Higher Ink Exercise
APPS – Differentiation/Integration
Calculators should only be used when necessary
1.) A window in the shape of a rectangle surmounted by a semicircle is being designed to
let in the maximum amount of light.
The glass to be used for the semi-circular part is stained glass which
lets in one unit of light per square metre; the rectangular part uses
clear glass which lets in 2 units of light per square metre.
The rectangle measures 2x metres by h metres.
(a) (i) If the perimeter of the whole window is 10 metres, express h in
terms of x.
(ii) Hence show that the amount of light, L, let in by the window is
given by
3
𝐿 = 20𝑥 − 4𝑥 2 − 2 𝜋𝑥 2 .
(b) Find the values of x and h that must be used to allow this design to
let in the maximum amount of light.
2.) Find the particular solution which satisfies this differential equation
𝑑𝑦
𝑑𝑥
2
= 4𝑥 3 + 𝑥 2 ,
y=3& x=1
3.) A point moves in a straight line such that its acceleration a is given by
a = 5(3 – t)3/2, 0 ≤ 𝑡 ≤ 4.
If it starts at rest, find an expression for the velocity v where 𝑎 =
𝑑𝑣
.
𝑑𝑡
4.) The diagram shows a sketch of the graph of y = x3 – 4x2 + x + 6.
a) Show that the graph cuts the x-axis at (3,0).
b) Hence or otherwise find the coordinates of A.
c) Find the shaded area.
5.) Acceleration is defined as the rate of change of velocity.
An object is traveling in a straight line.
The velocity, v m/s, of this object, t seconds after the start of the motion, is given by:
𝜋
v(t) = 8cos(2t - 2 )
a) Find a formula for a(t), the acceleration of this object, t seconds after the start of the
motion.
b) Determine whether the velocity is increasing or decreasing when t = 10.
c) Velocity is defined as the rate of change of displacement. Determine a formula for s(t), the
displacement of the object, given that s(t) = 4 when t = 0.
6a)
i.
ii.
Show that (x – 2) is a factor of x3 – 6x2 + 3x + 10
Hence factorise x3 – 6x2 + 3x + 10
C
B
A
0
x
The line with equation y = 3x + 10 intersects the curve with equation y = 6x2 – x3 at the points
A, B and C.
b)
Find the x-coordinates of the points A, B and C.
c)
Calculate the total shaded area shown in the diagram.