AP Calculus AB Summer Assignment

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Name______________ AP Calculus AB Summer Assignment

Mr. Linville DUE: First day of class

Dear AP Calculus AB Students,

Let me say again that I am very excited to be working with everyone this year! This assignment is meant to allow you to practice some of the prerequisite skills that will be necessary to be successful this year in AP Calculus AB. A deep understanding of these prerequisite skills is essential. We will not have much time in class to review these skills, as we will have to move through the curriculum quickly to be ready for the AP Exam. You should be familiar with all of the topics here, but we all need refreshers and review if you haven’t seen something for a while. I encourage you to use any resources necessary to answer the questions in the packet. You can use old notes, work together with others in your class or find resources online. There will be a quiz reflecting many of these topics within the first two weeks of school.

For some sections of this packet, you SHOULD NOT use a calculator if the directions tell you not to. You will need a graphing calculator for other sections of the packet. I recommend a TI-84 Plus for this course, as that is what I will use as a model in class, but other graphing calculators will also work. You will need to proficient in solving problems without a calculator, as some sections of the AP Exam will not allow the use of a calculator. As a result, a calculator will also not be allowed for many of the exams and quizzes this year.

I will collect this assignment on the first day of class. Make sure that you show all required work to support your solutions. You may want to attach loose-leaf paper (with the numbers of your work clearly label) if you need additional space.

--Mr. Linville

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Name: ______________________________________

Show all work – no credit will be awarded for answers missing appropriate work. No calculators!

Section I: Algebra Review (ln is the natural log function)

1. Solve xy + 2x + 1 = y for y. 2. Factor:

3. Solve ln (y – 1) – ln 2 = x + ln x for y 4. Factor:

Simplify each expression.

5. 6.

7. 8.

€

x²(x −1) − 4(x −1)

3x^3/2− 9x^1/2+ 6x^−1/2

€

(x²)³x x⁷

€

x ⋅ x³ ⋅ x

1 6

€

5(x + h)²− 5x² h

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1 x+ 4

x² 3 − 1

x

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Simplify, using factoring of binomial expressions. Leave answers in factored form.

Example:

9.

10.

Simplify by rationalizing the numerator.

Example:

11. 12.

[ ]

) 3 4 )(

1 (

) 1 )(

6 (

) 6 )(

3 4 ( ) 1 (

) 1 )(

6 (

) 18 21 4

( ) 1 (

) 1 )(

6 (

) 9 16 9 5 4 ( ) 1 (

) 1 )(

6 (

) 9 16 ( ) 9 4 )(

1 ( ) 1 ( )

1 )(

6 (

) 1 )(

9 16 ( ) 9 4 ( ) 1 (

2 2 2

2 2

3

+ +

=

+ -

- +

= +

+ -

- -

= +

+ -

- - - -

= +

+ -

+ - - +

= + +

-

+ +

= - +

x x

x x x

x x

x

x x

x x x

x

€

(x −1)³(2x − 3) − (2x +12)(x −1)²

€

(x −1)²(3x −1) − 2(x −1) (x −1)⁴

€

x + 4 − 2

x = x + 4 − 2

x • x + 4 + 2

x + 4 + 2 = x + 4 − 4 x

(

x + 4 + 2

)

⁼

x

(

x + 4 + 2

)

⁼

1 x + 4 + 2

€

x + 9 − 3 x

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x + h − x h

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Solve each equation or inequality for x over the set of real numbers.

13. 14.

15. 16.

17. [Your answer should be interval(s)] 18.

Solve each of the systems algebraically.

€

2x⁴ + 3x³− 2x² = 0

€

2x − 7 x +1 = 2x

x + 4

€

x²− 9 = x −1

€

2x − 3 = 14

€

x²− 2x − 8 < 0

€

3x + 5

(x −1)(x⁴+ 7) = 0

€

19. x + y = 8 ____________________ 20. y = x² − 3x _______________________

2x − y = 7 y = 2x − 6

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Section II: Trigonometry Review

Fill in the unit circle bellow for multiples of 𝝅/6, 𝝅/4, 𝝅/3 and 𝝅/2. (or 30°, 45°, 60° and 90°). Note:

you will be required to fill out this circle for a quiz early in the year.

