AP Calculus AB
Review Unit
2017-09-11
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Table of Contents
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Slopes
Equations of Lines
Functions
Graphing Functions
Piecewise Functions
Function Composition
Function Roots
Domain and Range
Inverse Functions
Trigonometry
Exponents
Logs and Exponential Functions
Slopes
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Slope
Recall from Algebra, The slope of a line is the ratio of the vertical
movement to the horizontal movement.
In other words, it describes both the steepness and direction of a
line.
Calculating Slope
One way to determine the slope is calculate it from two points.
Consider two points:
and
The slope, m, is:
*Note: a slope is not defined for a vertical line (where
)
Calculating Slope
Answer
Example: Calculate the slope of the line containing the points:
1 Find the slope of the line containing the points:
B
C
D
Answer
A
2 Find the slope of the line containing the points:
B
C
D
Answer
A
3 Find the slope of the line containing the points:
B
C
D
Answer
A
Equations of Lines
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Point-Slope Form
Once you have the slope of a line, it is important to be able to write
the equation for the line.
If you have the slope of the line, m, and any one point,
can write the equation of the line.
Let
, you
be a point, then
This form is called Point-Slope Form of an equation. Point-Slope
Form is extremely useful in Calculus and it is important that you are
comfortable using it.
Point-Slope Form
Answer
Example: Find the equation of the line that has a slope of 4 and
passes through the point (‒2,5). Write the answer in Point-Slope
form.
Answer
4 Write the equation for the line in point-slope form, that
has a slope of 4 and contains the point (5,‒8).
Answer
5 Write the equation of the line, in point-slope form, that
has a slope of -5 and contains the point (3,15).
Answer
6 Write the equation of the line, in point-slope form, that
contains the points (5,3) and (‒3,‒6).
Answer
7 Write the equation of the line, in point-slope form, that
contains the points (‒4,3) and (2,9).
Slope-Intercept Form
Recall from Algebra, another common way to express the equation
of a line is called slope-intercept form.
This is written as:
Where m is the slope, and the y-intercept is at (0,b).
Slope-Intercept Form
Example: Find the equation of the line with a slope of 3, containing
Teacher Notes
the point (4,5). Express your answer in slope-intercept form.
8 Write the equation of the line, in slope-intercept form, that
Answer
has a slope of 5 and contains the point (23,15).
9 Write the equation of the line, in slope-intercept form, that
Answer
has a slope of ‒3 and contains the point (6,8).
10 Write the equation of the line, in slope-intercept form,
Answer
that contains the points (16,14) and (‒2,‒7).
Functions
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What is a Function?
For example:
1.
2.
Teacher Notes
A function is a relationship between x and y such that for any
value x, there will be one and only one value of y.
For the function definition to be true, the function will also pass what
is called the Vertical Line Test.
This states a graph is a function if and only if there is no vertical line
that crosses the graph more than once.
For the same examples, let's look at their graphs:
1.
2.
Teacher Notes
Vertical Line Test
Functions as a Table
A third way to demonstrate functions is in tabular form. Sometimes
functions can be represented as a set of ordered pairs, or a
relation. This is used often when the equation itself is unknown.
Here is an example of how that would be expressed:
x
1
3
12
57
y
6
11
5
9
There is no given equation for this relation, but it is a function since
there is only one y value for each x value.
Equations Which are Not Functions
If for any input there is more than one output, it is not a function.
Here are examples of equations that are not functions:
1.
2.
Teacher Notes
Sometimes it is useful to consider relations that are not functions.
Failing the Vertical Line Test
1.
2.
Teacher Notes
You can see that both examples fail the Vertical Line Test:
11 Which of the following is NOT a function?
A
B
C
12 Which of the following is NOT a function?
A
B
C
Yes
No
Answer
13 Is the following relation a function?
14 Is the following relation a function?
Yes
No
-2
3
0
2
-1
-1
3
2
4
0
Consider the equation:
The following is the same equation written in "function notation":
Teacher Notes
Function Notation
Evaluating Functions
When asked to evaluate a function at a specific value, x = 2, we
use the notation:
Consider the same equation:
Evaluate:
15 Find the value of
given
.
16 Find the value of
given
given
Answer
17 Find the value of
Graphing Functions
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Graphing Functions
It is important to be able to graph functions. At this point, you
should be familiar with methods for doing so.
You should also be able to understand parent graphs, and identify
shapes and orientations of different, common functions.
