Name______________ AP Calculus AB Summer Packet Mr. Linville DUE: First day of class Dear AP Calc 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 on May 4. 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. I may post some resources on Google Classroom throughout the summer as well. 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.

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x 2(x −1) − 4(x −1)

3x3/2 − 9x1/2 + 6x−1/2

€

(x 2)3 xx 7

€

x ⋅ x3 ⋅ x16

€

5(x + h)2 − 5x 2

h

€

1x

+4x 2

3 − 1x

3

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

9.

10.

Simplify by rationalizing the numerator. Example:

11. 12.

[ ]

)34)(1()1)(6(

)6)(34()1()1)(6(

)18214()1()1)(6(

)916954()1()1)(6(

)916()94)(1()1()1)(6(

)1)(916()94()1(

2

22

22

223

++=+-

-++=

+---+

=

+-----+

=

+-+--++

=+-

++=-+

xxxx

xxxxx

xxxxx

xxxxxx

xxxxxx

xxxx

€

(x −1)3(2x − 3) − (2x +12)(x −1)2

€

(x −1)2(3x −1) − 2(x −1)(x −1)4

€

x + 4 − 2x

=x + 4 − 2x

•x + 4 + 2x + 4 + 2

=x + 4 − 4

x x + 4 + 2( )=

xx x + 4 + 2( )

=1

x + 4 + 2

€

x + 9 − 3x

€

x + h − xh

4

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.

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2x 4 + 3x 3 − 2x 2 = 0

€

2x − 7x +1

=2xx + 4

€

x 2 − 9 = x −1

€

2x − 3 =14

€

x 2 − 2x − 8 < 0

€

3x + 5(x −1)(x 4 + 7)

= 0

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19. x + y = 8 ____________________ 20. y = x 2 − 3x _______________________2x − y = 7 y = 2x − 6

5

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.

6

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. cos 2π3_____ 23. tan45 _____

24. sin −π6

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#$

%

&' _____ 25. tanπ ______ 26. csc 5π

6_____

27. cos 90( ) _____ 28. cos 3π4_____ 29. tan π

6_____

30. cos−1 12"

#$%

&' _____ 31. sin−1 2

2

"

#$

%

&' _____ 32. tan−1(1) _____

p20 ££ x

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33. sin x =32____________ 34. tan2 x =1 ___________

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35. cosx2

=22____________ 36. 2sin2 x + sin x −1 = 0 ____________

7

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.

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37. 5x =125 __________ 38. 8x+1 =16x __________ 39. 8134 = x _________

40. 8−23 = x __________ 41. ln x = 2 __________ 42. lne2 = x __________

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

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45. log3 5x2 ________________________ 46. ln

5xy 2_______________________

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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.

( ) ( )f x f x- =

2y x=

( ) ( )f x f x- = -

3y x=

( )f x-

( )f x ( )f x-

( ) 241xf x

x=

+

( ) ( )( )

( )2 2 2

4 4 41 11

x x xf x f xx xx

- -- = = = - = -

+ +- +( )f x

( )2

4 3xf xx

=+

( )1xf xx

=+

9

49. 50. II. Essential Graphs Sketch each graph. You should know the graphs of these functions. 51.

52.

53.

54.

55.

56.

57. For each graph above, state the domain, range, x-intercept(s) and y-intercept(s).

( ) 2 41 3 3f x x x= + + ( ) 3 51 3 3f x x x= + +

( )f x x=

-4 -2 2 4

4

2

-2

-4

( ) 3f x x=

-4 -2 2 4

4

2

-2

-4

( ) sinf x x=

-2π -π π 2π

( ) xf x e=

-4 -2 2 4

4

2

-2

-4

( ) ( )lnf x x=

-4 -2 2 4

4

2

-2

-4

( ) cosf x x=

-2π -π π 2π

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Graphing Skill #1: You should be able to graph a function in a viewing window that shows the important features. You should be familiar with the built-in zoom options for setting the window such as zoom-decimal and zoom-standard. You should also be able to set the window conditions to values you choose. 58. Graph using the built-in zoom-decimal and zoom-standard options. Draw each. 59. Find the appropriate viewing window to see the intercepts and the vertex defined by

. Use the window editor to enter the x and y values. Window: Xmin = _______ Xmax = _______ Xscl = _______ Ymin = _______ Ymax = _______ Yscl = _______

Graphing Skill #2: You should be able to graph a function in a viewing window that shows the x-intercepts (also called roots and zeros). You should be able to accurately estimate the x-intercepts to 3 decimal places. Use the built-in root or zero command. [You should use your graphing calculator] 60. Find the x-intercepts of . Window [-4.7, 4.7] x [-3.1, 3.1] (Write intercepts as points)

x-intercepts: ______________

= -2 3y x

= - +2 11 10y x x

= - -2 1y x x

11

61. Find the x-intercepts of .

x-intercepts: ______________ Graphing Skill #3: You should be able to graph two functions in a viewing window that shows the intersection points. Sometimes it is impossible to see all points of intersection in the same viewing window. You should be able to accurately estimate the coordinates of the intersection points to 3 decimal places. Use the built-in intersection command. 62. Find the coordinates of the intersection points for the functions:

.

Intersection points: ______________ 63. Find the coordinates of the intersection points of: . Intersection points: ______________

Graphing Skill #4: You should be able to graph a function and estimate the local maximum or minimum values to 3 decimal places. Use the built-in max/min command. 64. Find the relative maximum and relative minimum values of the function . (Value means the y-value)

Minimum value: ______________

Maximum value: ______________

65. Find the relative maximum and relative minimum values of the function .

66. Find the x-intercepts, relative maximum, and relative minimum of

= - -3 2 1y x x

= + = - - +2( ) 3 ( ) 7f x x g x x x

= =2( ) 4 ( ) 2xf x x g x

= - -3 4 1y x x

= - +3 24 4y x x x

3 22 1y x x= + -

12

67. Find the coordinates of the intersection points for the functions and

Section IV: Linear equations Write the equation for the line in both forms given a slope and a point. 68. m = 2/3 and P(3,5) 69. 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. 70. P(2,2) and Q(4,2) 71. P(3,-2) and Q(3,7) Point-Slope: Point-Slope: Slope-Intercept: Slope-Intercept:

2( ) 2 9f x x x= + -3( ) 34

g x x= - +

13

Section V: Functions and Models 72. (e) State the domain and range of f (f) State the domain and range of g 73. 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.