10 questions · Form 5 Additional Mathematics Bab 2: Differentiation
10 questions · plays in fullscreen
Prefer reading to quizzing? All 10 questions are listed below with the answer and explanation under each one.
1. Find the gradient of the normal to the curve y = x² - 4x + 5 at point (3, 2).
Answer: B
dy/dx = 2x - 4. At x = 3, m_t = 2(3) - 4 = 2. Gradient of normal m_n = -1m_t = -12.
2. If y = (3x - 2)⁴, find dy/dx.
Answer: A
Using Chain Rule: dy/dx = 4(3x - 2)³ × d/dx(3x - 2) = 4(3x - 2)³(3) = 12(3x - 2)³.
3. Find the minimum value of y = x² - 6x + 13.
Answer: A
dy/dx = 2x - 6 = 0 => x = 3. Substituting x = 3 into y: y = 3² - 6(3) + 13 = 9 - 18 + 13 = 4.
4. Variables x and y are related by y = 2x³ - 5. If x increases at a uniform rate of 3 units per second, find the rate of change of y when x = 2.
Answer: B
dy/dx = 6x². At x = 2, dy/dx = 6(2²) = 24. dy/dt = (dy/dx)(dx/dt) = 24 × 3 = 72... Wait: 6(4)=24; 24×3=72. Correct choice is 72.
5. 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.
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.
6. Find the gradient of the tangent to the curve y = 2x² - 3x + 1 at the point x = 2.
Answer: C
dy/dx = 4x - 3. At x = 2, dy/dx = 4(2) - 3 = 5.
7. Find the second derivative d²y/dx² for y = 2x⁵ - 4x³ + x.
Answer: A
dy/dx = 10x⁴ - 12x² + 1. d²y/dx² = d/dx(10x⁴ - 12x² + 1) = 40x³ - 24x.
8. If y = √2x + 5, find dy/dx.
Answer: A
y = (2x + 5)1/2). dy/dx = 12(2x + 5)-1/2) × 2 = 1 / √2x + 5.
9. Find the equation of the normal to the curve y = 3x² - 5 at x = 1.
Answer: B
At x = 1, y = 3(1)² - 5 = -2. dy/dx = 6x => m_t = 6(1) = 6. Gradient of normal m_n = -16. Equation: y - (-2) = -16(x - 1) => 6y + 12 = -x + 1 => x + 6y + 11 = 0.
10. Find the derivative of y = 2x + 1x - 3 with respect to x.
Answer: A
Using Quotient Rule (u = 2x+1, v = x-3): dy/dx = (x-3)(2) - (2x+1)(1)x-3² = 2x - 6 - 2x - 1x-3² = -7x-3².