Honors Calculus I Chapter P: Prerequisites Section P.1: Lines in the Plane.
Summer Reading AP Calculus - mpsri.net · Summer Reading AP Calculus Page 4 2017 Equations for...
Transcript of Summer Reading AP Calculus - mpsri.net · Summer Reading AP Calculus Page 4 2017 Equations for...
Summer Reading AP Calculus
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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
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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
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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
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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
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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)
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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
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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).
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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
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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.
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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
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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
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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
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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
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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