CALCULUS I



Mrs.Volynskaya AP Calculus Name: ___________________

Worksheet L.1-2

Refer to the graph to find each limit, if it exists:

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

1. 2.

a. _____ b. _____ c. _____ a. _____ b. _____ c. _____

d. _____ e. _____ f. _____ d. _____ e. _____ f. _____

3. 4.

a. _____ b. _____ c. _____ a. _____ b. _____ c. _____

d. _____ e. _____ f. _____ d. _____ e. _____ f. _____

5. 6.

a. _____ b. _____ c. _____ a. _____ b. _____ c. _____

d. _____ e. _____ f. _____ d. _____ e. _____ f. _____

7. 8. 9.

a. [pic]= _____ a. [pic]= _____ a. [pic]= _____

b. [pic]= _____ b. [pic]= _____ b. [pic]= _____

c. f(2) = _____ c. [pic]= _____

d. f(1) = _____

10. True or false?

_____ a. [pic]= -1

_____ b. [pic]= 1

_____ c. [pic]= 1

_____ d. [pic]exists

_____ e. [pic]= 1

_____ f. [pic] DNE

_____ g. [pic]= 1

_____ h. [pic]= [pic]

_____ i. [pic]exists

_____ j. [pic]= 1

_____ k. [pic]exists at every c on the interval (-1,1)

_____ l. [pic]exists at every c on the interval (1,3)

Calculus Name: _____________________________

Worksheet L.1-2

Based on the graph evaluate the following.

1. [pic]= _____ 11. [pic]= _____

2. [pic]= _____ 12. [pic]= _____

3. [pic]= _____ 13. [pic]= _____

4. [pic]= _____ 14. f(6) = _____

5. [pic]= _____ 15. [pic]= _____

6. [pic]= _____ 16. f(3) = _____

7. [pic]= _____ 17. [pic]( _____

8. f(1) = _____ 18. f(−1) ( _____

9. f(0) = _____ 19. True or False: [pic]exists at every c on (1,3)

10. f(−2) = _____ 20. True or False: [pic]exists at every c on (−2,1)

Evaluate the following.

21. [pic] = _____ 22. [pic] = _____ 23. [pic] = _____ 24. [pic] = _____

25. [pic] = _____ 26. [pic] = _____

27. [pic] = _____ 28. [pic] = _____

29. [pic] = _____ 30. [pic] = _____

31. [pic] = _____ 32. [pic] = _____

33. [pic] = _____ 34. [pic] = _____

35. [pic] = _____ 36. [pic] = _____

37. [pic] = _____ 38. [pic] = _____

39. [pic] = _____ 40. [pic] = _____

41. [pic] = _____ 42. [pic] = _____

1 – 2x, x ( 1

43. [pic]f(x) =

x – 3, x ( 1

(a graph may help)

x + 2, x ( −1

44. [pic]f(x) =

x2, x ( 1

(a graph may help)

45. Suppose [pic]and [pic]find the [pic]

Calculus Name: _____________________________

Worksheet L.2-1

Find the limits.

1. [pic] 2. [pic]

3. [pic] 4. [pic]

5. [pic] 6. [pic]

7. [pic] 8. [pic]

9. [pic] 10. [pic]

11. [pic] 12. [pic]

13. [pic] 14. [pic]

15. [pic] 16. [pic]

17. [pic] 18. [pic]

19. [pic] 20. [pic]

Calculus Name: _____________________________

Worksheet L.2-2

Evaluate the following.

1. [pic] = 2. [pic] = 3. [pic]=

4. [pic]= 5. [pic]= 6. [pic]=

7. [pic]= 8. [pic]= 9. [pic] =

10. [pic]= 11. [pic]= 12. [pic]=

13. [pic]= 14.[pic]= 15. [pic]=

16. [pic]= 17. [pic]= 18. [pic]=

19. [pic]= 20. [pic]= 21. [pic]=

22. [pic]= 23. [pic]= 24. [pic]=

25. [pic]= 26. [pic]= 27. [pic]=

28. [pic]= 29. [pic]= 30. [pic]=

31. [pic]= 32. [pic]= 33. [pic]=

34. [pic]= 35. [pic]= 36. [pic]=

37. [pic]= 38. [pic]= 39. [pic]=

40. [pic]= 41. [pic]= 42. [pic]=

43. [pic]= 44. [pic] = 45. [pic]=

Calculus Name: _____________________________

Worksheet L.3

Evaluate the following.

1. [pic] 2. [pic]

3. [pic] 4. [pic]

5. [pic] 6. [pic]

7. [pic] 8. [pic]

9. [pic] 10. [pic]

11. [pic] 12. [pic]

13. [pic] 14. [pic]

15. [pic] 16. [pic]

Calculus Name: _____________________________

Worksheet L.4-1

Refer to the graph to find each of the following:

a) the value(s) of x for which the function is discontinuous

b) why it is discontinuous at that value

c) the type of discontinuity

d) whether it is removable (R) or nonremovable (NR) discontinuity

1) 2) 3)

a) ______________ a) ______________ a) ______________

b) ______________ b) ______________ b) ______________

c) ______________ c) ______________ c) ______________

d) ______________ d) ______________ d) ______________

4) 5) 6)

a) ______________ a) ______________ a) ______________

b) ______________ b) ______________ b) ______________

c) ______________ c) ______________ c) ______________

d) ______________ d) ______________ d) ______________

Given the following graph, state for what values of x the function is discontinuous and state why it is discontinuous at that point. Also state what type of discontinuity it is and whether it is removable or nonremovable. Explain how any removable discontinuities should be defined or redefined to make the function continuous.

|Where? |Why? |Type |Removable (R) or Nonremovable |If R, what values makes it |

| | | |(NR) |continuous? |

| | | | | |

| | | | | |

| | | | | |

| | | | | |

Find the value of k that makes the function continuous.

