Higher Maths · Worksheet 12

Optimisation and Area Between Curves Worksheet | Higher Maths

Skills covered: Maximum and minimum values; optimisation in geometric and practical contexts; definite integrals for area; signed regions and areas between curves.

18 questions83 marks3 graded sectionsFree PDF

Download PDF Differentiation revision notes

Instructions: Attempt all questions. Show sufficient working to justify your answers. Give exact values unless a degree of accuracy is stated. Each section is separate, so you can target the part you need.

These are the same worksheets our pupils use at the Pollokshields tuition centre. Start with Section A if optimisation and area between curves still feels shaky; if you are confident, go straight to Section C and time yourself.

Section A

Prerequisite core skills

Revise earlier learning

Question 1

[4 marks]

Find the maximum value of

f(x) = -x^2 + 6x + 2

, and the value of

x

at which it occurs.

Question 2

[3 marks]

Evaluate

\displaystyle\int_0^3 (4x - x^2)\,dx

.

Question 3

[5 marks]

Find the total area bounded by

y = x^2 - 4x

and the

x

-axis.

Question 4

[2 marks]

Find the

x

-coordinates of the intersections of

y = 4x

and

y = x^2

.

Question 5

[4 marks]

Find the stationary point of

y = 3x^2 - 12x + 5

and state its nature.

Question 6

[4 marks]

A rectangle has perimeter

20

cm. Its sides are

x

cm and

y

cm. Show that its area is

A(x) = 10x - x^2

, and state the permitted values of

x

.

Section B

Topic practice

Build individual techniques

Question 7

[4 marks]

For the rectangle in question 6, find the maximum possible area and the dimensions of the rectangle.

Question 8

[6 marks]

An open box is made from a

20

cm by

12

cm card by cutting equal squares of side

x

cm from the corners and folding the sides.

(a)Show that $V = x(20 - 2x)(12 - 2x)$ .
(b)Find the value of $x$ giving maximum volume, to two decimal places.

Question 9

[4 marks]

The height of a ball,

h

metres, after

t

seconds is given by

h(t) = 12t - t^2

,

0 \le t \le 12

. Find the greatest height reached and the time taken to reach it.

Question 10

[5 marks]

Fencing of length

30

m is used to enclose three sides of a rectangular garden against a straight wall. Find the dimensions that give the greatest area.

Question 11

[4 marks]

Calculate the area enclosed by

y = 6x - x^2

and the

x

-axis.

Question 12

[5 marks]

Find the exact area enclosed by

y = 6

and

y = x^2

.

Question 13

[5 marks]

Find the area enclosed between the line

y = 2x + 3

and the parabola

y = x^2

.

Question 14

[5 marks]

Find the total area enclosed between

y = x^3 - 4x

and the

x

-axis, for

-2 \le x \le 2

.

Section C

Exam-style and challenge

Combine skills, exam-style

Question 15

[6 marks]

A closed cylindrical container has a fixed volume of

250\pi\ \text{cm}^3

. If its radius is

r

cm and height is

h

cm, show that its surface area is

S = 2\pi r^2 + \dfrac{500\pi}{r}

. Find the dimensions giving minimum surface area.

Question 16

[5 marks]

Find the exact area enclosed between

y = 4 - x^2

and

y = x + 2

.

Question 17

[6 marks]

A rectangle is placed symmetrically beneath the curve

y = 12 - x^2

, with one side on the

x

-axis and its two upper vertices on the curve. Find its maximum possible area.

Question 18

[6 marks]

Find the total area enclosed between the cubic

y = x^3 - 4x

and the line

y = 2x

.

Looking for the answers?

We do not publish the worked answers online. If you get stuck, bring the worksheet to a session and one of our Higher Maths tutors will go through the method with you, line by line, the way the SQA marking instructions award it.

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Frequently asked questions

Are the answers to these worksheets available online?

No. We keep the worked answers off the website on purpose. Pupils learn far more by attempting every question, checking against their notes and then going over the method with a tutor. Our students work through the full worked solutions in class, and if you are stuck on a particular question you can book a session and we will go through it with you step by step.

Are these official SQA or Qualifications Scotland papers?

No. Every question is newly written by Excel in MathSci to match the Higher Maths course specification. They are original practice resources, not official Qualifications Scotland papers, so use them alongside the real past papers rather than instead of them.

How should I use the three sections?

Section A checks the prerequisite skills you need before the topic makes sense, so start there if the topic feels shaky. Section B builds each technique on its own. Section C is exam-style and combines several skills, so treat it as a timed test once Sections A and B feel comfortable.

Can I print the worksheet?

Yes. Download the free PDF, which has space to write your working under every question, or print this page directly. The worksheet is free for personal study; please ask before reproducing it for a class or a commercial service.

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