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Applications of differentiation
6 ECAT Mathematics questions with answers and explanations.
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What is the value of the derivative of x^4 - 3x^2 + 2 at x = 1?
- 1 -2
- 2 2
- 3 0
- 4 -6
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Correct answer: -2
The derivative is 4x^3 - 6x. At x = 1 it equals 4 - 6 = -2.
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At which values of x does f(x) = x^3 - 3x have stationary points?
- 1 x = 0 only
- 2 x = 1 and x = -1
- 3 x = 3 and x = -3
- 4 x = 0 and x = 3
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Correct answer: x = 1 and x = -1
f'(x) = 3x^2 - 3 = 0 gives x^2 = 1, so x = 1 or x = -1.
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The function f(x) = x^3 - 3x has a local maximum at:
- 1 x = 1
- 2 x = -1
- 3 x = 0
- 4 x = 3
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Correct answer: x = -1
f''(x) = 6x. At x = -1 the second derivative is negative, so this is a local maximum.
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A particle moves so that its position is s = t^3 - 6t^2. What is its velocity at t = 2?
- 1 -12
- 2 12
- 3 0
- 4 -4
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Correct answer: -12
v = ds/dt = 3t^2 - 12t. At t = 2, v = 12 - 24 = -12.
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What is d/dx of x e^x?
- 1 e^x
- 2 x e^x
- 3 e^x (1 + x)
- 4 e^x (1 - x)
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Correct answer: e^x (1 + x)
By the product rule, the derivative is e^x + x e^x = e^x (1 + x).
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How fast is the area of a circle changing with respect to its radius when the radius is 3?
- 1 3 pi
- 2 6 pi
- 3 9 pi
- 4 pi