HighMarks

Calculus: rate of change — KCSE Mathematics

KCSE Mathematics · 113 practice questions · 3 syllabus objectives

38 easy38 medium37 hard

What You'll Learn

Key learning outcomes for this topic, aligned to the KNEC KCSE syllabus.

Define the gradient of a curve at a point as the limit of the gradient of a chord as its length tends to zero

Interpret the derivative as a rate of change; identify when a function is increasing (f'(x) > 0) or decreasing (f'(x) < 0)

Calculus: rate of change

Sample Questions

Try 3 questions free. Sign up to access all 113 questions with full marking schemes.

1
easySHORT ANSWER2 marks

State when the function g(t) = -2t^2 + 8t - 3 is decreasing. (2 marks)

View Marking Scheme
Part (a) — 2 marks
The derivative g'(t) must be less than 0 (1 mk)
Identify the interval where g'(t) < 0 (1 mk)
2
easySHORT ANSWER3 marks

State the conditions under which the function f(x) = 3x^2 - 12x + 5 is increasing. (3 marks)

View Marking Scheme
Part (a) — 3 marks
The derivative f'(x) must be greater than 0 (1 mk)
Identify the interval where f'(x) > 0 (1 mk)
For this function, x > 2 is the interval where f'(x) > 0 (1 mk)
3
easySHORT ANSWER4 marks

Consider the function g(t) = t^3 - 3t^2 + 2. (a) State the definition of the gradient of a curve at a point. (1 mark) (b) Calculate the gradient of the curve at t = 2. (2 marks) (c) Discuss the behaviour of the gradient as t approaches 2 from the left. (1 mark)

View Marking Scheme
Part (a) — 1 mark
The gradient at a point is the limit of the gradient of the chord as the length of the chord tends to zero (1 mk)
Part (b) — 2 marks
Differentiate g(t) to get g'(t) = 3t^2 - 6t (1 mk)
Substitute t = 2 into g'(t) to get g'(2) = 3(2^2) - 6(2) = 0 (1 mk)
Part (c) — 1 mark
As t approaches 2 from the left, the gradient g'(t) decreases and approaches 0 (1 mk)
4

Given the function f(x) = 3x^2 + 2x, (a) define the gradient of the curve at a point. (1 mark) (b) Find the gradient of the curve at the point where x = 1. (2 marks) (c) Explain how the gradient changes as x increases from 1. (1 mark)

+110 More Questions

Sign up free to access all 113 questions with marking schemes, track your progress, and get personalised recommendations.

Why Practise Calculus: rate of change?

KNEC Aligned

Questions match the KCSE syllabus objectives and exam format exactly.

Detailed Marking Schemes

Every answer shows exactly what examiners award marks for.

Track Your Mastery

See your score improve as you practise and identify remaining gaps.

Master Calculus: rate of change for KCSE

Sign up free to unlock all 113 questions, track your progress, and get a personalised study plan for Mathematics.