Transforming Functions
y = a f( bx ∓c) ± d
Teacher Notes
Functions, like equations, are transformed in a predictable manner.
Each letter below has a separate effect on a given function. Identify
how each letter transforms a function.
18 Which of the following is the graph of
A
B
C
D
?
19 Which of the following is the graph of f (x) = ln x ?
A
B
C
D
20 Which of the following is the graph of
A
B
C
D
?
21 Which of the following is the graph of
A
B
C
D
?
22 Which of the following is the graph of f (x) = sinx ?
A
B
C
D
23 Which of the following is the graph of f (x) = 3sinx + 1 ?
A
B
C
D
Further Challenge
B
C
D
Teacher Notes
From the previous slide's question, see if you can write the equations
for the other graphs:
24 Which of the following is a graph of
A
B
C
D
?
Further Challenge
From the previous question, see if you can you write the equations for
the three other graphs:
B
C
Teacher Notes
A
Piecewise Functions
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Piecewise Functions
Piecewise functions can be thought of as several functions, each
defined on a specific interval, or each used in a different region.
To graph a piecewise function you do not plot the entire graph of
each individual section - graph only the parts defined by x.
Piecewise Functions
A simple example of a piecewise function is the absolute value
function.
The graph of this function looks like this:
Note, that at the point x = 0 , the
two function pieces meet;
however, this is not always the
case.
Discontinuity Notation
Some piecewise functions can be discontinuous. When you have a
piecewise function in which the different sections do not meet, there is
special notation for the end points.
included endpoint/
solid circle
discluded endpoint/
open circle
Evaluating Piecewise Functions
Evaluating a piecewise function is the same as a continuous function,
however we must pay close attention to the endpoint definitions.
Example: Evaluate the piecewise function at the given points:
Graphing Piecewise Functions
Answer
Practice graphing the same piecewise function.
Evaluating Piecewise Functions
Example: Evaluate the piecewise function at the given values:
Graphing Piecewise Functions
Answer
Example: Graph the following piecewise function:
25 Given the piecewise function, find the value of
A
B
C
D
26 Given the piecewise function, find the value of
A
B
C
D
27 Given the piecewise function, find the value of
A
B
C
D
28 The graph accurately represents the piecewise function.
True or False
29 Given the piecewise function,
30 Given the piecewise function, find the value of
31 Given the piecewise function, find the value of
32 Given the piecewise function, find the value of
33 Given the piecewise function, find the value of
Function Composition
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Function Composition
Using Function Notation allows us to combine functions, where the
output from one function becomes the input for another.
"Function Composition" had the following notation:
In words, it is read "f of g of x".
When evaluating composite function at a number, it is written:
and read "f of g of 3"
Evaluating Composite Functions
For example, if
Answer
To evaluate composite functions, you must start from the innermost
"layer" and work your way out.
and
To evaluate
, first x passes through the function
that output is then substituted into
.
, and
Evaluating Composite Functions
Example: Given
and
Answer
Find
Order Matters
Answer
The order of Function Composition is important. The functions are
not interchangeable.
For example: Given:
Find:
and
34 What is the value of
A
B
C
D
if:
35 What is the value of
A
B
C
D
if:
36 What is the value of
A
B
C
D
if:
37 Find the value of
A
B
C
D
Combining Functions
Functions can also be combined to make other functions.
We can find the following:
Try this one also!
Answer
Given
38 Given
find h(x) if
A
B
C
D
and
,
39 Given
Find h(x) if
A
B
C
D
and
,
40 Given
find h(x) if
A
B
C
D
and
,
41 Given
find h(x) if
A
B
C
D
and
,
Function Roots
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Roots of a Function
Another important idea to understand regarding functions is the roots
of the function.
A root, sometimes called a zero or solution of f (x), is the value of x
such that f (x) = 0.
Calculating Roots
Example:
Find the roots of the polynomial:
Answer
One method for finding roots is to factor and use the zero product
property. For quadratics that are unfactorable, the quadratic formula
can be used.
Quadratic Formula
Recall:
;
Example: Find the roots of the equation:
Answer
Sometimes the equations are not as easily factorable, and the
quadratic formula is required.
42 Which of the choices are roots of the polynomial?
A
D
B
E
C
F
43 Which of the choices are roots of the polynomial?
A
D
B
E
C
F
44 Which of the choices are roots of the polynomial?
A
D
B
E
C
F
Domain and Range
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Domain and Range
Recall from Algebra II...
The Domain of a function is the set of all possible inputs for a
function, typically the x-values.
