Dear BC Calculus Student: This material is the A Calculus ......AP Calculus BC Summer 2019 Dear BC...

AP Calculus BC Summer 2019

Dear BC Calculus Student:

To be successful in AP Calculus BC, you must be proficient at every skill in this packet. This is a review of Limits, Derivatives, and Applications of Derivatives. This material is the “A” part of AP Calculus AB. Upon returning to school in the fall, you are expected to have completed this assignment, but it is not required. We will spend 3-4 classes reviewing the information in this packet, and then you will be tested on the content.

Calculators should not be used on this assignment unless specifically stated. Furthermore, you are expected to have memorized the formulas on the sheet attached. While 2/3 of the AP Calculus test is non-calculator, a calculator is required on 1/3 of the test. Therefore, it is highly recommended that you have a TI-84 when the school-year begins, since we will not have a full classroom set.

AP Calculus is a fast-paced course that is taught at the college-level. Therefore, you must maintain all pre-requisite skills. We will not have time to spend re-teaching the content in this packet. Make sure you are comfortable with this material.

Good luck!

Ms. Loy and Ms. Fitz

Formulas that must be memorized:

Position, Velocity, Acceleration

Speed is increasing when v(t) and a(t) have the same signs.

Speed is decreasing when v(t) and a(t) have different signs.

Section I: Limits

Strategies for finding limits.

Try direct substitution. Try to factor & cancel, then use direct substitution. Use a limit formula. Graph or make a table.

Find each limit algebraically. Do NOT use a calculator.

1. lim𝑥→∞

2𝑥2+5

3𝑥2−4

2. lim𝑥→0

√3𝑥+2−√2

𝑥

3. lim𝑥→

1

3

3𝑥2−7𝑥+2

−6𝑥2+5𝑥−1

4. lim𝑥→−1

𝑓(𝑥) = 𝑥2 − 4𝑥 − 12, 𝑥 < −1

𝑥 − 6, 𝑥 ≥ −1

5. limℎ→0

𝑓(𝑥+ℎ)−𝑓(𝑥)

ℎ 𝑓𝑜𝑟 𝑓(𝑥) = −𝑥2 + 3𝑥

6. lim𝑥→2.3

⟦𝑥⟧

7. lim𝑥→𝜋

sin 𝑥

8. Let 𝑓 be the function defined by the following:

cos 𝑥, 𝑥 < 0 f(x)= 𝑥3, 0 ≤ 𝑥 < 1 2 − 𝑥, 1 ≤ 𝑥 < 2 𝑥 − 4, 𝑥 ≥ 2 For what values of x is 𝑓 NOT continuous? A) 0 only B) 1 only C) 2 only D) 0 and 2 only E) 0, 1, and 2 9. For 𝑥 ≥ 0, the horizontal line 𝑦 = 2 is an asymptote for the graph of the function 𝑓. Which of the following statements must be true? A) 𝑓(0) = 2 B) 𝑓(𝑥) ≠ 2 𝑓𝑜𝑟 𝑎𝑙𝑙 𝑥 ≥ 0 C) 𝑓(2) 𝑖𝑠 𝑢𝑛𝑑𝑒𝑓𝑖𝑛𝑒𝑑 D) lim

𝑥→2𝑓(𝑥) = ∞

E) lim𝑥→∞

𝑓(𝑥) = 2

True or False. If false, explain why or give an example that shows it false. 10. If 𝑓(𝑐) = 𝐿, then lim

𝑥→𝑐𝑓(𝑥) = 𝐿.

11. If 𝑓 is undefined at 𝑥 = 𝑐, then the limit as x approaches c does not exist. 12. If 𝑓 is continuous on [2,3], 𝑓(2) > 0, and 𝑓(3) < 0, then there must be a point c in (2,3) such that 𝑓(𝑐) = 0.

Evaluate.

