summer reading ap calculus - mpsri.net · summer reading ap calculus page 4 2017 equations for...

Summer Reading AP Calculus

2017

AP CALCULUS (A/B) BASIC SUMMER REVIEW

As an advanced placement student you need to plan on 30 to 60 minutes a day of math

homework in AP Calculus. We have a specific amount of material that we MUST cover and

we should plan on several weeks of review and practice AP Test Questions.

Slope of a non vertical line:

Point Slope Equation: )( 11 xxmyy

The slope is m and a point on your line is ),( 11 yx .

Slope-Intercept Equation: bmxy slope= m y-intercept= b

General Linear Equation CByAx such that A and B both are not

zero.

What is a Function?

A function is a relation that assigns a single element of R to each element

of D.

A working definition of a function is that it is a devise that assigns an

output to every allowable input. The inputs make up the domain of the

function. The outputs make up the range. A Function must pass the vertical Line Test

Vertical Line Test

Identifying the Domain and Range: Remember, in the real number system you can not divide

by zero or find the even root of a negative number

Even and Odd Functions

A function y = f(x) is an even function of x if f(-x) = f(x) for every x in the function’s

domain.

A function is an odd function of x if f(-x) = - f(x) for every x in the function’s domain.

2 1

2 1

y yrise ym

run x x x


2017

Absolute Value Think of the absolute value function as a piecewise function.

The Greatest Integer Function xxf )( or

The greatest integer function represents the greatest integer less than or equal to x.

Composition of Functions

The composition of the functions f and g is defined by

The domain of consists of those x’s for which g(x) is in the

domain of f.

Geometric Transformations: Shifts, Reflections, Stretches, and Shrinks

Graph Shifting Formulas

Vertical shifts of the graph of )(xfy

cxfy )( Shifts graph of )(xfy down c units

cxfy )( Shifts graph of )(xfy up c units

Horizontal shifts of the graph of

)( cxfy Shifts graph of )(xfy right c units

)( cxfy Shifts graph of )(xfy left c units

How to stretch or shrink a graph

To stretch the parabola 2xy vertically by a factor of c (c>0), we must multiply each y-

coordinate by c.

– If you stretch the graph by a factor of two the new equation will be: 22xy

How to reflect a graph To reflect the graph of y=f(x) across the y-axis, we multiply each y

coordinate by -1.

Reflection Formulas:

– With respect to the y-axis )( xfy

– With respect to the x-axis )(xfy

Finding Vertex

Find the Vertex of the parabola y= 7164 2 xx by completing the square.

The Parabola cbxaxy 2

A parabola that opens in the positive y direction if a>0 and in the negative y direction if a<0.

The axis of symmetry is: a

bx

2

The vertex is at: ))2

(,2

(a

bf

a

b

if x 0( )

if x<0

xf x x

x

( ) int( )f x x

1.32 1 3.4 4

f g( )( ) ( ( ))f g x f g x

( )( ) ( ( ))f g x f g x


2017

POLYNOMIALS

Polynomial Expression: 01

2

2

1

1 ... axaxaxaxa n

n

n

n

n

n

Polynomial Function: 01

2

2

1

1 ...)( axaxaxaxaxf n

n

n

n

n

n

Polynomial Equation: 0... 01

2

2

1

1

axaxaxaxa n

n

n

n

n

n

Rational Zeros Theorem Suppose all the coefficients in the polynomial function

01

2

2

1

1 ...)( axaxaxaxaxf n

n

n

n

n

n

are integers.

If d

cx s a rational zero of f , where c and d have no common factors, then c is a factor

of 0a , and d is a factor of the leading coefficient na .

How to Solve f(x)= 0 using calculator or your own brain!!!!

1. Find the exact solution algebraically (often by factoring)

2. Draw a complete graph

a) Use ZOOM-IN

b) Use SOLVE

Steps for Solving a Problem 1. Find an algebraic representation involving variables.

2. Draw a complete graph of the function

3. Find the domain and range

4. Determine the values that make sense in the given situation

5. Draw a graph of the problem situation

6. Solve the problem using appropriate methods

For instance: Solve 012142 23 xxx

Factors of c: 12,6,4,3,2,1

Factors of d: 2,1

Possible zeros: d

c

2

3,

2

1,12,6,4,3,2,1

Look at the graph to see the zeroes must be between -2 and -1 or 3 and 4.