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Use your knowledge of the unit circle to evaluate each of the following. You MUST know your unit circle.

Leave your answers in radical form. DON’T use your Calculator.

Solve each trigonometric equation for .

21. sin(30^) _____ 22. cos2π

3 _____ 23. tan 45^_____

24. sin −π 6

"

#$ %

&

' _____ 25. tan π ______ 26. csc5π

6 _____

27. cos 90

( )

^ ^_____ ^{28. cos}^3π₄ ^_____ ^{29. tan}^π₆ ^_____

30. cos⁻¹ 1 2

"

#$ %

&

' _____ 31. sin⁻¹ 2

2

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' _____ 32. tan⁻¹(1) _____

p 2 0£ x£

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33. sin x = 3

2 ____________ 34. tan²x = 1 ___________

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35. cosx 2 = 2

2 ____________ 36. 2sin²x + sin x −1 = 0 ____________

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Solve each exponential or logarithmic equation. DON’T use your Calculator. Leave some of your answers in terms of e.

Expand each of the following using the laws of logs.

€

37. 5^x = 125 __________ 38. 8^{x +1} = 16^x __________ 39. 81

3

4 = x _________

40. 8

−2

3 = x __________ 41. ln x = 2 __________ 42. ln e²= x __________

43. ln x = 3 __________ 44. − 3+ 2 ln x = 0 ____________

€

45. log₃5x²________________________ 46. ln5x

y² _______________________

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Section III: Graphing Review

I. Symmetry – Even and Odd Functions

Quick Review

Even Function

Symmetric about the y axis for all x

Example:

Odd Function

Symmetric about the origin (equivalent to a rotation of 180

degrees)

for all x

Example:

To determine algebraically if a function is even, odd, or neither, find and determine if it is equal to , , or neither.

Example: Determine if is even or odd.

Therefore, is an odd function.

Determine if the following functions are even, odd, or neither.

47. 48.

49. 50.

( ) ( )

f - =x f x

y x= 2

( ) ( )

f - = -x f x

y x= 3

( )

f -x

( )

f x ^-^{f x}( )

( )

₂⁴

1 f x x

= x +

( ) ( )

( )

⁴ ₂ ₂⁴ ₂⁴

( )

1 1

1

x x x

f x f x

x x

x

- -

- = = = - = -

+ +

- + ^{f x}( )

( )

₄ ²

3 f x x

= x

+

( )

1 f x x

= x +

( ) ^{1 3} ² ³ ⁴

f x = + x + x ^{f x}( )^{= +}^{1 3}^x³⁺³^x⁵

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-2π -π π 2π

II. Essential Graphs

Sketch each graph. You should know the graphs of these functions.

51. 52. 53.

54. 55. 56.

57. f(x) = tanx

58. For each graph above, state the domain, range, x-intercept(s) (some functions will have infinite x- intercepts, but you should write your solution in a way that states all x-intercepts) and y-intercept (if three is one).

( )

f x = x

-4 -2 2 4 4

2

-2

-4

( ) ³

f x =x

-4 -2 2 4 4

2

-2

-4

( ) ^sin

f x = x

-2π -π π 2π

( ) ^x

f x =e

-4 -2 2 4 4

2

-2

-4

( ) ^ln( )

f x = x

-4 -2 2 4 4

2

-2

-4

( ) ^cos

f x = x

-2π -π π 2π

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Section IV: Linear equations

Point-slope form: y – y1 = m(x – x1) Example m = 2/3 P (2, -7) Equation: y + 7 = (2/3)(x – 2) Write the equation for the line in both forms given a slope and a point.

60. m = 2/3 and P(3,5) 61. m = -4/5 and P(1,2)

Point-Slope: Point-Slope:

Slope-Intercept: Slope-Intercept:

Write the equation for the line in both forms given two points.

62. P(2,2) and Q(4,2) 63. P(3,-2) and Q(3,7)

Point-Slope: Point-Slope:

Slope-Intercept: Slope-Intercept:

Section V: Functions and Models

64.

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65. The graph of y = f(x) is given and labeled as f below. Use your knowledge of transformations of functions to match each equation with the appropriate graph. Give a brief explanation for each answer.