Calculus - Georgetown ISD



AP Calculus BC Name ___________________________ Period _____

Fall Semester Review

Limits and Their Properties

This review provides you with a few problems over each topic we have covered. If you want to be fully prepared you should look back over your tests, quizzes, notes and homework assignments. Everything you have learned is fair game!! The exam will have 33 multiple choice questions on it. There will be a non-calculator and a calculator portion.

1. Use the graph of the function f(x) to answer the following questions.

m. State the x-values for all the discontinuities in the graph above. Classify each as removable or non-removable.

n. Clearly describe why the limit as x approaches 2 does not exist.

2. Evaluate the limit: [pic] 3. Evaluate the limit: [pic]

A. 5 B. 3 A. [pic] B. -[pic]

C. -3 D. -5 C. 1 D. 0

E. Limit does not exist E. Limit does not exist

4. Evaluate the limit: [pic] 5. Evaluate the limit: [pic]

A. [pic] B. -[pic] A. 0 B. DNE

C. 1 D. -1 C. [pic] D. 3 E. [pic]

E. Limit does not exist

6. Given the function: [pic]

what value of k will make this piecewise function continuous?

A. [pic] B. [pic] C. 0 D. [pic] E. [pic]

7. Identify the vertical asymptotes for [pic]

8. Find each of the following limits:

a. [pic] b. [pic] c. [pic] d. [pic]

9. [pic]

a. [pic] b. [pic]

c. What does this imply about [pic] ? Explain.

d. What is f(2)? Discuss the continuity of f(x).

10. If [pic], where L is a real number, which of the following must be true?

I. f(a) = L

II. [pic]

III. [pic]

A. I only B. I and II C. I and III D. II and III E. I, II and III

11. If the graph of [pic] has a horizontal asymptote y = -2, a vertical asymptote x = 4, and an x-intercept of 1.5, then a – b + c =

A. -3 B. 1 C. 5 D. -9 E. -1

Differentiation

1. Use the limit definition of the derivative[pic] to find the derivative of

f(x) = x2 + 2x

2. Find f’(5) if f(x) = x2 – 5x + 6

_____3. Determine all the numbers c which satisfy the conclusions of the Mean Value Theorem for the function [pic]on [-1, 2]

4. Find the derivative of the function [pic]. 5. Find [pic] if 3xy = 4x + y2

6. Find the second derivative of f(x) if 7. Find [pic] for y = 4sin2(3x)

f(x) = (2x + 3)4 A. 8sin(3x)

B. 24sin(3x)

C. 8sin(3x)cos(3x)

D. 12sin(3x)cos(3x)

E. 24sin(3x)cos(3x)

8. Find the derivative of each of the following. (Use separate paper if you need more space)

a. f(x) = x3 – 3x2 b. [pic] c. [pic] d. [pic]

e. [pic] f. f(x) = (3x2 + 7)(x2 – 2x + 3) g. [pic]

h. y = 3cos(3x + 1) i. y = 1 – cos(2x) + 2cos2(x) j. f(x) = csc(3x) + cot(3x)

k. [pic] l. [pic] m. [pic]

n. x2y3 – 3y = 5x o. [pic] p. [pic]

q. y = xe r. [pic] s. [pic] t. [pic]

9. The function f is differentiable for all real numbers. The point [pic] is on the graph of y = f(x), and the slope at each point (x,y) is given by [pic]. Find [pic] and evaluate it at the point [pic]

10. Suppose the function g is defined by [pic] where k and m are constants. If g is differentiable at x = 3, what are the values of k and m?

11. Suppose g(0) = 4, g’(0) = 8, and g”(0) = -12. If [pic], what is h”(0)?

A. -5 B. [pic] C. [pic] D. [pic] E. 1

12. Use the values listed in the chart to find the value of [pic]

when x = 2.

A. -8 B. -4 C. -4/3 D. 2/3 E. 8/3

____13. The equation of the tangent line to the function [pic] at x = 5 is

A. y = 3x + 27 B. y = x + 27

C. y = 3x + 17 D. y = 6x + 12

E. y = 6x + 2

_____14. If f(x) = 3 + |x – 2|, then f’(2) is

A. non existent B. 1 C. 2 D. 3 E. -1

_____15. A particle moves along the x-axis so that its position at time t is given by x(t) = t2 – 7t + 12. For what value of t is the velocity of the particle zero?

A. 2.5 B. 3 C. 3.5 D. 4 E. 4.5

16. In a right triangle ABC, point A is moving along a leg of the right triangle toward point C at a rate of [pic]cm/sec and point B is moving toward point C at a rate of [pic] cm/sec along a line containing the other leg of the right triangle as shown below. What is the rate of change in the area of ABC at the instant when

AC = 15 cm and BC = 20 cm?

