CALCULUS BC
CALCULUS AB
REVIEW FOR FIRST SEMESTER EXAM
Work these on notebook paper. Do not use your calculator.
Find the limit.
1. [pic] 2. [pic]
[pic] 4. [pic]
[pic] 5. [pic]
6. [pic] 7. [pic]
8. [pic] 9. [pic]
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Find the derivative. Do not leave negative exponents or complex fractions in your answers.
10. [pic] 11. [pic]
12. [pic] 13. [pic]
14. [pic] 15. [pic]
16. [pic] 17. [pic]
18. [pic] 19. [pic]
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Multiple Choice. Show all work.
20. At x = 3, the function given by [pic] is
(A) undefined (D) neither continuous nor differentiable
(B) continuous but not differentiable (E) both continuous and differentiable
(C) differentiable but not continuous
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21. An equation of the line tangent to [pic] at its point of inflection is
(A) [pic] (B) [pic] (C) [pic] (D) [pic] (E) [pic]
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22. If [pic] then there exists a number c in the interval [pic] that satisfies the
conclusion of the Mean Value Theorem. Which of the following could be c?
(A) [pic] (B) [pic] (C) [pic] (D) [pic] (E) [pic]
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23. If [pic] which of the following is true?
(A) f is increasing for all x greater than 0. (D) f is decreasing for all x between 1 and e.
(B) f is increasing for all x greater than 1. (E) f is decreasing for all x greater than e.
(C) f is decreasing for all x between 0 and 1.
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24. Let f and g be functions that are differentiable everywhere. If g is the inverse function of f and
if [pic]
(A) 2 (B) [pic] (C) [pic] (D) [pic] (E) – 2
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[pic] Find [pic]
(A) 24 (B) [pic] (C) [pic] (D) [pic] (E) [pic]
26. [pic]
(A) [pic] (B) [pic] (C) 1 (D) [pic] (E) 2
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27. If [pic]
(A) [pic] (B) [pic] (C) [pic] (D) 1 (E) [pic]
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28. Find [pic] given [pic]
(A) [pic] (B) [pic] (C) [pic] (D) [pic]
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29. Find the values of x that give relative extrema for [pic].
(A) relative maximum: [pic]; relative minimum: x = 0 and x = 4
(B) relative maximum: x = 0 ; relative minimum: [pic]and x = 4
(C) relative maximum: [pic]and x = 4; relative minimum: x = 0
(D) relative maximum: x = 4; relative minimum: x = 0 and [pic]
Free Response.
30. A snowball is in the shape of a sphere. Its volume is increasing at a constant rate of
10 [pic] How fast is the radius increasing when the volume is [pic]?
(Volume of a sphere: [pic])
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31. Water runs out of a conical tank at the constant rate of 2 cubic feet per minute. The radius at the top of the
tank is 5 feet, and the height of the tank is 10 feet. How fast is the water level sinking when the water
is 4 feet deep? (Volume of a cone: [pic])
32. Consider the curve defined by [pic]
(a) Find [pic] in terms of x and y. (b) Evaluate [pic] at the point [pic]
(c) Write the equation of the tangent line to the curve [pic] at the point [pic].
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33. A particle moves along a horizontal line so that its position at any time is given by
[pic] where s is measured in meters and t in seconds.
(a) Find the velocity of the particle at t = 5 seconds.
(b) When is the particle moving to the right? Justify your answer.
(c) Find the acceleration of the particle at t = 5 seconds.
(d) Is the particle speeding up or slowing down t = 5 seconds? Give a reason for your answer.
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34. The graph of a function f consists of a semicircle and two line segments
as shown. Let g be the function given by [pic]
(a) Find [pic]
(b) Find all values of x on the open interval [pic] at which
g has a relative maximum. Justify your answers.
(c) Write an equation for the line tangent to the graph of g at x = 3. Graph of f
(d) Find the x-coordinate of each point of inflection of the graph of g on [pic].
Justify your answer.
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35. The rate at which water is being pumped into a tank is given by the increasing function [pic].
A table of selected values of [pic], for the time interval [pic] minutes, is shown below.
|t (min.) |0 |4 |9 |17 |20 |
|[pic] (gal/min) |25 |28 |33 |42 |46 |
(a) Use a left Riemann sum with four subintervals to approximate the value of [pic].
(b) Use a right Riemann sum with four subintervals to approximate the value of [pic].
(c) Use a trapezoidal sum with four subintervals to approximate the value of [pic].
Graph of f Graph of g
Use the graphs above for problems 51 –53.
36. If [pic] 37. If [pic]
38. If [pic] find[pic]
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39. Find [pic] in terms of x and y, given [pic]
40. Find [pic] in terms of x and y, given _
41. Show whether or not the conditions of the Mean Value Theorem are met. If the theorem applies,
find the value of c that the Mean Value Theorem guarantees. [pic]
42. Given [pic], find the intervals where f is increasing and decreasing, and
identify all points that are relative maximum and minimum points. Justify your answers.
43. Given [pic], find the intervals where f is concave up and concave down, and
find the inflection points. Justify your answer.
44. Suppose that [pic] is continuous on [pic]. If [pic], what can you
conclude about f by the Second Derivative Test?
45. Given [pic]. Use the Second Derivative Test to find whether f has a
local maximum or a local minimum at x = [pic] Justify your answer.
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46. The graph of a function f is shown on the right.
Fill in the chart with +, [pic], or 0.
|Point | f | [pic] |[pic] |
| A | | | |
| B | | | |
| C | | | |
47. A fish is reeled in at a rate of 2 ft/sec from a bridge that is 16 ft. above
the water. At what rate is the angle between the line and the water changing 16 line
when there are 20 ft of line out? ft
fish
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48. The graph of [pic] is shown on the right.
(a) On what interval(s) is the graph of f decreasing? Justify
your answer.
(b) For what value(s) of x does the graph of f have a local maximum?
Justify your answer in sentence.
(c) On what interval(s) is the graph of f concave upward?
Justify your answer.
(d) For what value(s) of x does the graph of f have an inflection
point? Justify your answer in sentence.
Graph of [pic]
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49. The graph of [pic] is shown on the right.
(a) On what interval(s) is the graph of f concave downward?
Justify your answer.
(b) At what value(s) of x does the graph of f have an
inflection point? Justify your answer in a sentence.
Graph of [pic]
-----------------------
A
C
B
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