph with equation y = e 0.25x at the point (0, 1) is:
A y = 4x  1
B y = 4x + 1

C y= x
4

D y = 4 x + 1

E y=4x+1
6 Using the approximation f(x + h)  f(x) + h f (x) where f(x) = ex, with x = 0, the approximate value
of e0.025 is found to be:
A 0.025
B 0.975
C 1.025
D 1.0253
E 1.025315
7 The line with equation y = 3x + 5 and the curve with equation y = 2e–x + 3 intersect at the
point (0, 5). At this point, the acute angle between the line and the curve is  where tan  is
equal to:
1
A 2
B 1
C 2
D 3
E 5
8 If f(x) = loge 4x, then f (3) is equal to:
1
A 3
1
B 12
C loge 12
D 1.079
E 4 loge 4
9 The equation of the tangent to y = loge 3x at the point where x = 1 is:
A y = x  1 + loge 3
B y = x + loge 3
3
C y= x
1
D y= x
E y=
1
3x
10 The equation of the tangent to f(x) = 3e2x + 1 at the point where x = 0.5 is:
A y = 6e2x
B y = 6ex + 1
C y = 6xe  3e
D y = x6e  3e + 1
E y = 6ex  3e + 1
11 If f(x) = 2e(3x² + 1) then:
A f (1) = 2e4
B f (1) = 2e4 + 6
C f (1) = 6e4
D f (1) = 6e4 + 6
E f (1) = 12e4
12 Let y = 2xex where x  R. The minimum value of y is:
A 2
2
B (e )
2
C e
D 2
E 2e
13 If z = loge (x) then z is approximately equal to:
A loge (x + x)
B loge (x)
C
1
x
x
D x
1
E x
14 For y = sin x + cos x the maximum value of y is:
A 1
B
2
C 2
D 2 2
E 3
15 The equation of the tangent to the curve with equation y = ex 1 at the point where the curve
crosses the y-axis is:
A y=x
B y = x
1
C y=2x
1
D y = 2 x
E y = 2x
Short-answer questions (technology-free)
1 Find the derivative of each of the following with respect to x:
a 2 sin (x2)
b x3 sin x
c
sin x
ex
d
log e x
3x
e 5e3x cos (2x)
2 Find the gradient of the normal to the curve with equation y = 2 cos 2x at the point where x =
3 Find the equation of the tangent to the curve of y = 2e–x – 2 at the origin.
4 The volume V (litres) of water in a tank at time t (hours) is given by
V(t) = 3 sin
( 12t )
+
1
2
a Find the volume of water in the tank at time t = 24
b Find the rate of change of volume of the water in the tank when t = 24

