# BC Calculus: Midyear Review Worksheet 4

• 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

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1. If $$f(x) = 2 \sin ^2 {(5x)}, f'(x) =$$
1. $$10 \sin{(10x)}$$
2. $$20 \sin{(5x)}$$
3. $$10 \sin{(5x)}$$
4. $$4 \cos{(5x)}$$
5. $$20 \cos{(5x)}$$
2. Solution A
3. Find the x-intercept of the line tangent to $$y = \ln{ (\ln{ x})}$$ at $$x = e$$
1. $$x = -1$$
2. $$x = 0$$
3. $$x = 1$$
4. $$x = e$$
5. $$\displaystyle x = -\frac{1}{e}$$
4. Solution D
5. $$a(t) = 2e^t$$ and $$v(1) = 2$$. Find $$v(t)$$
1. $$2(e^t - e + 1)$$
2. $$2(e^{t - 1} + 1)$$
3. $$2t + 1$$
4. $$2e^t + e$$
5. $$2e^t$$
6. Solution A
7. If $$\displaystyle \lim_{x \to - \infty} \frac{\ln{ (1 - x)}}{x^2 + 1} =$$
1. $$-\infty$$
2. -1
3. 0
4. 1
5. $$\infty$$
8. Solution C
9. $$\displaystyle y = \frac{e^{2x - 1}}{x}$$ has
1. a relative maximum at $$\displaystyle x = \frac{1}{2}$$
2. a horizontal asymptote at $$y = 0$$
3. a vertical asymptote at $$x = 0$$
1. I only
2. I and II
3. I and III
4. II and III
5. I, II and III
10. Solution D
11. Given the piecewise function defined as $$\displaystyle f(x) = \begin{cases} x^2 + 2, & \text{if } x < 1 \\ 2x + 1, & \text{if }x \geq 1 \end{cases}$$, which of the following is true?
1. $$f(x)$$ is not continuous at $$x = 1$$.
2. $$f(x)$$ is continuous but not idfferentiable at $$x = 1$$.
3. $$f(x)$$ is differentiable but not continuous at $$x = 1$$.
4. $$f(x)$$ is continuous and differentiable at $$x = 1$$.
5. $$f(2) = 6$$
12. Solution D
13. $$\displaystyle \int_1^e x \ln{x} \, dx =$$
1. $$\displaystyle \frac{e^2+ 1}{4}$$
2. $$2e^2 - 1$$
3. $$\displaystyle \frac{e^2 - e - 1}{2}$$
4. $$2e^2 + 1$$
5. undefined
14. Solution A
15. Write an integral that represents the length of one arch of $$y = \sin{ x}$$.
1. $$\displaystyle \int_0^{\frac{\pi}{2}} \sqrt{1 + \cos^2{x}} \, dx =$$
2. $$\displaystyle \int_0^{\pi} \sqrt{1 + \cos^2{x}} \, dx =$$
3. $$\displaystyle \int_0^{\frac{\pi}{2}} ( 1 + \cos{ x}) \, dx =$$
4. $$\displaystyle \int_0^{\frac{\pi}{2}} \sin^2{x} \, dx =$$
5. $$\displaystyle \int_0^{\frac{\pi}{2}} \sqrt{1 + \sin^2{x}} \, dx =$$
16. Solution B
17. For what values of $$a$$ and $$c$$ is the piecewise function defined by $$\displaystyle f(x) = \begin{cases} x + c, & \text{if } x > 2 \\ ax^2, & \text{if }x \leq 2 \end{cases}$$ differentiable at x = 2?
1. $$\displaystyle a = \frac{1}{2}, c= 0$$
2. $$\displaystyle a = \frac{1}{4}, c = -1$$
3. $$a = 1, c = 6$$
4. $$a = 0, c = -2$$
5. no solution
18. Solution B
19. The slope field below depicts a certain differential equation. Which of the following choices could be a solution to that equation?
1. $$y = \ln{x}$$
2. $$y = e^{-x}$$
3. $$y = \sin{x}$$
4. $$y = e^x$$
5. $$y = x^{2\sin{ x}}$$
20. Solution B
21. $$\displaystyle h(x) = \frac{f(x)}{\left( g(x) \right)^2}$$. If $$\displaystyle f'(x) = g(x), g'(x) = \frac{1}{f(x)}, g(x) > 0$$ for all real $$x$$, and $$f(x) \neq 0$$ for all real $$x$$, then $$h'(x) =$$
1. $$\displaystyle \frac{2}{\left(g(x) \right)^3} - \frac{1}{g(x)}$$
2. $$\displaystyle \frac{1}{g(x)} - \frac{2}{\left(g(x)\right)^3}$$
3. $$\displaystyle \left(g(x)\right)^2 - 2$$
4. $$\displaystyle \frac{1}{g(x)} - \frac{2f(x)}{\left(g(x)\right)^3}$$
5. 0
22. Solution B
23. The maximum value of $$\displaystyle f(x) = \frac{\sqrt{x - 3}}{ x}$$ occurs at $$x =$$
1. -6
2. $$\displaystyle \frac{\sqrt{3}}{3}$$
3. 3
4. 6
5. 7
24. Solution D
25. $$y' = 2y + 5$$. Find $$y''$$ in terms of $$y$$.
