# BC Calculus: Midyear Review Worksheet 3

• 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

A Graphing Calculator is required for some of these questions.

1. Which of the following statements is true about the graph of $$f(x)$$ below:
1. $$\displaystyle \lim_{x \to 5} f(x)$$ exists
2. $$\displaystyle \lim_{x \to 1} f(x)$$ exists
3. $$\displaystyle \lim_{x \to 5} f(x) = f(5)$$
4. $$\displaystyle \lim_{x \to 1} f(x) = f(1)$$
5. $$\displaystyle \frac{f(5) - f(1)}{5 - 1} = f'(c)$$
2. Solution A
3. How many points of inflection are there for the function $$y = x + \cos{ (2x)}$$ on the interval $$[0, \pi]$$?
1. 0
2. 1
3. 2
4. 3
5. 4
4. Solution C
5. The graph of $$f(x)$$ consists of two segments, one joining the origin to the point (1, 1), and a second joining the point (1, 1) to the point (2, 1). If $$F'(x) = f(x)$$, and $$F(0) = -3$$, then $$F(2) =$$
1. -4.5
2. -1.5
3. 1.5
4. 3
5. 4.5
6. Solution B
7. If $$\displaystyle \lim_{h \to 0} \frac{f(3 + h) - f(3)}{h} = 0$$, then which of the following must be true?
1. $$f$$ has a derivative at $$x = 3$$
2. $$f$$ is continuous at $$x = 3$$
3. $$f$$ has a critical value at $$x = 3$$
1. I only
2. II only
3. I and II
4. I and III
5. I, II, and III
8. Solution E
9. Consider the function $$y = x^3 - x^2 -1$$. For what value(s) of $$x$$ is the slope of the tangent equal to 5?
1. -1 only
2. $$\displaystyle \frac{5}{3}$$ only
3. -1 and $$\displaystyle \frac{5}{3}$$
4. $$\displaystyle \frac{1}{3}$$
5. 2.219
10. Solution C
11. A pebble thrown into a pond creates circular ripples such that the rate of change of the circumference is $$12\pi$$ cm/sec. How fast is the area of the ripple changing when the radius is 3 cm?
1. $$6\pi \frac{\text{cm}^2}{\text{sec}}$$
2. $$2\pi \frac{\text{cm}^2}{\text{sec}}$$
3. $$12\pi \frac{\text{cm}^2}{\text{sec}}$$
4. $$36\pi \frac{\text{cm}^2}{\text{sec}}$$
5. $$6 \frac{\text{cm}^2}{\text{sec}}$$
12. Solution D
13. If $$y = x^2 + 1$$, what is the smallest positive value of $$x$$ such that $$\sin {y}$$ is a relative maximum?
1. 0.756
2. 0.841
3. 1
4. 1.463
5. 1.927
14. Solution A
15. Find the area in the first quadrant bounded by $$y = 2 \cos{ x}, y = 3 \tan {x}$$, and the y-axis.
1. 0.347
2. 0.374
3. 0.432
4. 0.568
5. 1.040
16. Solution D
17. $$f'(x) = x^3(x - 2)^4(x - 3)^2. f(x)$$ has a relative maximum at
1. 0
2. 2
3. 2 and 3
4. 0 and 3
5. There is no relative maximum.
18. Solution E
19. Find the average rate of change of $$f(x) = \sec {x}$$ on the interval $$[0, \frac{\pi}{3}]$$
1. 0.396
2. 0.955
3. 1.350
4. 1.910
5. undefined
20. Solution B
21. $$\displaystyle a(t) = \frac{5t^2+ 1}{5t}$$ and $$v(1) = 1$$. Find $$v(2)$$
1. 1.139
2. 2.10
3. 2.139
4. 2.639
5. undefined
22. Solution D
23. Use the table shown to approximate the area under the curve of $$y = f(x)$$ using trapezoids:
$$x$$ $$y$$
0
1
1
2
3
4
4
1
1. 5.5
2. 8
3. 10
4. 11
5. 20
24. Solution C
25. Given the graph of $$y = f(x)$$ below, which of the following statements is true?
1. $$\displaystyle \lim_{x \to 1} f(x) = 3$$
2. $$\displaystyle \lim_{x \to 1^+} f(x) = 3$$
3. $$f'(1) = 3$$
4. $$f(1) = 3$$
5. The average rate of change of $$f(x)$$ on [1, 3] is $$f'(2)$$
26. Solution E
27. The position of a particle on a line is given by $$x(t) = t^3 - t, t \geq 0$$. Find the distance traveled by the particle in the first two seconds.
1. 0.385
2. 3.385
3. 6
4. 6.385
5. 6.770
28. Solution E
29. $$\displaystyle \int_a^b \left| f(x) \right| \, dx = p$$ and $$\left| \int_a^b f(x) \, dx \right| = q$$. Which of the following must be true?
1. $$p = q$$
2. $$p\geq q$$
3. $$p \leq q$$
4. $$p > q$$
5. $$p < q$$
30. Solution B