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Applications of differentiation

Written by Hassan Raza (Mathematics)

ECAT · Mathematics · 6 questions

Name: ____________________   Date: ____________

  1. What is the value of the derivative of x⁴ - 3x² + 2 at x = 1?

    • A -2
    • B 0
    • C -6
    • D 2

  2. At which values of x does f(x) = x³ - 3x have stationary points?

    • A x = 1 and x = -1
    • B x = 0 and x = 3
    • C x = 0 only
    • D x = 3 and x = -3

  3. The function f(x) = x³ - 3x has a local maximum at:

    • A x = 3
    • B x = -1
    • C x = 1
    • D x = 0

  4. A particle moves so that its position is s = t³ - 6t². What is its velocity at t = 2?

    • A -12
    • B 12
    • C 0
    • D -4

  5. What is d/dx of x e^x?

    • A e^x
    • B x e^x
    • C e^x (1 + x)
    • D e^x (1 - x)

  6. How fast is the area of a circle changing with respect to its radius when the radius is 3?

    • A 3 pi
    • B 9 pi
    • C pi
    • D 6 pi

Answer key

1. A, 2. A, 3. B, 4. A, 5. C, 6. D

  1. The derivative is 4x³ - 6x. At x = 1 it equals 4 - 6 = -2.

  2. f'(x) = 3x² - 3 = 0 gives x² = 1, so x = 1 or x = -1.

  3. f''(x) = 6x. At x = -1 the second derivative is negative, so this is a local maximum.

  4. v = ds/dt = 3t² - 12t. At t = 2, v = 12 - 24 = -12.

  5. By the product rule, the derivative is e^x + x e^x = e^x (1 + x).

  6. A = pi r², so dA/dr = 2 pi r = 6 pi at r = 3.