Applications of derivatives (related rates, MVT, optimization, L'Hôpital's rule) Knowledge-Based Practice

Questions

1. 1.

   A spherical balloon is inflated so that its volume increases at a constant rate of $36\pi\text{ cm}^3/\text{s}$. At what rate is the surface area of the balloon increasing at the instant when its radius is 3 cm?

   • A) $12\pi\text{ cm}^2/\text{s}$
   • B) $18\pi\text{ cm}^2/\text{s}$
   • C) $24\pi\text{ cm}^2/\text{s}$
   • D) $36\pi\text{ cm}^2/\text{s}$
2. 2.

   A rectangular garden is to be enclosed using an existing straight stone wall for one side and 120 meters of fencing for the remaining three sides. What is the maximum possible area that can be enclosed?

   • A) 900 m²
   • B) 1200 m²
   • C) 1800 m²
   • D) 2400 m²
3. 3.

   Evaluate the limit $\lim_{x \to 0} \frac{x - \sin(x)}{x^3}$.

   • A) $0$
   • B) $\frac{1}{2}$
   • C) $\frac{1}{3}$
   • D) $\frac{1}{6}$
4. 4.

   A 10-foot ladder leans against a vertical wall. If the bottom of the ladder slides away from the wall at a rate of 2 feet per second, at what rate is the top of the ladder sliding down the wall when the bottom is 6 feet from the wall?

   • A) $\frac{4}{3}$ feet per second
   • B) $\frac{8}{3}$ feet per second
   • C) $\frac{3}{2}$ feet per second
   • D) 2 feet per second
5. 5.

   Find the value of $c$ that satisfies the conclusion of the Mean Value Theorem for the function $f(x) = x^3 - 3x$ on the closed interval $[0, 3]$.

   • A) $c = 1$
   • B) $c = \frac{3}{2}$
   • C) $c = \sqrt{2}$
   • D) $c = \sqrt{3}$

Answers and step-by-step solutions appear after you finish the test.