AP Calculus BC Unit 5 Practice Test & Exam | 33 Questions

1. Let $f$ be the function defined by $f(x) = x^3 - 4x$. The Mean Value Theorem is applied to $f$ on the closed interval $[-1, 2]$. What is the value of $c$ in the open interval $(-1, 2)$ guaranteed by the theorem?

(A)1 only(B)-1 only(C)-1 and 1(D)0

2. Let $g$ be the function defined by $g(x) = |x - 2|$. Which of the following statements best explains why the Mean Value Theorem does NOT apply to the function $g$ on the interval $[0, 4]$?

(A)The function $g$ is not continuous on the interval $[0, 4]$.(B)The function $g$ is not differentiable at $x = 2$, which is in the interval $(0, 4)$.(C)The average rate of change of $g$ on $[0, 4]$ is 0.(D)The values of $g(0)$ and $g(4)$ are equal.

3. The function $h$ is continuous on the closed interval $[0, 9]$ and differentiable on the open interval $(0, 9)$. Selected values of $h(t)$ are given in the table below. | $t$ | 0 | 2 | 5 | 9 | | :--- | :--- | :--- | :--- | :--- | | $h(t)$ | 4 | 10 | 16 | 20 | Based on the table, which of the following statements must be true?

(A)There exists a value $c$ in the interval $(0, 9)$ such that $h'(c) = 4$.(B)There exists a value $c$ in the interval $(2, 5)$ such that $h'(c) = 2$.(C)There exists a value $c$ in the interval $(5, 9)$ such that $h'(c) = 0$.(D)For all $t$ in the interval $(0, 9)$, $h'(t) > 0$.

4. A rectangular storage container with an open top is to have a volume of 108 cubic meters. The base of the container is a square. To minimize the cost of materials, the surface area of the container must be minimized. What is the height of the container that minimizes the surface area?

(A)3 meters(B)4 meters(C)6 meters(D)12 meters

5. A particle moves along the x-axis so that at any time $t \geq 0$, its position is given by $x(t) = t^3 - 6t^2 + 9t + 2$. What is the absolute maximum position of the particle on the interval $0 \leq t \leq 5$?

(A)2(B)6(C)22(D)27

6. A piece of wire of length 20 is cut into two pieces. One piece is bent into the shape of a square and the other is bent into the shape of a circle. If the wire is cut so that the total area enclosed by the two shapes is maximized, what is the length of the piece of wire used for the circle?

(A)0(B)$\frac{20\pi}{\pi + 4}$(C)10(D)20

7. A rectangle is inscribed in the region bounded by the x-axis and the semicircle defined by the function $y = \sqrt{12 - x^2}$. One side of the rectangle lies along the x-axis. What is the maximum possible area of this rectangle?

(A)6(B)12(C)$6\sqrt{2}$(D)$12\sqrt{2}$

8. A cylindrical container with an open top is to be constructed to hold a volume of $16\pi$ cubic meters. The material for the circular base costs 4 dollars per square meter, and the material for the curved vertical side costs 2 dollars per square meter. Which of the following equations represents the cost function $C(r)$ solely in terms of the radius $r$, and what is the radius that minimizes this cost?

(A)$C(r) = 4\pi r^2 + \frac{32\pi}{r}$; minimal radius $r = \sqrt[3]{4}$(B)$C(r) = 4\pi r^2 + \frac{64\pi}{r}$; minimal radius $r = 2$(C)$C(r) = 4\pi r^2 + \frac{32\pi}{r^2}$; minimal radius $r = 2$(D)$C(r) = \pi r^2 + \frac{32\pi}{r}$; minimal radius $r = 4$

9. A manufacturing company wants to construct an open-top box from a rectangular sheet of cardboard measuring 15 inches by 24 inches. They will cut congruent squares of side length $x$ from each of the four corners and fold up the sides. What is the value of $x$ that maximizes the volume of the box?

(A)2(B)3(C)5(D)10

10. Consider the hyperbola defined by the equation $x^2 - y^2 = 9$. What is the value of $\frac{d^2y}{dx^2}$ at the point $(5, 4)$?

(A)-$\frac{9}{64}$(B)$\frac{9}{64}$(C)-$\frac{7}{64}$(D)-$\frac{9}{16}$

11. The curve $C$ is defined by the implicit relation $y^2 - xy = 8$. Which of the following correctly describes the behavior of the curve at the point $(2, 4)$?

