AP Calculus AB Unit 5: Analytical Applications of Differentiation Practice Exam

1. Let $f$ be the function given by $f(x) = x^3 - 3x^2$. What is the value of $c$ in the open interval $(0, 3)$ that satisfies the conclusion of the Mean Value Theorem for $f$ on the closed interval $[0, 3]$?

2. A farmer plans to fence a rectangular pasture adjacent to a river. The pasture must contain 180,000 square meters in order to provide enough grass for the herd. No fencing is needed along the river. What dimensions will require the least amount of fencing?

3. A manufacturing company wishes to design a closed rectangular box with a square base and a fixed volume of 32 cubic feet. The material for the top and bottom of the box costs 3 dollars per square foot, while the material for the sides costs 6 dollars per square foot. If $x$ represents the side length of the square base, which of the following equations gives the derivative of the cost function, $C'(x)$, equal to zero?

4. Which of the following values of $x$ produces the minimum distance between the point $(4, 0)$ and a point $(x, y)$ on the graph of the function $y = \sqrt{x}$?

5. [Skill: 1.E | Topic: 5.11] A farmer plans to enclose a rectangular pasture adjacent to a river. To provide enough grass for the herd, the pasture must contain 1,800 square meters of area. No fencing is required along the river. What is the minimum amount of fencing needed to enclose the rectangular area?

6. [Skill: 2.A | Topic: 5.11] Let $f$ be the function given by $f(x) = x^2$. Which of the following is the $y$-coordinate of the point on the graph of $f$ that is closest to the point $(0, 3)$?

7. Consider the curve defined by the equation $x^2 + xy + y^2 = 3$. At which of the following $x$-coordinates does the curve have a horizontal tangent line?

8. If $y^2 - 2x = 4$, what is $\frac{d^2y}{dx^2}$ in terms of $y$?

9. The curve defined by the implicit relation $xy + y^2 - x^2 = 1$ has a first derivative given by $\frac{dy}{dx} = \frac{2x - y}{x + 2y}$. Which of the following statements is true about the curve at the point $(1, 1)$?

10. [Skill: 1.E | Topic: 5.2] Let $f$ be the function defined by $f(x) = 3x^{2/3} - 2x$. Which of the following is the set of all critical points of $f$?

11. [Skill: 3.E | Topic: 5.2] Consider the piecewise function $h$ defined below: $$ h(x) = \begin{cases} x^2 - 2x & \text{for } -1 \le x < 2 \\ kx - 4 & \text{for } 2 \le x \le 5 \end{cases} $$ For what value of $k$ does the Extreme Value Theorem guarantee the existence of an absolute minimum and an absolute maximum for $h$ on the interval $[-1, 5]$?

12. Let $f$ be the function given by $f(x) = \frac{1}{3}x^3 - \frac{1}{2}x^2 - 6x + 4$. On which of the following intervals is $f$ decreasing?

13. The derivative of a function $f$ is given by $f'(x) = (x-2)^2 (x+5)$. On which of the following intervals is $f$ increasing?

14. Let $f$ be the function defined by $f(x) = x^3 - 6x^2 - 15x + 4$. At which x-value does $f$ have a relative minimum?

15. The derivative of a function $g$ is given by $g'(x) = (x-2)^2(x+3)$. Which of the following statements is true about $g$ at $x = 2$?

16. Let $f$ be the function defined by $f(x) = 2x^3 - 3x^2 - 12x + 10$. What is the absolute minimum value of $f$ on the closed interval $[-2, 4]$?

17. Let $h$ be the function defined by $h(x) = (x-1)e^x$. Which of the following statements is true regarding the absolute extrema of $h$ on the closed interval $[-2, 2]$?

18. Let $f$ be the function defined by $f(x) = x^5 - 5x^4$. For which value(s) of $x$ does the graph of $f$ have a point of inflection?

19. The second derivative of a function $g$ is given by $g''(x) = x(x-2)^2(x+3)$. How many points of inflection does the graph of $g$ have?

20. Let $f$ be the function defined by $f(x) = 2x^3 - 3x^2 - 12x + 4$. Which of the following statements is true regarding the relative extrema of $f$?

21. The figure above shows the graph of $f'$, the derivative of a twice-differentiable function $f$, on the interval $[-4, 6]$. The graph of $f'$ has horizontal tangents at $x = -1$ and $x = 3$. At which values of $x$ does the graph of $f$ have a point of inflection?

22. The first derivative of the function $f$ is defined by $f'(x) = x^2 e^{-x}$. On which of the following intervals is the graph of $f$ concave up?

23. Let $f$ be the function defined by $f(x) = x^3 - 3x$. What are all values of $c$ that satisfy the conclusion of the Mean Value Theorem for $f$ on the closed interval $[-1, 2]$?

24. The graph of $f'$, the derivative of a twice-differentiable function $f$, is shown above for the interval $-3 < x < 4$. The graph of $f'$ has horizontal tangent lines at $x = -1$ and $x = 2$, and intersects the x-axis at $x = -2.5$, $x = 0.5$, and $x = 3.5$. Which of the following statements is true regarding the graph of $f$?