13. lim𝑥→0

𝑥+tan 𝑥

sin 𝑥 14. lim

𝑥→0

sin(5𝑥)

2𝑥 15. lim

𝜃→0

1−cos 𝜃

2𝑠𝑖𝑛2𝜃

16. lim𝜃→0

𝜃 sec 𝜃 17. lim𝜃→0

cos 𝜃 tan 𝜃

𝜃 18. lim

𝑥→0

3(1−cos 𝑥)

𝑥

19. lim𝑥→0

tan2 𝑥

𝑥 20. lim

𝑡→0

sin 3𝑡

𝑡 21. lim

𝑥→2

𝑥−2

|𝑥−2|

22. lim𝑥→0

√2+𝑥−√2

𝑥 23. lim

𝑥→16

4−√𝑥

𝑥−16 24. lim

𝑥→𝜋

4

1−tan 𝑥

sin 𝑥−cos 𝑥

1

2

D C B F E A 0

3

Use the graph to answer # 25-32. 25. )(lim xf

Ax

26. )(lim xfBx

27. )(lim xfCx

28. )(lim xfDx

29. )(lim xfBx

30. )(lim xfEx

31. )(Ef 32. )(Af

33. lim𝑥→∞

2𝑥+5

3𝑥2+1 34. lim

𝑥→∞

2𝑥3+5

3𝑥2+1

35. lim𝑥→∞

sin 𝑥 36. lim𝑥→∞

sin 𝑥

𝑥 (Squeeze Thm)

37. lim𝑥→∞

𝑎−𝑏𝑥4

𝑐𝑥4+𝑥2 38. lim

𝑥→∞(5 −

2

𝑥2)

39. lim𝑥→∞

3𝑥−2

√2𝑥2+1 40. lim

𝑥→1

ln 𝑥

𝑥

Section II. Derivatives.

41. Use the limit definition of the derivative to find the derivative of 𝑓(𝑥) = −3𝑥2 + 5𝑥 + 4.

42. Find limℎ→0

𝑠𝑖𝑛(3𝜋

4+ℎ)−𝑠𝑖𝑛(

3𝜋

4)

ℎ

43. Given 𝑓(𝑥) = 𝑥3 − 8 , 𝑥 < 2

3𝑥2 − 12 , 𝑥 ≥ 2 is 𝑓 continuous at 𝑥 = 2? Is it differentiable at 𝑥 = 2?

44. Write an equation of the tangent line to the graph of 𝑓(𝑥) = −3𝑥2 + 5𝑥 − 2 at 𝑥 = 3. Find each derivative.

45. 𝑦 = 2𝑥 − 𝑥2 sin 𝑥 46. 𝑦 = 2𝑥√3𝑥 − 4 47. 𝑦 =2𝑥3+3𝑥

𝑥2−2

48. 𝑦 = 3(4 − 5𝑥2)3 49. 𝑦 = 5𝑐𝑜𝑠3(2𝑥4) 50. 𝑦 = (2𝑥 − 4)3(𝑥2 + 1)

51. 𝑦 =2𝑥3−2√𝑥+1

√𝑥3 52. 𝑦 = sin (cos 𝑥2) 53. 𝑦 =

1

√2𝑥+7

54. Find the points at which the function has horizontal and vertical tangent lines. 𝑓(𝑥) =2𝑥−4

𝑥2−3𝑥

55. Find the average rate of change of 𝑓(𝑥) = 3𝑥2 + 4𝑥 − 2 on the interval [0, 3].

56. Find the instantaneous rate of change of 𝑓(𝑥) = 3𝑥2 + 4𝑥 − 2 at 𝑥 =3

2.

57. Find 𝑑𝑦

𝑑𝑥. 𝑥3 − 2𝑥𝑦2 + 3𝑦3 = 12.

58. Write the equation of the tangent line to the curve 2𝑦2 − 3𝑥3 = 8 at (2, −4). 59. 60. 61. 62. 63.

64. 65.

66.

The table below gives both the function values and derivative values for three functions.

𝑥 𝑓(𝑥) 𝑓′(𝑥) 𝑔(𝑥) 𝑔′(𝑥) ℎ(𝑥) ℎ′(𝑥)

0 2 𝟏

𝟐

1 𝟏

𝟑

5 0

1 3 4 2 −𝟎. 𝟕𝟓 −𝟏 −𝟏

2 5 −

𝟏

𝟐

3 4 1 1

3 1 3 5 −𝟑. 𝟐 0 −𝟐

4 7 9 −𝟒 7 4 𝟏

𝟐

Use the chart to complete each of the following. Show all work.