The Range of a function is the set of all possible outputs for a
function, typically the y-values.
Domain and Range
Certain conditions must be avoided in order for the Domains and
Ranges of functions to be real.
Watch for values which may cause:
–zero in the denominator
–square roots of negative numbers
–logs of zero
–logs of negative numbers
Domain and Range
Teacher Notes
Example: Find the Domain and Range of the function:
Domain and Range
Teacher Notes
Example: Find the Domain and Range of the function:
45 What is the Domain and Range for the function:
A
C
B
D
46 What is the Domain and Range of the function:
A
C
B
D
47 What is the Domain and Range for the function:
A
C
B
D
Sometimes more complicated functions are presented. In this
case, finding ranges might be very difficult, and finding domains
are more important.
Example: Find the Domain for the function:
Teacher Notes
A More Challenging Example
48 What is the Domain (only) for the function:
A
Domain: All real numbers
B
Domain: All real numbers except
C Domain: All real numbers except
D Domain: All real numbers except
,
and
and
and
A
B
C
D
Answer
49 What is the Domain (only) for the function:
Inverse Functions
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In order to study inverse functions, it is first necessary to specify
which kind of functions are appropriate.
We know that for a relation to be a function, every value in the
domain must have exactly one value in the range.
For a function to have an inverse, we further require that every
value in the range must have exactly one value in the domain.
In other words, no two values of x yield the same y.
This relationship is called a One-to-One Function.
Teacher Notes
One-to-One Functions
Horizontal Line Test
You must determine if a function is One-to-One, in order for you to
then find it's inverse.
If given ordered pairs, simply look to see if there are no repeated yvalues.
If given an equation that is easy to plot, you can use the Horizontal
Line Test.
If it is possible to draw a Horizontal line anywhere on the graph, and
it crosses the graph more than once, it fails the One-to-One
requirement.
Failing the Horizontal Line Test
Example:
Notice:
The line crosses the graph twice
and fails the Horizontal Line Test.
Therefore, it is not a One-to-One
function.
Passing the Horizontal Line Test
Example:
Notice:
The line does not cross the
graph more than once and
Passes the Horizontal Line Test.
Therefore it is a One-to-One
function.
One-to-One by Definition
If neither of those methods are viable, you must determine if the
function is One-to-One using its definition:
50 Is the graph a One-to-One function?
Yes
No
51 Is the graph a One-to-One function?
Yes
No
Steps for Finding Inverse Functions
Given a function
, to find the Inverse Function
1. Replace the notation
2. Switch each
with an
with
:
.
, and vice versa .
3. Solve this new equation for
.
4. Change notation again, and replace the
with
This is the Inverse Function.
5. You can verify the accuracy by checking to see if:
.
Finding the Inverse
Answer
Example: Find the inverse of
Inverse Definition
Step 5 involves the previously discussed Function Composition.
(click for link)
Inverse Functions can be defined as:
Given two One-to-One Functions
if:
then
and
and
and
are Inverses of each other.
Inverses
Are these two functions inverses of each other?
Check to make sure it follows the definition.
Answer
Example: Given:
The Inverse of
is written as
The Inverse of
is written as
Teacher Notes
Notation
52 Which is the correct notation for the Inverse Function
of
?
A
B
C
D
E
53 Given the function, which is its inverse function?
A
B
C
D
54 Given
A
B
C
D
, Find
55 Given
A
B
C
D
, Find
Graphs of Inverses
Another special relationship that you may recall about functions
and their inverses is that their graphs are a reflection across
the line y = x.
Trigonometry
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Trig Functions
Teacher Notes
These are the six trigonometry functions you are familiar with from
Geometry and Precalculus.
Trig - Right Triangles
All these trig functions are defined in terms of a right triangle:
Opposite
Hypotenuse
Adjacent
The graphs of these functions should be easily recognizable:
Range of Trig Functions
The ranges for these functions can also be determined.
The range of sin θ and cos θ ?
The range of csc θ and sec θ ?
The range of tan θ and cot θ ?
Answer
What is:
Another important matter is the sign of the trig functions in each
quadrant. The letters A-S-T-C represent the positive values. All
other trig functions will be negative in those quadrants.
A: All trig functions are positive in the 1st quadrant.
S: sin values are positive in the 2nd quadrant.
T: tan values are positive in the 3rd quadrant.
C: cos values are positive in the 4th quadrant.
Teacher Notes
A-S-T-C
Radians
This table shows the "special" angles, in both, which you should be
familiar with.
