Module 4 Quiz: Applications of Derivatives I

Answer each question, then reveal the answer. Aim for at least 7 out of 10 correct.

1. A circle's radius increases at 4 cm/s. How fast is the area increasing when r = 5?

dA/dt = 2πr(dr/dt) = 2π(5)(4) = 40π cm²/s.

2. Find the linearization of f(x) = 1/x at a = 2.

f(2) = 1/2, f'(2) = −1/4. L(x) = 1/2 − (1/4)(x − 2) = 1 − x/4.

3. Find all critical points of f(x) = x³ − 12x.

f'(x) = 3x² − 12 = 3(x − 2)(x + 2). Critical points: x = ±2.

4. Find the absolute max and min of f(x) = sin x on [0, 2π].

f'(x) = cos x = 0 at x = π/2, 3π/2. f(0) = 0, f(π/2) = 1, f(3π/2) = −1, f(2π) = 0. Max = 1 at π/2; min = −1 at 3π/2.

5. True or false: The MVT requires f to be differentiable at the endpoints.

False. The MVT requires continuity on [a,b] and differentiability on the open interval (a,b). Differentiability at the endpoints is not required.

6. Find c satisfying the MVT for f(x) = √x on [1, 4].

Average rate: (2 − 1)/3 = 1/3. f'(c) = 1/(2√c) = 1/3. √c = 3/2, c = 9/4.

7. If y = x³ and x changes from 3 to 3.01, estimate dy.

dy = 3x² dx = 3(9)(0.01) = 0.27.

8. A 20-ft ladder slides: base moves at 3 ft/s. How fast does the top slide when base is 12 ft out?

y = √(400 − 144) = 16. 2(12)(3) + 2(16)(dy/dt) = 0. dy/dt = −9/4 ft/s.

9. Find the critical points of g(x) = x e^−x.

g'(x) = e^−x − x e^−x = e^−x(1 − x) = 0 when x = 1.

10. Apply Rolle's Theorem to f(x) = x² − 5x + 6 on [2, 3].

f(2) = 0, f(3) = 0. f is a polynomial. Rolle's applies: f'(x) = 2x − 5 = 0, c = 5/2 in (2, 3).