So 0)2/3( f )32(2

1)2/3( xx

So )32( x is a factor.

By division 0)42)(32( 2 xxx

Use the Quadratic Formula to find 51x

Equations with Absolute Values:

Solve this equation algebraically: 732 x

The equation says that 732 x

So, solve 732 x and 732 x such that x = 5, -2

Inequalities with Absolute Values:

Solve this Inequality Algebraically 7 10x

This means 10710 x Then you add 7 to all three parts of inequality to get

173 x


2017

Equations for Circles in the Plane Circle is the set of points in a plane whose distance from a fixed point in the plane is a constant.

The fixed point is the center of the circle. The constant distance is the radius of the circle.

Equation: 222 )()( rkyhx

Inverse Relations and Functions:

Inverse Relation: Let R be a relation. The inverse relation 1R of R consists of all those

ordered pairs (b,a) for which (a,b) belongs to R. So the domain of 1R = the range of R

and the range of 1R = the domain of R.

Horizontal Line Test :

The inverse relation 1R of the relation R is a function if and only if every horizontal

line intersects the graph of R in at most one point.

Notice that the inverse of 16)( 2 xxf is not a function since f(x) fails the

horizontal line test.

One-to-One:

The inverse 1f of a function f is a function if and only if f is a one-to-one function.

Exponential Functions:

Definition: Let a be a positive real number other than 1. The function xaxf )(

whose domain is ),( and whose range is ),0( is the exponential function with

base a.

Examples:

1. xxf 2)(

2.

x

xf

2

1)(

Logarithmic Function

Definition: Let a be a positive real number other than 1. The function xxf alog)(

with domain ),0( and range ),( is the inverse of the exponential function xaxf )( and is called the logarithmic function with base a.

y

a axxy log

The number xalog is the logarithm of x to the base a.

Properties of Logarithms: Let 0a and 1a . Then the following are true.

1. 01log a

2. 1log aa

3. xaxa

log

4. xa x

a log


2017

MORE Properties of Logarithms: Let ,,ra and s be positive real numbers with 1a

Then the following are true.

1. srrs aaa logloglog 2. srs

raaa logloglog

3. rcr a

c

a loglog

Change of Base Formula : Let a and b be positive real numbers with 1a and

1b . Then b

aab

log

loglog

Solving Log Equations: Solve the following equations using properties of logs.

1. )2log(log0 xx

2. 3)2(log)5(log 22 xx

Trigonometric Functions:

Unit Circle

Graph of the Sine Curve: y = sin(x)

Graph of the Cosine Curve: y = cos(x)

Graph of the Tangent Curve: y = tan(x)


2017

POLAR COORDINATES

The polar coordinates (the radial coordinate) and (the angular coordinate, often called the polar

angle) are defined in terms of Cartesian coordinates by

where is the radial distance from the origin, and is the counterclockwise angle from the x-axis.

In terms of and ,

The equation of a curve expressed in polar coordinates is known as a polar equation, and a plot

of a curve in polar coordinates is known as a polar plot. A polar curve is symmetric about the x-

axis if replacing by in its equation produces an equivalent equation, symmetric about the y-

axis if replacing by in its equation produces an equivalent equation, and symmetric about

the origin if replacing by in its equation produces an equivalent equation.

Conic Sections

http://mathworld.wolfram.com/PolarAngle.html

http://mathworld.wolfram.com/PolarAngle.html

http://mathworld.wolfram.com/CartesianCoordinates.html

http://mathworld.wolfram.com/Origin.html

http://mathworld.wolfram.com/x-Axis.html

http://mathworld.wolfram.com/PolarEquation.html

http://mathworld.wolfram.com/PolarPlot.html



http://mathworld.wolfram.com/y-Axis.html

http://mathworld.wolfram.com/y-Axis.html


2017

Circle

Ellipse (h)

Parabola (h)

Hyperbola (h)

Definition: A conic section is

the intersection of

a plane and a cone.

Ellipse (v)

Parabola (v)

Hyperbola (v)

By changing the angle and location of intersection, we can produce a circle, ellipse, parabola or

hyperbola; or in the special case when the plane touches the vertex: a point, line or 2 intersecting

lines.

Point

Line

Double Line

The General Equation for a Conic Section: Ax2 + Bxy + Cy2 + Dx + Ey + F = 0

The type of section can be found from the sign of: B2 - 4AC

If B2 - 4AC is... then the curve is a...