67. 𝐹′(1) when 𝐹(𝑥) = 4𝑓(𝑥) + 3𝑔(𝑥) 68. 𝐹′(2) when 𝐹(𝑥) = 𝑓(𝑥)𝑔(𝑥)

69. 𝐹′(0) when 𝐹(𝑥) =𝑓(𝑥)

𝑔(𝑥) 70. 𝐹′(2) when 𝐹(𝑥) = 𝑓(𝑔(𝑥))

71. 𝐹′(1) when 𝐹(𝑥) = 𝑓(𝑥2)[𝑔(𝑥)]2 72. 𝐹′(2) when 𝐹(𝑥) = 𝑓(𝑔(ℎ(𝑥)))

AP-Style Multiple Choice Practice Part 1: No Calculator

1. Suppose 𝑓(𝑥) = 𝑥4 + 𝑎𝑥2 . What is the value of 𝑎 if 𝑓 has a local minimum at 𝑥 = 2?

A) −24 B) −8 C) −4 D) −1

2 E) −

1

6

2. Suppose 𝑔(𝑥) = 𝑥3 + 3𝑥2 − 2𝑥 𝑖𝑓 𝑥 ≤ 0

−𝑥2 𝑖𝑓 𝑥 > 0 . Then

A) 𝑔 is continuous and differentiable at 𝑥 = 0

B) 𝑔 is continuous but not differentiable at 𝑥 = 0

C) 𝑔 is not continuous but is differentiable at 𝑥 = 0

D) 𝑔 is not continuous and not differentiable at 𝑥 = 0

E) Nothing can be said about the differentiability of 𝑔 at 𝑥 = 0

3. Which of the following is an equation for the tangent line to the graph of 𝑓(𝑥) = sin(cos 𝑥)

at =𝜋

2 ?

A) 𝑦 = 𝑥 −𝜋

2 B) 𝑦 = −𝑥 +

𝜋

2 C) 𝑦 =

𝜋

2

D) 𝑦 = sin 1 (𝑥 −𝜋

2) E) 𝑦 = cos 1 (𝑥 −

𝜋

2)

4. If 𝑓(𝑥) = sec (4𝑥) , then 𝑓′ (𝜋

16)=

A) 4√2 B) √2 C) 0 D) 1

√2 E)

4

√2

5. Let 𝑓 be the function with derivative given by 𝑓′(𝑥) = 𝑥2 −2

𝑥 . On which of the following intervals is

𝑓 decreasing?

A) (−∞, −1] B) (−∞, 0) C) [−1, 0) D) (0, √23

] E) [√23

, ∞)

6. The derivative of 𝑓 is 𝑥4(𝑥 − 2)(𝑥 + 3). At how many points will the graph of 𝑓 have a relative maximum?

A) None B) One C) Two D) Three E) Four

7. What is the derivative of 𝑓(𝑡) = sec √𝑡 ?

A) 𝑡𝑎𝑛2√𝑡 B) 𝑠𝑒𝑐1

2√𝑡 𝑡𝑎𝑛

1

2√𝑡 C)

𝑠𝑒𝑐√𝑡 𝑡𝑎𝑛√𝑡

2√𝑡 D) 𝑠𝑒𝑐√𝑡 𝑡𝑎𝑛√𝑡 E) None of these

8. Given 𝑓(𝒙) = 𝟗 −𝟏𝟒

𝒙 , find all 𝑐 in the interval (2 , 7) such that 𝑓′(𝑐) is parallel to the secant line as

guaranteed by the Mean Value Theorem.

A) 9

2 B)

√6

2 C) √14 D)

14

9 E) None of these

9. Evaluate the lim𝑥→0

𝑥

sin 3𝑥 .

A) 0 B) 1

3 C) 1 D) 3 E) Nonexistent

10. Let 𝑓(𝑥) = √𝑥 . What is the equation of the tangent line to 𝑓 at the point (4 , 2) ?

A) 𝑦 =1

4𝑥 + 1 B) 𝑦 = −

1

2𝑥 + 3 C) 𝑦 =

1

2𝑥 D) 𝑦 = 2𝑥 − 6 E) None of these

11. If lim𝑥→𝑐

𝑓(𝑥) = −6 and lim𝑥→𝑐

𝑔(𝑥) = 3 , find lim𝑥→𝑐

([𝑓(𝑥)]2 − 2𝑓(𝑥)𝑔(𝑥) + [𝑔(𝑥)]2).

