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]

________________________________________________________________________________

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]

__________________________________________________________________________________

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

__________________________________________________________________________________________

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]

_________________________________________________________________________________________

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]

_________________________________________________________________________________________

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.

_________________________________________________________________________________________

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

_______________________________________________________________________________________

[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

__________________________________________________________________________________________

27. If [pic]

(A) [pic] (B) [pic] (C) [pic] (D) 1 (E) [pic]

__________________________________________________________________________________________

28. Find [pic] given [pic]

(A) [pic] (B) [pic] (C) [pic] (D) [pic]

__________________________________________________________________________________________

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])

__________________________________________________________________________________________

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].

__________________________________________________________________________________________

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.

_________________________________________________________________________________________

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.

_________________________________________________________________________________________

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]

_________________________________________________________________________________

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.

_________________________________________________________________________________

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

__________________________________________________________________________________________

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]

__________________________________________________________________________________

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

................
................

In order to avoid copyright disputes, this page is only a partial summary.

Google Online Preview   Download