CHAPTER 5 REVIEW – SL
1.
Let f(x) = cos 2x and g(x) = ln(3x – 5).
(a) Find f′(x).
(2)
(b)
Find g′(x).
(c)
Let h(x) = f(x) × g(x). Find h′(x).
(2)
(2)
(Total 6 marks)
2.
Let f(x) = ex cos x. Find the gradient of the normal to the curve of f at x = π.
(Total 6 marks)
3.
π

Let f(x) = e–3x and g(x) = sin  x   .
3

(a) Write down
(i)
f′(x);
(ii) g′(x).
(2)
(b)
π

π
Let h(x) = e–3x sin  x   . Find the exact value of h′   .
3

3
(4)
(Total 6 marks)
4.
Differentiate each of the following with respect to x.
(a) y = sin 3x
(1)
(b)
y = x tan x
(c)
ln x
y=
x
(2)
(3)
(Total 6 marks)
5.
(a)
(b)
(c)
Let f (x) = e5x. Write down f ′ (x).
Let g (x) = sin 2x. Write down g′ (x).
Let h (x) = e5x sin 2x. Find h′ (x).
(Total 6 marks)
6.
Let h(x) =
6x
. Find h′(0).
cos x
(Total 6 marks)
7.
Let g(x) = 2x sin x.
(a) Find g′(x).
(4)
(b)
Find the gradient of the graph of g at x = π.
(3)
(Total 7 marks)
IB Questionbank Maths SL
1
8.
Let f(x) = x cos x, for 0 ≤ x ≤ 6.
(a) Find f′(x).
(3)
(b)
On the grid below, sketch the graph of y = f′(x).
(4)
(Total 7 marks)
9.
Let f(x) =
cos x
, for sin x ≠ 0.
sin x
(a)
Use the quotient rule to show that f′(x) =
(b)
Find f′′(x).
1
sin 2 x
.
(5)
(3)
π
π
In the following table, f′   = p and f′′   = q. The table also gives approximate values of
2
2
π
f′(x) and f′′(x) near x = .
2
π
 0.1
2
f′(x)
–1.01
f″(x)
0.203
Find the value of p and of q.
x
(c)
π
2
p
q
π
 0.1
2
–1.01
–0.203
(3)
(d)
Use information from the table to explain why there is a point of inflexion on the graph of
π
f where x = .
2
(2)
(Total 13 marks)
IB Questionbank Maths SL
2
10.
The following diagram shows a waterwheel with a bucket. The wheel rotates at a constant rate
in an anticlockwise (counterclockwise) direction.
diagram not to scale
The diameter of the wheel is 8 metres. The centre of the wheel, A, is 2 metres above the water
level. After t seconds, the height of the bucket above the water level is given by h = a sin bt + 2.
(a) Show that a = 4.
(2)
The wheel turns at a rate of one rotation every 30 seconds.
π
(b) Show that b =
.
15
(2)
In the first rotation, there are two values of t when the bucket is descending at a rate of
0.5 m s–1.
(c) Find these values of t.
(6)
(d)
Determine whether the bucket is underwater at the second value of t.
(4)
(Total 14 marks)
11.
The diagram shows part of the graph of the curve with equation
y = e2x cos x.
y
P(a, b)
0
(a)
Show that
x
dy
= e2x (2 cos x – sin x).
dx
(2)
2
(b)
Find
d y
.
dx 2
(4)
There is an inflexion point at P (a, b).
(c) Use the results from parts (a) and (b) to prove that:
(i)
tan a = 3 ;
4
(3)
(ii)
the gradient of the curve at P is e2a.
(5)
(Total 14 marks)
IB Questionbank Maths SL
3
12.
The diagram below shows a plan for a window in the shape of a trapezium.
Three sides of the window are 2 m long. The angle between the sloping sides of the window and
π
the base is θ, where 0 < θ < .
2
(a) Show that the area of the window is given by y = 4 sin θ + 2 sin 2θ.
(5)
2
(b)
Zoe wants a window to have an area of 5 m . Find the two possible values of θ.
(c)
John wants two windows which have the same area A but different values of θ.
Find all possible values for A.
(4)
(7)
(Total 16 marks)
IB Questionbank Maths SL
4