(A)The curve is increasing and concave up.(B)The curve is increasing and concave down.(C)The curve is decreasing and concave up.(D)The curve is decreasing and concave down.

12. For the curve defined by $x^2 + xy + y^2 = 3$, at which $x$-coordinates does the curve have a horizontal tangent line?

(A)$x = 0$ only(B)$x = 1$ and $x = -1$(C)$x = \sqrt{3}$ and $x = -\sqrt{3}$(D)There are no horizontal tangent lines.

13. Let $f$ be the function defined by $f(x) = x^3 - 3x^2 - 9x + 5$. What is the absolute maximum value of $f$ on the closed interval $[-2, 4]$?

(A)10(B)5(C)-15(D)-22

14. The graph of $f'$, the derivative of the continuous function $f$, is shown on the closed interval $[-2, 6]$. The graph of $f'$ has x-intercepts at $x=1$ and $x=4$. Furthermore, $f'(x) > 0$ on the intervals $[-2, 1)$ and $(4, 6]$, and $f'(x) < 0$ on the interval $(1, 4)$. Which of the following sets contains all values of $x$ at which the absolute maximum of $f$ could occur on the interval $[-2, 6]$?

(A){1}(B){6}(C){1, 6}(D){-2, 4}

15. Let f be a function with a first derivative given by f'(x) = (x-2)^2(x+4)(x-5) . At which value(s) of x does f have a relative maximum?

(A) x = -4 only(B) x = 2 only(C) x = 5 only(D) x = -4 and x = 5

16. The function g is defined by g(x) = x^2 e^{-x} . Which of the following statements is true regarding the relative extrema of g ?

(A) g has a relative minimum at x = 0 and a relative maximum at x = 2 .(B) g has a relative maximum at x = 0 and a relative minimum at x = 2 .(C) g has a relative minimum at x = 0 but no relative maximum.(D) g has a relative maximum at x = 2 but no relative minimum.

Refer to the figure below.

17. The graph of f' , the derivative of the function f , is shown above for -2 < x < 6 . The graph of f' has x-intercepts at x = 0 , x = 3 , and x = 5 . If f' is positive on the interval (0, 3) and negative on the intervals (-2, 0) , (3, 5) , and (5, 6) , which of the following describes the relative extrema of f ?

(A) f has a relative minimum at x = 0 and a relative maximum at x = 3 .(B) f has a relative minimum at x = 0 and relative maxima at x = 3 and x = 5 .(C) f has a relative maximum at x = 0 and a relative minimum at x = 3 .(D) f has relative minima at x = 0 and x = 5 and a relative maximum at x = 3 .

18. Let $f$ be the function given by $f(x) = 2x^3 - 3x^2 - 12x + 5$. What is the absolute minimum value of $f$ on the closed interval $[-2, 3]$?

(A)-15(B)-4(C)1(D)2

Refer to the figure below.

19. The graph of $f'$, the derivative of the function $f$, is defined on the closed interval $[0, 7]$. The graph consists of line segments and a semi-circle, and has $x$-intercepts at $x=2$ and $x=5$. The areas of the regions between the graph of $f'$ and the $x$-axis are as follows: - The area of the region between $x=0$ and $x=2$ is 10. - The area of the region between $x=2$ and $x=5$ is 12. - The area of the region between $x=5$ and $x=7$ is 5. If $f(0) = 4$, at which $x$-value does the absolute minimum of $f$ occur on the interval $[0, 7]$?

(A)x = 0(B)x = 2(C)x = 5(D)x = 7

20. A particle moves along the $x$-axis such that its position at time $t$ is given by $x(t) = t^3 - 6t^2 + 9t + 2$ for $0 \le t \le 4$. What is the absolute maximum position of the particle on the interval $0 \le t \le 4$?

(A)2(B)4(C)6(D)22

Refer to the figure below.

21. The graph of $f'$, the first derivative of the function $f$, is shown above for the interval $-2 < x < 5$. The graph of $f'$ has horizontal tangent lines at $x=0$ and $x=3$, and intersects the x-axis at $x=-1.5$ and $x=4.5$. On which of the following intervals is the graph of $f$ concave down?