[pic]

At what points, if any, is the function f(x) =[pic] discontinuous?

Calculus Name: _____________________________

Worksheet L.4-2

Draw a graph with the following conditions.

Function #1

□ f(0) = 0 ( [pic]

□ f(1) = 2 ( [pic]

□ f(-1) = -2

□ at f(3) there is a non-removable discontinuity

□ at f(-4) there is a removable discontinuity

Function #2

□ [pic] ( [pic]

□ [pic] ( [pic]

□ f(0) = 0

□ at f(-4) there is a removable discontinuity

□ [pic]exists, but the graph is discontinuous

Function #3

□ [pic] ( [pic]

□ [pic] ( [pic]

□ f(0) = 0 ( [pic]

□ [pic]exists, but the graph is discontinuous

Function #4

□ [pic] ( [pic]

□ [pic] ( [pic]

□ f(-1) = 0 ( f(2) = 1

□ [pic] does not exist

Function #5

□ f(-3) = 0 ( f(0) = 3

□ [pic] ( [pic]

□ [pic]

□ at f(5) there is a removable discontinuity

□ [pic] does not exist

Function #6

□ [pic] ( [pic]

□ [pic] ( [pic]

□ f(0) = 1

□ [pic]exists, but the graph is discontinuous

Function #7

□ f(0) = 0 ( [pic]

□ f(2) = 1 ( [pic]

□ f(-2) = -4

□ [pic] does not exist

□ [pic]exists, but the graph is discontinuous

Function #8

□ [pic] ( [pic]

□ [pic] ( [pic]

□ [pic] ( [pic]

□ f(0) = 2

□ at f(-5) there is a non-removable discontinuity

Calculus Name: _____________________________

Limits Practice Test

Refer to the graph to evaluate the following:

1. _______ [pic] 2. _______ [pic] 3. _______ [pic]

4. _______ [pic] 5. _______ [pic] 6. _______ [pic]

7. _______ [pic] 8. _______ [pic] 9. _______ [pic]

10. _______ [pic] 11. _______ [pic] 12. _______ [pic]

13. _______ [pic] 14. _______ [pic] 15. _______ [pic]

16. _______ [pic] 17. _______ [pic] 18. _______ [pic]

19. _______ True or False: [pic] exists for every c in the interval (2,10)

20. _______ True or False: [pic] exists for every c in the interval (−2,2)

Refer to the graph to find the following:

a) the value(s) of x for which the function is discontinuous

b) why it is discontinuous at that value

c) the type of discontinuity

d) whether it is a removable (R) or nonremovable (NR) discontinuity

e) if it is removable, how could you make the function continuous at that point?

| |Where? |Why? |Type? |Removable (R) or Nonremovable (NR)? |If R, what values make it |

| | | | | |continuous? |

|21. | | | | | |

|22. | | | | | |

|23. | | | | | |

|24. | | | | | |

Evaluate the following.

25. _______ [pic] 26. _______ [pic]

27. _______ [pic] 28. _______ [pic]

29. _______ [pic] 30. _______ [pic]

31. _______ [pic] 32. _______ [pic]

33. _______ [pic] 34. _______ [pic]

35. _______ [pic] 36. _______ [pic]

37. _______ [pic] 38. _______ [pic]

39. _______ [pic] 40. _______ [pic]

41. _______ [pic] 42. _______ [pic]

43. _______ [pic] 44. _______ [pic]

45. _______ [pic] 46. _______ [pic]

47. _______ [pic] 48. _______ [pic]

49. _______ [pic] 50. _______ [pic]

51. _______ [pic] 52. _______ [pic]

53. _______ [pic] 54. _______ [pic]

55. _______ [pic] 56. _______ [pic]

57. _______ [pic] 58. _______ [pic]

59. _______ [pic] 60. _______ [pic]

61. _______ [pic] 62. _______ [pic]

63. _______ [pic] 64. _______ [pic]

Find the points of discontinuity. If there are no such points, write “none”.

65. _______ [pic] 66. _______ [pic]

67. _______ [pic] 68. _______ [pic]

69. _______ [pic] 70. _______ [pic]

71. _______ [pic] 72. _______ [pic]

Find the values of a and k that make the function continuous.

[pic] [pic]

73. a = _______ k = _______ 74. a = _______ k = _______

75. Draw a graph with the following conditions:

( [pic]

( [pic]

( [pic]

( [pic]

( at f(−3) the limit exists but the graph is discontinuous

( at f(3) there is a non-removable discontinuity

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