Exam Questions & Step-by-Step Markschemes (5 Problems)
Question 1 • 2024
5 Marks
A manufacturing company finds that its total cost function for producing x units of a product is given by: C(x) = \frac{1}{3}x^3 - 5x^2 + 28x + 50 (i) Find the Average Cost (AC) and Marginal Cost (MC) functions. (ii) Find the output level x at which the Average Cost is minimized. (iii) Verify that at this output level, Marginal Cost equals Average Cost ( MC = AC ).
Answer: (i) \; AC = \frac{1}{3}x^2 - 5x + 28 + \frac{50}{x}, \; MC = x^2 - 10x + 28 \quad (ii) \; \text{Min at } MC = AC
Question 2 • 2024
4 Marks
Evaluate the limit: \lim_{x \to 0} \left( \frac{1}{x} - \frac{1}{\sin x} \right)
Answer: \lim_{x \to 0} \left( \frac{1}{x} - \frac{1}{\sin x} \right) = 0
Question 3 • 2023
4 Marks
Verify Rolle's Theorem for the function f(x) = x^3 - 4x on the interval [-2, 2] .
Answer: c = \pm \frac{2}{\sqrt{3}} \in (-2, 2) \quad (\text{Rolle's Theorem Verified})
Question 4 • 2024
5 Marks
Find the absolute maximum and minimum values of f(x) = 2\cos 2x - \cos 4x in the interval [0, \pi] .
Answer: \text{Absolute Maximum} = \frac{3}{2}, \quad \text{Absolute Minimum} = -3
Question 5 • 2024
4 Marks
Evaluate: \lim_{x \to 0} \left( \frac{1}{x} - \frac{1}{\sin x} \right)
Answer: 0