17. The side of a cube is increasing at a constant rate of 0.2 cm/sec. In terms of the surface area S, what is the rate of change of the volume of the cube, in cubic centimeters per second?

18. The base of the right triangle pictured below is 8 cm and the angle [pic]is increasing at the constant rate of 0.03 radians/second. How fast, in cm/sec, is the altitude h of the triangle increasing when h = 13?

A. 0.458 cm/sec

B. 0.744 cm/sec

C. 0.874 cm/sec

D. 12.626 cm/sec

E. 29.125 cm/sec

Graphical Analysis

_____1. Which of the following equations has y = 1 as an asymptote?

A. y = cosx B. [pic] C. [pic] D. [pic] E. [pic]

2. Given the function [pic].

a. Find the zeros of f(x) b. Write the equation of any vertical asymptotes.

c. Write the equation of any horizontal asymptotes

3. We need to enclose a field with a fence.  We have 500 feet of fencing material and a building is on one side of the field and so won’t need any fencing for that side.  Determine the maximum area of the field that we can enclose. Show your calculus.

4. Find the point (x,y) on the graph of [pic] nearest to the point (4,0)

5. To the nearest whole number, over what interval(s) will the graph

of f’ have negative values?

A. [pic] B. [pic]

C. (0,2) D. [pic]

E. Never

6. A function f(x) exists such that f”(x) = (x – 2)2(x + 1). How many points of inflection does f(x) have?

A. None B. One C. Two D. Three E. Cannot be Determined

7. What is the x-coordinate of the point of inflection on the graph of [pic]

A. -4 B. -3 C. -1 D. [pic] E. 0

8. Is it possible for f’(x) to equal zero at a point x = m but f not have a local maximum or minimum at x = m? Explain?

9. Is it possible for f’(x) to fail to exist at x = a but f still have a maximum or minimum at x = a? Explain.

10. Without a calculator, find the absolute maximum value of [pic] on the interval [-1,4]. Justify your answer.

11. On the interval [-5,5], f is continuous and differentiable. If f’(x) = (x – 1)(2x + 1)(x + 3)2, briefly explain the reasons for the following conclusions.

a. There is a local maximum on f at x = -1/2

b. There is a horizontal tangent but no extrema at x = -3.

c. If f(2) = 7, then f(3) > 7

12. The function f is given by f(x) = -x6 + x3 – 2. On which of the following intervals is f decreasing?

A. [pic] B. [pic] C. [pic] D. [pic] E. [pic]

13. The graph of a twice differentiable function f is shown. Which of the following is true?

14. Which of the following is an equation of the line tangent to the graph of f(x) = x6 – x4 at the point where f’(x) = -1?

A. y = -x – 1.031 B. y = -x – 0.836

C. y = -x + 0.836 D. y = -x + 0.934

E. y = -x + 1.031

15. The graph of the function [pic] changes concavity at x =

A. 3.29 B. 2.21 C. 1.34

D. 0.41 E. -0.39

16. Use the graph of the derivative f’(x) to answer the following questions. (Estimate when necessary)

d. When is the graph of f(x) increasing/decreasing. Justify

e. When is the graph of f(x) concave up/down. Justify.

17. Find the area of the rectangle of largest area which can be inscribed in the closed region bounded by the x-axis, y-axis, and graph of y = 8 – x3.

18. A manufacturer of compact disc players can produce at most 100 players per week. Past experience shows that their weekly profit can be modeled by P(x) = -3.5x2 + 450x – 1000. How many players should they produce each week to maximize their profit?

Integration

1. [pic] 2. [pic]

3. [pic] 4. [pic]

5. [pic] 6. [pic]

7. [pic] 8. [pic]

9. [pic] 10. [pic]

-----------------------

a. f(-2) = b. f(0) = c. f(2) =

d. [pic] e. [pic] f. [pic]

g. [pic] h. [pic] i. [pic]

j. [pic] k. [pic] l. [pic]

|x |f(x) |g(x) |f’(x) |g’(x) |

|1 |3 |4 |2/3 |-5/2 |

|2 |4 |2 |4/3 |-3/2 |

|4 |8 |1 |8/3 |-1/2 |

C

B

A

z

h

8 cm

[pic]

f(x)

A. f(2) < f’(2) < f”(2) B. f(2) < f”(2) < f’(2)

C. f’(2) < f(2) < f”(2) D. f”(2) < f(2) < f’(2)

E. f”(2) < f’(2) < f(2)

2

f’(x)

a. State the critical values of f(x) and classify them as relative maximums, minimums or neither.

b. On the graph, box the places where f”(x) = 0. Are these points of inflection? How do you know?

c. Find the equation of the line tangent to f(x) at x = -2 if

f(-2) = 7.

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