AP Calculus AB Unit 4: Contextual Applications of Differentiation Practice Exam

1. The temperature of a pot of soup placed on a counter to cool is modeled by the function $H$, where $H(t)$ is measured in degrees Fahrenheit and $t$ is measured in minutes for $0 \le t \le 30$. Which of the following is the best interpretation of the statement $H'(10) = -2.3$?

2. The cost to produce $x$ units of a specialized electronic component is given by the function $C(x)$, where $C(x)$ is measured in dollars. What are the units of $C'(x)$, and which of the following best interprets $C'(500) = 25$?

3. The volume of water in a cylindrical tank is modeled by the function $V(h)$, where $V$ is the volume in liters and $h$ is the depth of the water in meters. Which of the following equations correctly expresses the statement: "When the water is 3 meters deep, the volume is increasing at a rate of 50 liters per meter"?

4. A particle moves along the x-axis so that its position at time t ≥ 0 is given by x(t) = t³ - 9t² + 15t + 2. For which of the following intervals is the speed of the particle increasing?

5. The temperature of a chemical compound during a reaction, $H(t)$, measured in degrees Celsius, is a differentiable function of time $t$, measured in minutes, for $0 \le t \le 20$. Which of the following is the best interpretation of the statement $H'(5) = -2.3$?

6. The total mass of a thin metal rod of length $x$ meters is modeled by the function $M(x) = \sqrt{4x + 9}$ kilograms for $x \ge 0$. The linear density of the rod is defined as the rate of change of the mass with respect to the length. What is the linear density of the rod, in kilograms per meter, when the length is 10 meters?

7. The radius $r$ of a circle is increasing at a constant rate of $0.5$ centimeters per second. At the instant when the circumference of the circle is $10\pi$ centimeters, what is the rate of change of the area of the circle, in square centimeters per second?

8. The length $x$ of a rectangle is increasing at a rate of 2 units per second, while the width $y$ is decreasing at a rate of 3 units per second. At the instant when $x = 10$ units and $y = 6$ units, what is the rate of change of the area of the rectangle?

9. Water is draining from a conical tank with its vertex pointing downward. The tank has a height of 10 feet and a base radius of 5 feet. The water level is changing such that the volume of water in the tank is decreasing at a constant rate of 2 cubic feet per minute. What is the rate of change of the water level, in feet per minute, when the depth of the water is 4 feet? (The volume of a cone is given by $V = \frac{1}{3}\pi r^2 h$.)

10. An observer stands on level ground 300 meters from a launch pad. A rocket rises vertically from the pad at a constant velocity of 4 meters per second. What is the rate of change of the angle of elevation $\theta$ of the rocket from the observer, in radians per second, at the instant when the rocket is 400 meters above the ground?

11. Let $f$ be a differentiable function such that $f(5) = 3$ and $f'(5) = -2$. What is the approximation for $f(5.04)$ found by using the line tangent to the graph of $f$ at $x = 5$ ? [Skill: 1.E | Topic: 4.6]

12. The function $h$ is twice differentiable on the closed interval $[1, 5]$. It is known that $h'(x) > 0$ and $h''(x) < 0$ for all $x$ in the interval. The value of $h(3.2)$ is approximated using the local linearization of $h$ at $x=3$. Which of the following statements is true regarding this approximation? [Skill: 2.D | Topic: 4.6]

13. Calculate the value of $\lim_{x \to 0} \frac{1 - \cos(2x)}{3x^2}$.