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

1. The function $P(h)$ models the atmospheric pressure, in pounds per square inch (psi), at an altitude of $h$ feet above sea level. Which of the following describes the units of the derivative $P'(h)$?

2. The temperature of a substance during a chemical reaction is modeled by the differentiable function $T(m)$, where $T(m)$ is measured in degrees Celsius and $m$ is measured in minutes since the reaction began. Which of the following is the correct interpretation of the statement $T'(15) = -4$?

3. The total cost, in dollars, to manufacture $k$ kilograms of a specialized polymer is given by the function $C(k)$. Which of the following is the best interpretation of $C'(200) = 50$?

4. [Skill: 1.C | Topic: 4.2] A particle moves along the x-axis so that at any time $t \ge 0$, its position is given by the function $x(t) = t^3 - 6t^2 + 9t$. For what values of $t$ is the speed of the particle increasing?

5. [Skill: 2.B | Topic: 4.2] A particle moves along the x-axis. The graph of the particle's velocity $v(t)$ is shown above for $0 \le t \le 5$. The graph consists of two line segments connecting the points $(0, 2)$, $(2, -2)$, and $(5, 4)$. On which of the following intervals is the particle moving to the left and slowing down?

6. The total cost, in dollars, to manufacture $x$ meters of a specialized fiber optic cable is modeled by the function $C(x) = 500 + 40\sqrt{x}$, where $x > 0$. What is the value of $C'(100)$, and what is its interpretation in the context of the problem?

7. The temperature of a substance during a chemical reaction is modeled by the differentiable function $H(t)$, where $H(t)$ is measured in degrees Celsius and $t$ is measured in minutes since the reaction began. Which of the following explains the meaning of $H'(15) = -4$?

8. 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 $8\pi$ centimeters, what is the rate of change of the area of the circle, in square centimeters per second?

9. The length of a rectangle, $x$, is increasing at a rate of 4 units per minute, while the width of the rectangle, $y$, is decreasing at a rate of 3 units per minute. Which of the following expressions gives the rate of change of the area of the rectangle, $A$, with respect to time $t$, in square units per minute?

10. Water is draining from a conical tank at a rate of 2 cubic feet per minute. The tank stands vertex down and has a height of 12 feet and a radius of 4 feet at the top. How fast is the water level dropping when the water is 6 feet deep?

11. A rocket is launched vertically from a launchpad. A camera is positioned on the ground 3,000 feet away from the launchpad to record the flight. When the rocket is 4,000 feet above the ground, its velocity is 500 feet per second. At this instant, what is the rate of change of the angle of elevation of the camera?

12. Let $f$ be a function that is twice differentiable for all real numbers. The table below gives values for $f$, $f'$, and $f''$ at selected values of $x$. | $x$ | $f(x)$ | $f'(x)$ | $f''(x)$ | |:---:|:---:|:---:|:---:| | 2 | 4 | $-3$ | 2 | | 3 | 5 | $-2$ | $-4$ | Using the local linear approximation of $f$ at $x=3$, what is the approximation of $f(3.1)$, and is this approximation an underestimate or an overestimate of the actual value of $f(3.1)$?

13. Let $f$ be the function defined by $f(x) = e^{2x}$. Which of the following is the linear approximation of $f$ at $x=0$, and does this approximation provide an underestimate or an overestimate of $f(0.1)$?

14. What is the value of $\lim_{x \to 0} \frac{\int_0^{2x} \cos(t^2) \, dt}{3x}$?