# BC Calculus: Midyear Review Worksheet 2

• Class: 5H: BC Calculus
• Author: Peter Atlas
• Text: Calculus Finney, Demana, Waits, Kennedy

Questions have been taken from Lifshitz, Maxine, Calculus AB/BC Preparing for the Advanced Placement Examinations; Amsco, 2004

The following should be done without a calculator .

1. If $$\displaystyle xy - y = 2x + 4, \frac{dy}{dx} =$$
1. $$\displaystyle \frac{y - 2}{x - 1}$$
2. $$\displaystyle \frac{2 - y}{x - 1}$$
3. $$\displaystyle \frac{y - 6}{x - 1}$$
4. $$\displaystyle \frac{2}{x - 1}$$
5. $$\displaystyle \frac{y - 2}{x + 1}$$
2. Solution B
3. Let $$f(x)$$ be an odd function and $$g(x)$$ be even. Which of the following statements are true?
1. $$\displaystyle \int_{-2}^{2} f(x) \, dx = 0$$
2. $$\displaystyle \int_{-2}^{2} g(x) \, dx = 0$$
3. $$\displaystyle \int_{-2}^{2} f(x)g(x) \, = 0$$
1. II
2. III
3. I and II
4. I and III
5. I, II, and III
4. Solution D
5. $$\displaystyle \lim_{x \to 0} \frac{\sin{3x}}{5x} =$$
1. $$\displaystyle \frac{3}{5}$$
2. $$\displaystyle \frac{5}{3}$$
3. 3
4. 5
5. no limit exists
6. Solution A
7. For which of the following x-values on the graph of $$y = 2x - x^2$$ is $$\displaystyle \frac{dy}{dx}$$ the largest?
1. -2.7
2. -2.2
3. 0
4. 1
5. 2.7
8. Solution A
9. If $$\displaystyle y = e^{2x} + \tan {(2x)}$$, then $$y'( \pi ) =$$
1. $$\displaystyle 2e^{2\pi}$$
2. $$\displaystyle e^{2\pi} + 1$$
3. $$\displaystyle 2e^{2\pi} + 2$$
4. $$\displaystyle 2e^{\pi} - 2$$
5. 0
10. Solution C
11. Write the equation of the line tangent to $$y = e^{x+1}$$ at $$x = 0$$.
1. $$y = ex + e$$
2. $$y = x$$
3. $$y = x + 1$$
4. $$y = x + e$$
5. $$y = ex + 1$$
12. Solution A
13. If the acceleration of a particle is given by $$a(t) = 2e^t$$ and at $$t = 1$$ the velocity is 2, then $$v(0)$$ is
1. 0
2. $$2 - 2e$$
3. $$\displaystyle 2 - \frac{e}{2}$$
4. $$4 - 2e$$
5. 2
14. Solution D
15. If $$3xy + 2y^2 = 5$$, find $$\displaystyle \frac{dy}{dx}$$ at (1, 1)
1. $$\displaystyle -\frac{3}{7}$$
2. 0
3. $$\displaystyle \frac{3}{7}$$
4. 7
5. undefined
16. Solution A
17. The graph of $$\displaystyle y = \ln{\frac{1 - x}{x + 1}}$$ has vertical asymptote(s) at
1. $$x = 1$$
2. $$x = 0$$
3. $$x = \pm 1$$
4. $$x = -1$$
5. no vertical asymptotes
18. Solution C
19. $$\displaystyle \frac{d}{dt} \int_{t}^{t^2} \frac{1}{x} \, dx =$$
1. $$\displaystyle \frac{1}{t^2}$$
2. $$\ln {t}$$
3. $$\ln{(t^2)}$$
4. $$\displaystyle \frac{1}{t}$$
5. $$\displaystyle \ln{\left(\frac{1}{t}\right)}$$
20. Solution D
21. $$\displaystyle \lim_{h \to 0} \frac{\cos{ \left( \frac{\pi}{4} + h \right) } - \cos {\left( \frac{\pi}{4} \right)}}{h}$$
1. $$\displaystyle -\frac{\sqrt{3}}{2}$$
2. $$\displaystyle -\frac{\sqrt{2}}{2}$$
3. 0
4. $$\displaystyle \frac{\sqrt{2}}{2}$$
5. 1
22. Solution B
23. $$\displaystyle \int (3x^2 - \cos{x }) \, dx =$$
1. $$6x +\sin{x} + C$$
2. $$x^3+\sin{ x} + C$$
3. $$x^3-\sin{ x} + C$$
4. $$6x -\sin{ x} + C$$
5. $$3x^3-\sin{ x} + C$$
24. Solution C
25. Find the area under the curve $$y = x^2+ 1$$ on the interval [1, 2].
1. $$\displaystyle \frac{7}{3}$$
2. 3
3. $$\displaystyle \frac{10}{3}$$
4. 4
5. 7
26. Solution C
27. $$\displaystyle \int_{-1}^{2} \frac{x}{x^2+ 1} \, dx =$$
1. $$\displaystyle \ln {\frac{2}{5}}$$