Degrees
0
Radians
0
30
45
60
90 180 270 360
Teacher Notes
In Calculus class almost all problems are in radians, not in degrees.
In Geometry and Pre-calculus you learned quite a bit
about trigonometry. To be successful in calculus, it is
very important that you know how to evaluate trig
functions at various angles. Many real life situations
behave in a trigonometric pattern (i.e. traffic flow),
therefore you will see that trig functions are very
prevalent in the course and on the AP Exam.
Teacher Notes
Trigonometry
Method #1: The Unit Circle
This method requires you to
memorize values for each
ordered pair. Recall that the
x value of each ordered pair
is the cosine value, while the
y value of the ordered pair is
the sine value.
The Unit Circle
The Unit Circle is divided into 4 quadrants. They are listed below.
II
III
I
IV
Special Angles in the II, III, and IV Quadrants
The x and y coordinates for special angles in the other quadrants
can be determined by knowing the similar 1st quadrant angle's
value. The x and y values will be the same, but the signs will (or
can) be different.
Method #2: The Trig Table
0
0
1
0
This method requires you memorize values from the table and
remember:
This method requires you to draw any of the above triangles on a
set of axes depending on given angle, and remember:
Teacher Notes
Method #3: Special Right Triangles
56 Using method of your choice, evaluate
A
B
C
D
E
57 Evaluate
A
B
C
D
E
58 Evaluate
A
B
C
D
E
59 Evaluate
A
B
C
D
E
60 Evaluate
A
B
C
D
E
61 Evaluate
A
B
C
D
E
62 Evaluate
A
B
C
D
E
63 Evaluate
A
B
C
D
E
Trig Identities
The following Trig Identities are some of the more common ones,
you may recall from Pre-calculus.
Pythagorean Identity
Sum Identities
Double Angle Formulas
Half Angle Formulas
Trig Identities
using the double angle formula.
Answer
Example: Evaluate
64 Evaluate
A
B
C
D
65 Evaluate
A
B
C
D
66 Evaluate
A
B
C
D
Inverse Trig Functions
They "undo" what the trig function does. For example if the function
is
then the inverse trig function is
.
You may also see the following terminology.
Teacher Notes
Inverse Trig Functions follow the same rules as other Inverse
Functions we learned earlier. (Click here)
Inverse Trig Functions
Remember that Inverse Functions must be One-to-One. Recalling
our basic trig graphs, (Click here) we can see that none of them
are One-to-One. Therefore, we must restrict the range.
For sin x:
For cos x:
For tan x:
Evaluating Inverse Trig Functions
Example: Evaluate
,
Answer
In other words, we must find what angles have sin values of
remembering our range restrictions.
67 Evaluate
A
B
C
D
68 Evaluate
A
B
C
D
69 Evaluate:
A
B
C
D
Exponents
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Properties of Exponents
Practice
Answer
Simplify each expression.
70 Simplify:
A
B
C
D
71 Simplify.
A
B
C
D
72 Simplify:
A
B
C
D
73 Simplify:
A
B
C
D
74 Simplify:
A
B
C
D
75 Simplify:
A
B
C
D
76 Simplify:
A
B
C
D
77 Simplify:
A
B
C
D
Logs and Exponential
Functions
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Logarithmic Functions
If
then
The variable b is called the base, and b > 0, b ≠ 0 .
(If the base is not specified, it is defined to be 10.)
The domain of log is
of positive numbers.
, meaning, we can only take the logarithm
Logarithms
Answer
Example: Evaluate
78 Find
A
B
C
D
79 Find
A
B
C
D
80 Find
A
B
C
D
81 Find
A
B
C
D
Teacher Notes
Log Properties:
Change of Base Formula:
Logarithms
Answer
Example: Find
82 Find
A
B
C
D
83 Find
A
B
C
D
84 Find
A
B
C
D
85 Find
A
B
C
D
This is called the natural log, and it has a base of e.
ln x follows the same rules and has the same properties as log x.
Note that:
Teacher Notes
Special Case of Log
Exponential and Logarithm Equations
Using what we learned about the relationships between logs
and exponents, we can now solve equations containing them.
Exponential and Logarithm Equations
(remember domain
requirements for log)
Answer
Example: Solve for x:
86 Solve for x:
B
C
D
Answer
A
87 Solve for x:
B
C
D
Answer
A
88 Solve for x:
B
C
D
Answer
A
89 Solve for x:
B
C
D
Answer
A
90 Solve for x:
B
C
D
Answer
A