GCSE Maths · Higher · Chapter 23: Graphs · Lesson 2

Area under a curve and rates of change

Grades 5–8About 35 minutesCalculator allowed31 marks

Hints online, answers free with an account:
mathswariz.uk/worksheets/gcse-area-under-a-curve-and-rates-of-change/

NameClassDate

Do Now

Grades 4–6

The first question is what today's lesson needs. The others bring back earlier topics.

A1
Needed today
A parallelogram has base 12 cm and perpendicular height 8 cm. Find its area.
[1 mark]
Answer
A2
Last lesson
A distance-time graph rises 150 m in 5 s in a straight line. Find the speed.
[1 mark]
Answer
A3
From chapter 7
Translate $(2, 3)$ by $(22)$ .
[1 mark]
Answer
A4
From chapter 14
Simplify $(x^{4})^{4}$ .
[1 mark]
Answer

Grades 5–7

Your teacher works through each example with you. Then try the one beside it.

Example 1

Use 3 equal strips to estimate the area under $y = 2 x^{2} + 1$ from $x = 0$ to $x = 3$ . Is it an over- or underestimate?

Hint Find

y

x = 0, 1, 2, 3

; add three trapezia.

B1
Use 4 equal strips to estimate the area under $y = x^{2} + 4 x$ from $x = 0$ to $x = 2$ . Is it an over- or underestimate? Give a reason.
[4 marks]
Answer

Example 2

A tangent is drawn to a distance-time graph at $t = 4$ s. It passes through $(3, 8)$ and $(7, 40)$ . Estimate the speed at $t = 4$ s.

Hint The tangent's gradient is the rate of change there.

B2
A tangent is drawn to a velocity-time graph at $t = 6$ s. It passes through $(2, 4)$ and $(10, 22)$ . Estimate the acceleration at $t = 6$ s.
[2 marks]
Answer

Grades 5–7

C1
A curve has heights 4, 9, 16 and 25 at $x = 0, 1, 2, 3$ . Use 3 strips to estimate the area under it.
[2 marks]
Answer
C2
A tank holds $V = 40 + 3 t^{2}$ litres after $t$ minutes. Find the average rate of flow from $t = 1$ to $t = 4$ .
[2 marks]
Answer

C3
A curve bends downwards. Is a trapezium estimate of its area too big or too small? Why?
[2 marks]
C4
On a graph of depth $h$ cm after $t$ min, a tangent goes through $(0, 30)$ and $(12, 6)$ . Find the rate $h$ falls.
[2 marks]
Answer

Grades 6–8

D2
Ali uses 4 strips of equal width to estimate the area under $y = 16 - x^{2}$ between $x = 0$ and $x = 4$ . He gets 42. He says, "The true area is less than 42, because the curve slopes downwards."
Show that 4 strips give 42, and explain why Ali is wrong.
[4 marks]

Grade 8 and beyond

No route is given. The hints are at the end of the sheet. Use one at a time.

Colour one circle on each line.

	Not yet	Nearly	Yes
I can estimate the area under a curve with trapezium strips and say if it is too big or too small.
I can find a rate of change from a chord or from a tangent.

The worked answers and the full mark scheme are free with a Mathswariz account. Teachers and students can both sign up.