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

Questions

1. 1.

   What is the absolute maximum value of the function $f(x) = 2x^3 - 9x^2 + 12x + 1$ on the closed interval $[0, 3]$?

   • A) 5
   • B) 6
   • C) 8
   • D) 10
2. 2.

   A rectangular pen is to be built with one side along a straight stone wall, requiring no fencing. If 120 meters of fencing are available for the remaining three sides, what is the maximum area, in square meters, that can be enclosed?

   • A) 1800
   • B) 1600
   • C) 2400
   • D) 3600
3. 3.

   What is the value of the limit $\lim_{x \to 0} \frac{e^{2x} - 2x - 1}{1 - \cos(3x)}$?

   • A) $\frac{2}{3}$
   • B) $\frac{2}{9}$
   • C) $\frac{4}{3}$
   • D) $\frac{4}{9}$
4. 4.

   Find the value of $c$ that satisfies the conclusion of the Mean Value Theorem for the function $f(x) = \sqrt{x}$ on the closed interval $[1, 9]$.

   • A) $c = 2$
   • B) $c = 3$
   • C) $c = 5$
   • D) $c = 4$
5. 5.

   A 10-meter ladder leans against a vertical wall. If the bottom of the ladder slides away from the wall at a rate of 2 m/s, at what rate is the top of the ladder sliding down the wall when the bottom is 6 meters from the wall?

   • A) $\frac{2}{3}\text{ m/s}$
   • B) $\frac{4}{3}\text{ m/s}$
   • C) $\frac{3}{2}\text{ m/s}$
   • D) $\frac{8}{3}\text{ m/s}$