1. 0
2. 2
3. $$4y + 5$$
4. $$4y + 10$$
5. $$4y + 20$$
26. Solution D
27. For what value of $$c$$ does $$\displaystyle y = cx + \frac{3}{x}$$ have a relative minimum at $$x = 2$$?
1. $$\displaystyle -\frac{2}{3} \ln {2}$$
2. 0
3. $$\displaystyle \frac{3}{8}$$
4. $$\displaystyle \frac{1}{2}$$
5. $$\displaystyle \frac{3}{4}$$
28. Solution E
29. $$f(x) = 3(x - 2)^2 + 6(x - 2) + 1$$. Find the equation of the line tangent to $$f(x)$$ at $$x = 2$$.
1. $$6x - y = 11$$
2. $$y = 0$$
3. $$6x - y = 12$$
4. $$6x - y = 13$$
5. $$7x - y = 13$$
30. Solution A
31. Which of the following formulas could be used to calculate the average rate of change of $$f$$ on the closed interval [0, 4]?
1. $$\displaystyle \frac{f(4) - f(0)}{2}$$
2. $$\displaystyle \frac{f(0) + f(4)}{2}$$
3. $$\displaystyle \frac{f(4) - f(0)}{4 }$$
4. $$\displaystyle \frac{f(0) + f(4)}{4}$$
5. $$\displaystyle \frac{f'(4) - f'(0)}{4}$$
32. Solution C
33. $$\displaystyle f(x) = \frac{x}{x - 3}$$. If $$f^{-1}(x)$$ is the inverse of $$f(x)$$, find the derivative of $$f^{-1}(x)$$ at $$x = -2$$
1. -3
2. $$\displaystyle -\frac{1}{3}$$
3. $$\displaystyle \frac{1}{3}$$
4. 1
5. 3
34. Solution B
35. $$f(x) = 3x^3 - 4$$. Find $$\displaystyle \lim_{x \to 2} \frac{f(x) - f(2)}{x - 2}$$
1. 18
2. 20
3. 24
4. 32
5. 36
36. Solution E
37. Given the graph of $$y = f(x)$$ shown below, if $$\displaystyle g(x) = f\left(\frac{1}{x}\right)$$, find the value of $$\displaystyle \lim_{x \to 0^+} g(x)$$
1. -1
2. 0
3. 1
4. $$\infty$$
5. does not exist
38. Solution C
39. $$y = \ln{ (\cos^2{x})}. y' =$$
1. $$-2 \tan{ x}$$
2. $$\sec^2 {x}$$
3. $$2 \sec {x}$$
4. $$2 \tan{ x}$$
5. $$-2 \sin{x} \cos{ x}$$
40. Solution A
41. Write the equation of the line perpendicular to the tangent of the curve represented by the equation $$y = e^{x+1}$$ at $$x = 0$$
1. $$\displaystyle y = -\frac{x}{e}$$
2. $$\displaystyle y = -\frac{x}{e} + e$$
3. $$y = ex + e$$
4. $$\displaystyle y = \frac{x}{e} + e$$
5. $$y = ex$$
42. Solution B
43. Find the area bounded by the graph of $$\displaystyle y = \frac{x}{x^2 - 1}$$ and the x-axis on the interval $$\displaystyle \left[-\frac{1}{2}, \frac{1}{2} \right]$$.
1. $$\displaystyle \ln{ \left(\frac{3}{4}\right)}$$
2. 0
3. $$\displaystyle \frac{1}{2}\ln{ \left(\frac{4}{3}\right)}$$
4. $$\displaystyle \ln{ \left(\frac{4}{3}\right)}$$
5. $$\ln {2}$$
44. Solution D
45. $$F(x)$$ is the antiderivative of $$f(x), F(5) = 7$$, and $$\displaystyle \int_2^5 f(x) \, dx = 9$$. Find $$F(2)$$
1. -16
2. -7
3. -2
4. 2
5. 16
46. Solution C