2. 0
3. $$\displaystyle \frac{1}{2} \ln {\frac{5}{2}}$$
4. $$\displaystyle \frac{1}{2} \ln {3}$$
5. undefined
28. Solution C
29. $$\displaystyle \int_{0}^{1} e^{3x + 2} \, dx =$$
1. $$\displaystyle \frac{1}{3} e^5$$
2. $$\displaystyle \frac{1}{3}(e^5 - e^2)$$
3. $$\displaystyle \frac{1}{3}(e^5 - 1 )$$
4. $$e^5 - 1$$
5. $$e^5 - e^2$$
30. Solution B
31. If $$\displaystyle y =\sin^2{(5x)}, \frac{dy}{dx} =$$
1. $$5\sin{ (10x)}$$
2. $$5\cos {(5x)}$$
3. $$5\sin{(5x)}$$
4. $$10\sin{(10x)}$$
5. $$10\sin^2{(5x)}$$
32. Solution A
33. For what values of $$x$$ is $$f(x) = 2x^3- x^2 + 2x$$ concave up?
1. $$\displaystyle x < \frac{1}{6}$$
2. $$x < 0$$
3. $$x > 0$$
4. $$\displaystyle x > \frac{1}{6}$$
5. $$x > 6$$
34. Solution D
35. This problem refers to the graph of the velocity, $$v(t)$$ of an object at time $$t$$. The graph consists of two segments: one joining (0, 20) to (5, 20), and a second joining (5, 20) to (10, -20). If $$x(t)$$ is the position of the object at time $$t$$, and $$a(t)$$ is the acceleration of the object at time $$t$$, which of the following is true?
1. $$v(5) > v(2)$$
2. $$x(5) > x(2)$$
3. $$a(6) > a(2)$$
4. $$x(10) < x(5)$$
5. $$a(9) > a(6)$$
36. Solution B
37. Referring to the velocity as described in problem 18, $$\int_{0}^{10} v(t) \, dt =$$
1. 0
2. 5
3. 50
4. 75
5. 100
38. Solution E
39. If $$f(1) = 2$$ and $$f'(1) = 5$$, use the equation of the line tangent to the graph of $$f$$ at $$x = 1$$ to approximate $$f(1.2)$$
1. 1
2. 1.2
3. 3
4. 5.4
5. 9
40. Solution C
41. The equation of the line tangent to $$y = \tan^2{(3x)}$$ at $$\displaystyle x = \frac{\pi}{4}$$ is
1. $$y = -12x + 3 \pi - 1$$
2. $$y = -12x + 3 \pi + 1$$
3. $$\displaystyle y = -6x + \frac{3\pi}{2} + 1$$
4. $$y = -12x + 3 \pi + 3$$
5. $$y = -12x + \pi + 3$$
42. Solution B
43. The graph of $$f'(x)$$ is shown below. Which of the following choices could be the graph of $$f$$?
44. Solution D
45. At what value of $$x$$ is the line tangent to the graph of $$y = x^2 + 3x + 5$$ perpendicular to the line $$x - 2y = 5$$?
1. $$\displaystyle -\frac{5}{2}$$
2. -2
3. $$\displaystyle -\frac{1}{2}$$
4. $$\displaystyle \frac{1}{2}$$
5. $$\displaystyle \frac{5}{2}$$
46. Solution A
47. The function $$f(x)$$ below is called a sawtooth wave. Which of the following statements about this function is true?
1. $$f(x)$$ is continuous everywhere
2. $$f(x)$$ is differentiable everywhere
3. $$f(x)$$ is continuous everywhere but the values of x for which $$f(x) = 0$$
4. $$f(x)$$ is an even function
5. $$f(x)$$ is a one-to-one function.
48. Solution A
49. Find the derivative of $$y = x^2e^{x^2}$$
1. $$2xe^{x^2}(x^2 + 1)$$
2. $$2xe^{x^2}$$
3. $$2x^3e^{x^2}$$
4. $$4x^2e^{x^2}$$
5. $$x^4e^{x^2}$$
50. Solution A
51. $$\displaystyle \int_{0}^{ \frac{\pi}{2}} e^{2 - \cos{x}} \sin{ x} \, dx =$$
1. $$\displaystyle e^2 - e$$
2. $$\displaystyle 1$$
3. $$\displaystyle 0$$
4. $$\displaystyle e^2$$
5. does not exist
52. Solution A
53. The average value of $$\displaystyle f(x) = -\frac{1}{x^2}$$ on $$[\frac{1}{2}, 1]$$ is
1. -4
2. -2
3. $$\displaystyle -\frac{1}{2}$$
4. 2
5. undefined
54. Solution B
55. $$f(x) = \left| x^2 - 3x \right|$$. Find $$f'(1)$$
1. -3
2. -1
3. 1
4. 3
5. does not exist
56. Solution C