(A)$(-2, 0)$ and $(3, 5)$(B)$(0, 3)$(C)$(-1.5, 4.5)$(D)$(-2, -1.5)$ and $(4.5, 5)$

22. The second derivative of a function $f$ is given by $f''(x) = x^2(x-2)(x+4)$. At which values of $x$ does the graph of $f$ have a point of inflection?

(A)$x = -4, x = 0,$ and $x = 2$(B)$x = -4$ and $x = 2$ only(C)$x = 0$ and $x = 2$ only(D)$x = 2$ only

23. The derivative of the function $f$ is given by $f'(x) = (x-2)\sin(x)$. For the interval $0 < x < 4$, at which value(s) of $x$ does $f$ have a relative maximum?

(A)$x = 2$ only(B)$x = \pi$ only(C)$x = 2$ and $x = \pi$(D)$f$ has no relative maximums on this interval.

24. Let $g$ be a twice-differentiable function. Selected values for $g$, $g'$, and $g''$ are given in the table below. | $x$ | $g(x)$ | $g'(x)$ | $g''(x)$ | |:---:|:---:|:---:|:---:| | 5 | 12 | 0 | -3 | Based on the information in the table, which of the following statements must be true about the function $g$ at $x=5$?

(A)$g$ has a relative minimum at $x=5$.(B)$g$ has a relative maximum at $x=5$.(C)$g$ has a point of inflection at $x=5$.(D)The Second Derivative Test is inconclusive.

25. The function $h$ is twice differentiable on the open interval $(0, 8)$. It is known that $x=4$ is the **only** critical point of $h$ on this interval. If $h'(4) = 0$ and $h''(4) = 2$, which of the following describes the absolute extremum of $h$ on $(0, 8)$?

(A)$h$ has an absolute maximum at $x=4$.(B)$h$ has an absolute minimum at $x=4$.(C)$h$ has a relative maximum at $x=4$, but it is not necessarily the absolute maximum.(D)$h$ has a relative minimum at $x=4$, but it is not necessarily the absolute minimum.

Refer to the figure below.

26. The graph of $f'$, the derivative of the function $f$, is shown above for $-2 < x < 6$. The graph of $f'$ has x-intercepts at $x = 0$ and $x = 4$, and horizontal tangent lines at $x = 2$ and $x = 5$. At which of the following x-values does $f$ have a relative maximum?

(A)0 only(B)4 only(C)0 and 4(D)2 and 5

27. Let $f$ be a function defined on the interval $(0, 10)$. It is known that $f'(x) > 0$ for all $x$ in the interval and that $f''(x) < 0$ for $x < 5$ and $f''(x) > 0$ for $x > 5$. Which of the following best describes the graph of $f$?

(A)The graph of $f$ is increasing and concave up everywhere.(B)The graph of $f$ is increasing and concave down everywhere.(C)The graph of $f$ is increasing, concave down for $x < 5$, and concave up for $x > 5$.(D)The graph of $f$ is increasing, concave up for $x < 5$, and concave down for $x > 5$.

28. The graph of f', the first derivative of the function f, is shown above for the interval -2 < x < 6. The graph of f' has x-intercepts at x = 0 and x = 4, and a local minimum at x = 2. On which of the following intervals is the graph of f both decreasing and concave up?

(A)(-2, 0)(B)(0, 2)(C)(2, 4)(D)(4, 6)

29. The second derivative of a function f is given by f''(x) = x^2(x-3)(x+2). At how many points does the graph of f have a point of inflection?

(A)Zero(B)One(C)Two(D)Three

30. The function $f$ is continuous on the closed interval $[-2, 6]$ with an initial value of $f(-2) = 10$. The graph of $f'$, the derivative of $f$, consists of two line segments: one connecting the points $(-2, 2)$ and $(2, -2)$, and the other connecting the points $(2, -2)$ and $(6, 6)$. What is the absolute minimum value of $f$ on the interval $[-2, 6]$?

(A)3(B)9(C)10(D)18

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

Exam Details

A. Multiple-Choice Questions (30 Questions)

B. Short Answer Questions (3 Questions)

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

Exam Details

A. Multiple-Choice Questions (30 Questions)

B. Short Answer Questions (3 Questions)