6
5 If f(x) = 2x loge
( 3x2 )
:
a find f (x)
b find f (1)
c find f (e)
6 The cross-section of a fault in a
y
floor can be modelled by the
function f: [0, 3]  R where f(x)
= 3 – 3 cos
( 2x3 )
y=
f(x)
and where
x cm is measured from a point O
on the floor in a straight line and
0
3
x
f(x) cm is the height of the fault
at the point x.
a How far from O does the fault reach a height of:
i
3 cm?
ii
1.5 cm?
iii
4.5 cm?
b What is the gradient of y = f(x) at each of these points?
Extended-response questions
The volume of water, w megalitres, at time t months in a dam is modelled by the formula
t
t
w(t) = 4 0.5 sin ( 3 )  cos ( 6 ), t ≥ 0 where t is measured from the first of January.
1 Find the volume of water in the dam on 1 January.
2 State the period of w.
dw
3 a Find the exact rate of change of volume, dt , when t = 2
b Find the values of t for which the rate of change of volume is zero, t  [0,12].
4 The graph of y = w(t) is shown.
y
t
dw
On the same set of axes sketch the graph of y = dt .
Applications on differentiations
Question 1
Icicle
A cone-shaped icicle is dripping from the roof. The radius of the icicle is decreasing at a
rate of 0.2 cm per hour, while the height is increasing at a rate of 0.8 cm per hour. If the icicle
is currently 4 cm in radius and 20 cm long, is the volume of the icicle increasing or
decreasing, and at what rate?
(answer: volume is decreasing at a rate of 20 cm3 per hour)
Question 2
Sliding Ladder
A 15 m ladder is placed against a large building. The base of the ladder is resting on an oil
spill, and it slips to the right in (figure above) at the rate of 3 m per minute. Find the rate of
change of the height of the top of the ladder above the ground at the instant when the base
of the ladder is 3 m from the base of the building.
Question 3
Revenue
A company is increasing production of peanuts at the rate of 50 cases per day. All cases
produced can be sold. The daily demand function is given by
Question 4
Blood Flow
Blood flows faster the closer it is to the centre of a blood vessel. According to Poiseuille’s
laws, the velocity V of blood is given by
where R is the radius of the blood vessel, r is the distance of a layer of blood flow from the
center of the vessel, and k is a constant, assumed here to equal 375.
Suppose a skier’s blood vessel has radius R=0.08 mm and that cold weather is causing the
vessel to contract at a rate of dR/dt = - 0.01 linear unit per minute. How fast is the velocity of
blood changing?
Question 5
Shadow Length
A man 6 ft tall is walking away from a lamp post at the rate of 0.5 m per minute. When the
man is 2.5 m from the lamp post, his shadow is 3 m long. Find the rate at which the length of
the shadow is increasing when he is 7.5 m from the lamp post. (See the figure.)
1.75 m
Question 6
Water Level
A trough has a triangular cross section. The trough is 6 m across the top, 6 m deep, and 16 m
long. Water is being pumped into the trough at the rate of 2 m3 per minute.
Find the rate at which the height of the water is increasing at the instant that the height is 4 m.
Question 7
Kite Flying
Christine O’Brien is flying her kite in a wind that is blowing it east at a rate of 15 m/ minute.
She has already let out 200 m of string, and the kite is flying 100 m above her hand.
How fast must she let out string at this moment to keep the kite flying with the same speed
and altitude?
Question 8
 x
4
a) Sketch the graph of f : [ ,6 ]  R, f ( x)  2 cos 2    1
HINT: The equation for y=f(x) can be written in the form y =cos(ax) +b . Find the values of
a and b.
b) Find f   and f 3 
c) i. Find the gradient of the tangent to the tangent to the curve at any point x.
ii. Find exact coordinates of the points on the curve y = f(x) where the gradient of the
normal to the curve is equal to 2.
d) Find the exact equations of the tangent to the curve at the points where x  3 and
x  5 .
e) The graph of y = f(x) is transformed to give the graph of y = f(x) +b . Find the exact
value of b, such that the graph of the tangent at x  3 and the graph of the tangent
at x  5 intersect on the x-axis.
Question 9
Water is poured into a container of irregular shape to a depth of 1.20 metres in 2
minutes at 25 litres per minute. Let the volume of water in the container be V and the
depth of water be h at time t.
dV
a. Find
in cm3 per minute.
dt
h
b. Find
in metres per minute.
t
h
c. Find
in cm-2.
V
Answers to Chapter 11 Test A
Answers to multiple-choice questions
Q1
Q2
Q3
Q4
Q5
Q6
Q7
Q8
Q9
Q10 Q11 Q12 Q13 Q14 Q15
B
E
C
C
B
C
B
A
A
B
E
B
D
B
Answers to short-answer (technology-free) questions
1 a 4x cos (x2)
b x3 cos (x) + 3x2 sin (x)
c (cos (x) – sin (x))e–x
1 – loge (x)
3x2
d
e (15 cos (2x) – 10 sin (2x))e3x
2
1
y = –2x
3
2 3
4
a
1
2
b
π
4
2
5 a 2 loge(3x ) – 2
2
b 2 loge( ) – 2
3
2
c 2 loge(3 ) – 4
6 a i 0.75 cm and 2.25 cm
ii 0.5 cm and 2.5 cm
iii 1 cm and 2 cm
b i
ii
2π and –2π
3 π and – 3 π
iii π 3 and – π 3
Answers to extended-response questions
1 3 megalitres
2
12 months
4
y
y=
w(t)
t
1
5
9
y=
dw
dt
3 a
( 3 +1)π
b t = 1, 5 and 9 months
12
B