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Calculus: applications of differentiation (gradients, maxima and minima) — KCSE Mathematics

KCSE Mathematics · 110 practice questions · 4 syllabus objectives

24 easy31 medium55 hard

What You'll Learn

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

Find stationary points of a polynomial function by solving f'(x) = 0; classify them as maxima or minima using the second derivative test

Apply differentiation to solve optimisation problems (maximum area, minimum cost) in practical contexts

Apply integration to find the area under a curve between two limits; evaluate definite integrals of simple polynomials

Calculus: applications of differentiation (gradients, maxima and minima)

Sample Questions

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1
easySHORT ANSWER3 marks

State the method to find the point where the curve y = x^3 - 6x^2 + 9x has a local maximum. (3 marks)

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Part (a) — 3 marks
Differentiate the function to find y' = 3x^2 - 12x + 9 (1 mk)
Set y' = 0 to find critical points (1 mk)
Use the second derivative test to confirm local maximum (1 mk)
2
easySHORT ANSWER4 marks

Define the steps to evaluate the definite integral of f(x) = 3x^3 - 2 from x = 0 to x = 2. (4 marks)

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Part (a) — 4 marks
Integrate f(x) to find F(x) = (3/4)x^4 - 2x + C (1 mk)
Substitute the upper limit x = 2 into F(x) (1 mk)
Substitute the lower limit x = 0 into F(x) (1 mk)
Calculate the definite integral as F(2) - F(0) (1 mk)
3
easySHORT ANSWER4 marks

Explain how to find the minimum value of the function f(x) = x^2 - 4x + 5. (4 marks)

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Part (a) — 4 marks
Differentiate f(x) to find f'(x) = 2x - 4 (1 mk)
Set f'(x) = 0 and solve for x (1 mk)
Determine the second derivative f''(x) = 2 (1 mk)
Conclude minimum value occurs at that x (1 mk)
4

State how to determine the maximum area of a rectangle inscribed under the curve y = 12 - x^2. (3 marks)

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