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Differential calculus: board exam questions with solutions

On the ECE exam under Mathematics and the REE exam under Mathematics. Try each question before you open its solution, then practise with fresh numbers below: 2 computed question types on this topic.

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Questions

1. Find d/dx (x³eˣ).

  1. eˣ(x³ + 3x²)
  2. 3x²eˣ
  3. x³eˣ
  4. eˣ(3x² − x³)
Show the answer and solution

Answer: A. eˣ(x³ + 3x²)

By the product rule, 3x²·eˣ + x³·eˣ = eˣ(x³ + 3x²).

2. Evaluate the limit of sin(3x)/x as x → 0.

  1. 0
  2. 1
  3. 3
  4. 1/3
Show the answer and solution

Answer: C. 3

sin(3x)/x = 3·sin(3x)/(3x), and sin u/u → 1, so the limit is 3.

3. If f(x) = ln(x² + 1), find f′(2).

  1. 0.25
  2. 0.2
  3. 4
  4. 0.8
Show the answer and solution

Answer: D. 0.8

f′(x) = 2x/(x² + 1), so f′(2) = 4/5 = 0.8.

4. Find the local maximum of f(x) = x³ − 6x² + 9x + 1.

  1. 1, at x = 3
  2. 5, at x = 3
  3. 9, at x = 0
  4. 5, at x = 1
Show the answer and solution

Answer: D. 5, at x = 1

f′(x) = 3x² − 12x + 9 = 3(x − 1)(x − 3) = 0 gives x = 1 or x = 3. f″(1) = −6 < 0, so x = 1 is a maximum, with f(1) = 1 − 6 + 9 + 1 = 5.

5. Evaluate the limit of (x² − 4)/(x − 2) as x → 2.

  1. 0
  2. 2
  3. 4
  4. Does not exist
Show the answer and solution

Answer: C. 4

Factor: (x − 2)(x + 2)/(x − 2) = x + 2 for x ≠ 2, so the limit is 2 + 2 = 4.

6. The radius of a circular ripple grows at 2 cm/s. How fast is its area increasing when r = 5 cm?

  1. 31.42 cm²/s
  2. 62.83 cm²/s
  3. 20.00 cm²/s
  4. 157.08 cm²/s
Show the answer and solution

Answer: B. 62.83 cm²/s

dA/dt = 2πr·dr/dt = 2π(5)(2) = 20π = 62.83 cm²/s.

7. A farmer has 200 m of fencing for a rectangular plot along a straight river. No fence is needed along the river. What is the maximum area he can enclose?

  1. 2500 m²
  2. 5000 m²
  3. 10 000 m²
  4. 4444 m²
Show the answer and solution

Answer: B. 5000 m²

Let the two sides perpendicular to the river be y and the side parallel to it be x = 200 − 2y. A = y(200 − 2y), and dA/dy = 200 − 4y = 0 gives y = 50 m and x = 100 m, so A = 5000 m².

8. What is the slope of the tangent to y = √x at x = 4?

  1. 0.25
  2. 0.5
  3. 2
  4. 4
Show the answer and solution

Answer: A. 0.25

dy/dx = 1/(2√x) = 1/(2 × 2) = 0.25.

Worked examples with computed numbers

These numbers are fixed on this page; in practice they change every time.

9. If f(x) = 2x³ + 3x² + 9x, what is f′(-3)?

  1. 54
  2. 45
  3. -12
  4. -54
Show the answer and solution

Answer: B. 45

f′(x) = 6x² + 6x + 9; at x = -3: 45.

10. A rectangular lot is fenced with 40 m of fence. What is the largest area it can enclose, in m²?

  1. 200
  2. 150
  3. 100
  4. 400
Show the answer and solution

Answer: C. 100

For a fixed perimeter the largest rectangle is a square of side P/4 = 10 m, area 100 m².

Other topics in the same subjects

Algebra · Trigonometry · Analytic geometry · Integral calculus · Differential equations · Complex numbers · Probability and statistics · Matrices and determinants · Laplace transforms and Fourier series

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