Applications of Derivatives — Practice

Question 1

Find the intervals where f(x) = 2x^3 - 15x^2 + 36x + 7 is increasing or decreasing.

Accepted Answer

f'(x) = 6(x-2)(x-3). f'(x) > 0 for x < 2 or x > 3 (increasing), and f'(x) < 0 for 2 < x < 3 (decreasing). So increasing on (-inf,2) and (3,inf), decreasing on (2,3).

Question 2

Find the tangent and normal to x^2 + y^2 = 25 at (3,4).

Accepted Answer

dy/dx = -x/y = -3/4 at (3,4). Tangent: y - 4 = (-3/4)(x-3), i.e. 3x + 4y = 25. Normal: y - 4 = (4/3)(x-3), i.e. 4x - 3y = 0.

Question 3

Find two positive numbers with sum 16 and maximum product.

Accepted Answer

P = x(16-x) = 16x - x^2. P'(x) = 16 - 2x = 0 gives x = 8. P''(x) = -2 < 0, a maximum. Numbers 8 and 8, product 64.

Question 4

A disc's radius increases at 0.02 cm/s. Find the rate of increase of area at r = 7 cm.

Accepted Answer

A = pi r^2, dA/dt = 2 pi r (dr/dt) = 2 pi (7)(0.02) = 0.28 pi cm^2/s.

Question 5

Find the absolute maximum and minimum of f(x) = 2x^3 - 3x^2 - 12x + 5 on [-2, 3].

Accepted Answer

f'(x) = 6(x-2)(x+1), critical points x = 2, -1. f(-2) = 1, f(-1) = 12, f(2) = -15, f(3) = -4. Absolute max = 12 at x = -1; absolute min = -15 at x = 2.

Practice — Applications of Derivatives

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