< 0 ellipse, circle, point or no curve.

= 0 parabola, 2 parallel lines, 1 line or no curve.

> 0 hyperbola or 2 intersecting lines.

The Conic Sections. For any of the below with a center (j, k) instead of (0, 0), replace each x

term with (x-j) and each y term with (y-k).


2017

Circle Ellipse Parabola Hyperbola

Equation (horiz.

vertex): x2 + y2 = r2

x2 / a2 + y2 / b2

= 1 4px = y2 x2 / a2 - y2 / b2 = 1

Equations of

Asymptotes: y = ± (b/a)x

Equation (vert. vertex): x2 + y2 = r2 y2 / a2 + x2 / b2

= 1 4py = x2 y2 / a2 - x2 / b2 = 1

Equations of

Asymptotes: x = ± (b/a)y

Variables: r = circle

radius

a = major radius

(= 1/2 length

major axis)

b = minor

radius (= 1/2

length minor

axis)

c = distance

center to focus

p = distance from

vertex to focus

(or directrix)

a = 1/2 length major

axis

b = 1/2 length

minor axis

c = distance center

to focus

Eccentricity: 0 c/a c/a

Relation to Focus: p = 0 a2 - b2 = c2 p = p a2 + b2 = c2

Definition: is the locus

of all points which

meet the condition...

distance to

the origin is

constant

sum of

distances to

each focus is

constant

distance to focus

= distance to

directrix

difference between

distances to each

foci is constant


2017

NAME_________________________

The following exercises will give you an opportunity to refresh your prior knowledge and skills

from Algebra and Geometry, in preparation for Trig/Pre-calculus. For the following problems,

write out your steps or show how you arrived at answers for each problem. Keep these

exercises in your notebook for quick reference.

Prerequisite Skills:

1. Order of operations with integers, fractions and exponents.

2. Simplify exponents, including fractional and negative exponents.

3. Simplify radicals.

4. Solve equations including linear, literal, absolute value, quadratic, and radical.

5. Solve systems of linear equations.

6. Solve and graph linear and compound (system of) inequalities.

7. Determine slope, write linear equations in various forms, perpendicular and parallel lines.

8. Graph equations, functions, and inequalities and shifts/transformations from parent graphs.

9. Indentify domain, range, and asymptotes.

10. Write and sketch inverse function and composition of functions.

11. Factoring including greatest common term, difference of squares, trinomials.

12. Operations with polynomials.

13. Basic Geometry concepts for triangles, polygons, and circles.

14. Area and Volume.

15. Right triangle concepts including basic trigonometry ratios and Pythagorean Theorem.

If you have difficulty with any of these topics, review your notes from prior classes. You can

also look on the internet for tutorials on specific topics on websites such as

www.purplemath.com, or any other websites from you preferred search engine. Be persistent and

resourceful until you find a tutorial that is helpful, understandable, and provides good examples

with answers for you to follow. Don’t accept just “getting an answer” as it is important that you

understand how to successfully complete these types of review problems.

If you are still having questions about any of the above topics, please ask your teacher for

assistance when school begins.


2017

Simplify the following algebraic and numeric expressions.

1. 27 (9 3 5) =

2.

1 2

5 31

2

=

3. 22 (4 3 5) =

4. 1 0 0(5 3 ) a =

5. (7 2 ) (3 5 )x y x y =

6. (7 2 )(3 5 )x y x y =

7. (4 2) (3 33)i i =

8. (4 2) (3 33)i i =

9. (4 2)(3 3)i i =

10. 5[4( 2) 2( 3)]y y =

11. 2 2

2

2 3 2

2 1

x x x x

x x

=

12. 1

1

y z

z y

=

13.

11

11

t

t

Simplify without a calculator, giving answer in exact form (not decimal). In your answer,

express all exponents as positive values and convert any fractional powers to radical form.

14.

5

2

5

8

15

12

t

t

15.

13

12

8

16

16. 2 8 2

2

2 6( )x y z

xy z

17. 4 3 0 2( 2 ) (6 )x x x

18.

