Essential Mathematical Methods 3 & 4 CAS
Chapter 11 Differentiation of transcendental functions: SAC 4 Revision
Multiple-choice questions
π
3
1 If y = |cos (x)|, the rate of change of y with respect to x at x = k, < k <
, is:
2
2
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= x+
e
e
E y = ex
3 The minimum value of e –x + 2ex occurs where x =
A 0
B 2 2
1
C –2 loge 2
D 1
E 2.82
1
Essential Mathematical Methods 3 & 4 CAS
Chapter 11 Differentiation of transcendental functions: SAC 4 Revision
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
2
Essential Mathematical Methods 3 & 4 CAS
Chapter 11 Differentiation of transcendental functions: SAC 4 Revision
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
1
E y = 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
3
Essential Mathematical Methods 3 & 4 CAS
Chapter 11 Differentiation of transcendental functions: SAC 4 Revision
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
4
Essential Mathematical Methods 3 & 4 CAS
Chapter 11 Differentiation of transcendental functions: SAC 4 Revision
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 the gradient of the normal to the curve with equation y = 2 cos 2x at the point
where x =

6
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
5
Essential Mathematical Methods 3 & 4 CAS
Chapter 11 Differentiation of transcendental functions: SAC 4 Revision
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 floor can
y
be modelled by the function f: [0, 3]  R
where f(x) = 3 – 3 cos
( 2x3 )
and where
y = f(x)
x cm is measured from a point O on the
floor in a straight line and f(x) cm is the
height of the fault at the point x.
0
3
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.
6
Essential Mathematical Methods 3 & 4 CAS
Chapter 11 Differentiation of transcendental functions: SAC 4 Revision
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 .
7
Essential Mathematical Methods 3 & 4 CAS
Chapter 11 Differentiation of transcendental functions: SAC 4 Revision
Answers to Chapter 11 Test A
Answers to multiple-choice questions
1 B
2 E
3 C
4 C
5 B
6 C
7 B
8 A
9 A
10 B
11 E
12 B
13 D
14 B
15 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
2 3
3 y = –2x
1
4 a 2
π
b 4
2
5 a 2 loge(3x ) – 2
2
b 2 loge(3 ) – 2
8
Essential Mathematical Methods 3 & 4 CAS
Chapter 11 Differentiation of transcendental functions: SAC 4 Revision
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
3 a
( 3 +1)π
12
b t = 1, 5 and 9 months
4
y
y = w(t)
1
5
9
t
y=
dw
dt
9