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Quiz Chapter 2: Differentiation

10 questions · Form 5 Additional Mathematics Bab 2: Differentiation

Question 1 of 10Score: 0

Find the second derivative d²y/dx² for y = 2x⁵ - 4x³ + x.

Full Question List & Answer Key

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1. Find the second derivative d²y/dx² for y = 2x⁵ - 4x³ + x.

  1. 40x³ - 24x
  2. 10x⁴ - 12x² + 1
  3. 40x⁴ - 24x²
  4. 20x³ - 12x
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Answer: A

dy/dx = 10x⁴ - 12x² + 1. d²y/dx² = d/dx(10x⁴ - 12x² + 1) = 40x³ - 24x.

2. Evaluate lim (x → 3) [x² - 9x - 3].

  1. 3
  2. 0
  3. 6
  4. Undefined
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Answer: C

lim (x → 3) [(x - 3)x + 3x - 3] = lim (x → 3) (x + 3) = 3 + 3 = 6.

3. Find the minimum value of y = x² - 6x + 13.

  1. 4
  2. 3
  3. 13
  4. 7
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Answer: A

dy/dx = 2x - 6 = 0 => x = 3. Substituting x = 3 into y: y = 3² - 6(3) + 13 = 9 - 18 + 13 = 4.

4. Given y = x³ - 6x² + 9x + 2, find the x-coordinates of the stationary points.

  1. x = -1 and x = -3
  2. x = 0 and x = 2
  3. x = 1 and x = 3
  4. x = 2 and x = 4
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Answer: C

dy/dx = 3x² - 12x + 9 = 0 => x² - 4x + 3 = 0 => (x - 1)(x - 3) = 0 => x = 1, x = 3.

5. Given y = x³, find the small change in y, δy, when x increases from 2 to 2.01.

  1. 0.12
  2. 0.06
  3. 0.01
  4. 0.24
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Answer: A

dy/dx = 3x². At x = 2, dy/dx = 3(2²) = 12. δx = 2.01 - 2 = 0.01. δy ≈ (dy/dx) · δx = 12 × 0.01 = 0.12.

6. Given y = 3x², evaluate dy/dx when x = 2.

  1. -34
  2. 34
  3. -38
  4. -32
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Answer: A

y = 3x⁻² => dy/dx = -6x⁻³ = -6x³. At x = 2, dy/dx = -62³ = -68 = -34.

7. Find the gradient of the normal to the curve y = x² - 4x + 5 at point (3, 2).

  1. 2
  2. -12
  3. -2
  4. 12
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Answer: B

dy/dx = 2x - 4. At x = 3, m_t = 2(3) - 4 = 2. Gradient of normal m_n = -1m_t = -12.

8. Differentiate y = 4x³ - 5x² + 7x - 9 with respect to x.

  1. 12x² - 10x
  2. 12x² - 10x + 7
  3. 4x² - 5x + 7
  4. 12x³ - 10x² + 7
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Answer: B

dy/dx = d/dx(4x³) - d/dx(5x²) + d/dx(7x) - d/dx(9) = 12x² - 10x + 7.

9. The radius r of a circle increases at a rate of 0.2 cm s⁻¹. Find the rate of change of its area A when r = 5 cm.

  1. 1.0π cm² s⁻¹
  2. 4.0π cm² s⁻¹
  3. 0.5π cm² s⁻¹
  4. 2.0π cm² s⁻¹
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Answer: D

A = π r² => dA/dr = 2π r. Using Chain Rule: dA/dt = (dA/dr)(dr/dt) = (2π × 5)(0.2) = 2.0π cm² s⁻¹.

10. If y = √(2x + 5), find dy/dx.

  1. 1 / √(2x + 5)
  2. 2 / √(2x + 5)
  3. 12√(2x + 5)
  4. √(2x + 5)
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Answer: A

y = (2x + 5)1/2). dy/dx = 12(2x + 5)-1/2) × 2 = 1 / √(2x + 5).

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