1

223

( ) 14

19. 1 1

2 272 98

20. 3 700 2 7

21. 4 2 3

5 2

22. 20

27

23. (2 6)(3 15)

24. 6 3

5 3

Summer Reading AP Calculus:

11

2010

25. 3( 7) 5 2 8x x

26. 1 3 3

4 8 4y

27. 1 2

53 7

x x

28. 2 12 0x x

29. 2 3 1x x

30. 2( 1) ( 7)( 1)x x x

31. Solve by completing the square: 2 4 10 0x x

32. 15 2x x

33. 1 4 5x

34. 2

23 5

x

35. 5

23

x

solve for x in terms

of .

36. x x

ca b , where

0, 0, ,a b b a solve for x.

37. 1

as

r

, solve for r.

Solve the system of equations

38. 3 7

5 6 19

x y

x y

39. 3 2 22

9 8 4

x y

x y

40. Solve: 5 4 3 12x


12

2010

Given Point A (2,-3) and point B (4,-5) find….

41. The slope of AB .

42. Distance from A to B

43. Write the equation of the perpendicular bisector of the line AB .

44. Write in slope-intercept form the equation of the line containing the point (-1, 2)

and parallel to the given line y = 2x + 4.

45. Write in slope-intercept form the equation of the line containing the point (4, 5)

and perpendicular to the given line y = 6x – 1.

You should know how to quickly sketch the graphs of these five basic “parent functions:

a) y x b) 2y x c) y x d) 3y x e) 1

yx

46. From the parent graph of 2y x describe the shift to obtain the new graph of 2( 3) 5y x and graph the function.

47. From the parent graph of y x describe the shift to obtain the new graph of

2 3 1y x and graph the function.

48. State whether the given set of points is a relation or a

function{(1,2),(3,10),(2,20),(3,11)(6,2)}.

49. For the points given F= {(1,2),(3,10),(2,20),(3,11), (a, b)}, state the domain and

range.

Find the domain and range of the following functions.

50.

51.

52.

53. State the domain and range of the function 2

1( )

1

xh x

x

and vertical and

horizontal asymptotes if any exist.

54. Find the domain, range, zero(s), and y-intercept of ( ) 4g x x and verify by

graphing.

55. Find the domain, range, zero(s), and y-intercept of 3 2( ) 3 1h x x x x and

verify by graphing.

Are the following functions even or odd? Determine your answer algebraically and then

verify using a graphing calculator.

1( )

4f x

x

2( ) 4f x x

2

2

2 8( )

2 8

xf x

x x


13

2010

56.

57.

58.

59. Given 1

( ) 12

f x x find its inverse 1( )f x and then sketch the graph of both.

60. Given 2( ) 2 4g x x find its inverse 1( )g x and then sketch the graph of both.

Given: 52)( xxf and 12)( 2 xxxh

61. Find ))1((hf

62. Find ))0((hh

63. Find ))(( xhf

If 22( ) ; ( ) 2

4f x g x x

x

64. Find ( ( ))f g x

65. Find ( ( ))g f x

Write f(x) as a piecewise function.

66. 67.

68. Graph the following:

69. For the function above find the following:

Factor the following.

70. 29 900y

71. 2 7 6x x

72. 2 24 4xy xz

73. 212 36 27x x

74. 3x3 – 15x + 2x2y – 10y

75. Find the Vertex of the parabola y= 7164 2 xx by completing the square.

Complete the indicated operation to simplify the polynomials. Rational answers should

have a common denominator.

3 2( ) 4f x x x

2( )

1

xf x

x

2 4( ) 4f x x x

2

2 x 1

( ) 3 -1<x 3

3 15 x>3

x

f x x

x

( 1)f (1)f

( ) 4f x x

( ) 3 1f x x x


14

2010

76.

2 2

2

6 8 5 4

4 3 5 10

x x x x

x x x

77. 64.

2 2

2

8 64

9 3

x x x

x x

78. 2

4 2

5 6 2

x

x x x

79.

3 2

6 2

x

x x

80. Find the center and radius of the circle. 424 22 yyxx . (complete the

square)

81. Graph the following:

922 yx

2522 yx

Which of the following functions are one-to-one?

81. )sin()( xxf

82. x

xf1

)(

83. 32)( xxf

84. Find the inverse of 5

2)(

x

xxf

Graph the following

85. xy 3log

86. 1)2(log3 xy

87. 1)2(log3 xy

88. 6)(2sin5)( xxf

89. )tan(5)( xxg

90. )(tan)( 1 xxh

91. sin2r

92. 2r

summer reading ap calculus - mpsri.net · summer reading ap calculus page 4 2017 equations for...

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