A) −9 B) 45 C) 63 D) 81 E) None of these

12. Which of the following is not true given 𝑓(𝑥) = 1 + 𝑥2/3 ?

A) 𝑓 is continuous for all real numbers

B) 𝑓 has a minimum at 𝑥 = 0

C) 𝑓 is increasing for 𝑥 > 0

D) 𝑓′(𝑥) exists for all 𝑥

E) 𝑓′′(𝑥) is negative for 𝑥 > 0

13. If 𝑓(𝑥) = cos 𝑥 sin 3𝑥 , then 𝑓′ (𝜋

6) =

A) 1

2 B) −

√3

2 C) 0 D) 1 E) −

1

2

14. Rolle’s Theorem does not apply to 𝑓(𝑥) = |𝑥 − 3| on [1, 4] because of which of the following reasons? I. 𝑓(𝑥) is not continuous on [1, 4]

II. 𝑓(𝑥) is not differentiable on (1, 4)

III. 𝑓(1) ≠ 𝑓(4)

A) I only B) II only C) III only D) I and II E) II and III

15. If ℎ(𝑥) = 𝑓2(𝑥) − 𝑔2(𝑥) and 𝑓′(𝑥) = −𝑔(𝑥) , then ℎ′(𝑥) =

A) 0 B) 2 ( )( '( ) '( ))g x f x g x C) 4 ( ) ( )f x g x D) 2 2

( ) ( )g x f x E) 2 ( )( ( ) '( ))g x f x g x

Free Response-No Calculator

1.

Let 𝑓 be a function that has domain the closed interval [−1, 4] and range the closed interval [−1, 2]. Let 𝑓(−1) = −1, 𝑓(0) = 0, and 𝑓(4) = 1. Also let 𝑓 have the derivative function 𝑓′ that is continuous and shown in the figure above.

a) Find all values of 𝑥 for which 𝑓 assumes a relative maximum. Justify your answer.

b) Find all values of 𝑥 for which 𝑓 assumes an absolute minimum. Justify your answer.

c) Find the intervals on which 𝑓 is concave downward.

d) Give all the values of 𝑥 for which 𝑓 has a point of inflection.

e) On the axes provided, sketch the graph of 𝑓.

2. Consider the curve given by 𝑦2 = 2 + 𝑥𝑦.

a) Find 𝑑𝑦

𝑑𝑥.

b) Find all points (x , y) on the curve where the line tangent to the curve has a slope 1

2 .

c) Show that there are no points (x , y) on the curve where the line tangent to the curve is horizontal.

d) Let 𝑥 and 𝑦 be functions of time 𝑡 that are related by the equation 𝑦2 = 2 + 𝑥𝑦. At the time 𝑡 = 5,

the value of 𝑦 is 3 and 𝑑𝑦

𝑑𝑡= 6. Find the value of

𝑑𝑥

𝑑𝑡 at time 𝑡 = 5.



r h V r h

dzdt

dx dydt dt

x y dxdt



f x x f x c xc



b h

A

AAA b hA b hA

C

C

CC

C

C

AP Calculus AB Optimization Worksheet B Name_________________________________

Show all work on separate paper. Do not use a calculator unless specifically stated.

1.

2.

3. Which point on the graph of 𝑦𝑦 = √𝑥𝑥 is closest to the point (5, 0)?

4. A contractor is building an additional room that is 800 𝑓𝑓𝑓𝑓2 to an existing house. Therefore only 3 walls need to be built. What dimensions will require the least amount of building materials?

5. Find the largest possible area for a rectangle with its base on the x-axis and upper vertices on the curve 𝑦𝑦 = 4 − 𝑥𝑥2.

6. Calculator OK.

Squares of equal size are cut off the corners of an 8x10 piece of carboard. The sides are then turned up to form an open box. Find the largest possible volume of the box.

7. Calculator OK.

Find the shortest distance from (3, 0) to a point on the curve 𝑦𝑦 = 𝑥𝑥2 − 2𝑥𝑥.

8. Calculator OK.

You are to make a one quart oil can in the shape of a right circular cylinder. What dimensions (radius and height) will use the least material? (1 quart of oil=57.75 in3)

9.

Dear BC Calculus Student: This material is the A Calculus ......AP Calculus BC Summer 2019 Dear BC...

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Transcript of Dear BC Calculus Student: This material is the A Calculus ......AP Calculus BC Summer 2019 Dear BC...