Slide 1 / 180content.njctl.org/.../review-2-2015-10-20-1-slide-per-page-w-answers.p… · 20/10/2015 · Slide 12 / 180 Once you have the slope of a line, it is important to be able

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AP Calculus AB

Review Unit

2015-10-20

www.njctl.org

http://www.njctl.org

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Table of ContentsClick on the topic to go to that section

SlopesEquations of LinesFunctionsGraphing FunctionsPiecewise FunctionsFunction CompositionFunction RootsDomain and RangeInverse FunctionsTrigonometryExponentsLogs and Exponential Functions

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Slopes

Return toTable ofContents

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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.

Slope

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One way to determine the slope is calculate it from two points.

Consider two points, (x1,y1) and (x2,y2)

The slope, m, is:

*Note: a slope is not defined for a vertical line (where x1=x2)

Calculating Slope

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Example: Calculate the slope of the line containing the points

(3,4) and (2,8)

Calculating Slope

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Example: Calculate the slope of the line containing the points

(3,4) and (2,8)

Calculating Slope

[This object is a pull tab]A

nsw

erslope m= -4

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1 What is the slope of the line containing the points:

(15,-7) and (3,5) ?

A m= 1

B m= -1

C m= -1/11

D m= 2/8



(15,-7) and (3,5) ?

A m= 1

B m= -1

C m= -1/11

D m= 2/8

[This object is a pull tab]

Ans

wer B m= -1

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(2,2) and (8,3) ?

A m= 6

B m= 5

C m= 1/2

D m= 1/6



(2,2) and (8,3) ?

A m= 6

B m= 5

C m= 1/2

D m= 1/6


Ans

wer

D m= 1/6

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(17,23) and (-6,-18) ?

A m= 41/23

B m= 23/41

C m= -2

D m= -23/41



(17,23) and (-6,-18) ?

A m= 41/23

B m= 23/41

C m= -2

D m= -23/41[This object is a pull tab]

Ans

wer A m= 41/23

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Equations of Lines


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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, (x1, y1), you can write the equation of the line.

Let 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

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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.

Point-Slope Form


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.

Point-Slope Form


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- Write the equation for the line in point-slope form, that has a slope of 4 and contains the point (5,-8).


- Write the equation for the line in point-slope form, that has a slope of 4 and contains the point (5,-8).


Ans

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- Write the equation of the line, in point-slope form, that has a slope of -5 and contains the point (3,15).


- Write the equation of the line, in point-slope form, that has a slope of -5 and contains the point (3,15).


Ans

wer

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- Write the equation of the line, in point-slope form, that contains the points (5,3) and (-3,-6).


- Write the equation of the line, in point-slope form, that contains the points (5,3) and (-3,-6).


Ans

wer

First, find slope:

Next, choose point and plug into Point-Slope form:

or

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- Write the equation of the line, in point-slope form, that contains the points (-4,3) and (2,9).


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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

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Example: Find the equation of the line with a slope of 3, containing the point (4,5). Express your answer in slope-intercept form.



Example: Find the equation of the line with a slope of 3, containing the point (4,5). Express your answer in slope-intercept form.



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form that:

Converting to slope-intercept:

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- Write the equation of the line, in slope-intercept form, that has a slope of 5 and contains the point (23,15).


- Write the equation of the line, in slope-intercept form, that has a slope of 5 and contains the point (23,15).


Ans

wer

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- Write the equation of the line, in slope-intercept form, that has a slope of -3 and contains the point (6,8).


- Write the equation of the line, in slope-intercept form, that has a slope of -3 and contains the point (6,8).


Ans

wer

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- Write the equation of the line, in slope-intercept form, that contains the points (16,14) and (-2,-7).


- Write the equation of the line, in slope-intercept form, that contains the points (16,14) and (-2,-7).


Ans

wer

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Functions


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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 example:

1.2.

What is a Function?


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 example:

1.2.

What is a Function?


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We can show that these are functions, evaluate the equations. Put in any value for x, there will be only one value for y.

1. if x is 7, y is 30. if x is 1, y is 24 etc.

2. if x is 3, y is 24. if x is 2, y is 19 etc.

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For the function definition given on the previous slide to be true, the function will also pass what is called the Vertical Line Test. This states that a graph is of 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.

Vertical Line Test


For the function definition given on the previous slide to be true, the function will also pass what is called the Vertical Line Test. This states that a graph is of 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.

Vertical Line Test


Teac

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and notice it will not cross the graph more than once, for

both examples.

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Here is an example of how that would be expressed:

x y1 63 11

12 557 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.

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.

Functions as a Table

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Sometimes it is useful to consider relations that 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.

Equations Which are Not Functions


Sometimes it is useful to consider relations that 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.

Equations Which are Not Functions


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Show students that there is more than one possible y-value for each x, in both examples.

Example 1: if we choose x=19,

then y=4 or y=-4

Example 2: if we choose x=3,

then y=5 or y=-5

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You can see that both examples do not pass the Vertical Line Test:

1. 2.

Failing the Vertical Line Test


You can see that both examples do not pass the Vertical Line Test:

1. 2.

Failing the Vertical Line Test


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Note: The purple vertical line crosses both graphs more than once.

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6 Is the following relation a function?

Yes

No

-103

-2-1 0

x y



Yes

No

-103

-2-1 0

x y


Ans

wer

No, x=-1 has 2 values for y

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Yes

No

-2 3

0 2

-1 -1

3 2

4 0

Ans

wer

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10 What is the value of f(x+2) given


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Graphing Functions


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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.

Graphing Functions

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y = a f( bx ∓ c) ± d

Transforming FunctionsFunctions, 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.


y = a f( bx ∓ c) ± d

Transforming FunctionsFunctions, 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.

horizontal slidemoves opposite direction than sign

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11 Which of the following is the graph of ?

A B

C D



A B

C D


Ans

wer

B

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A B

C D



A B

C D


Ans

wer

C

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A B

C D



A B

C D


Ans

wer

A

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14 Which of the following is a graph of ?

A B

C D



A B

C D


Ans

wer

D

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A B

C D



A B

C D


Ans

wer

C

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A B

C D



A B

C D


Ans

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A

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From the previous slide's question, see if you can write the equations for the other graphs:

B C D

Further Challenge


From the previous slide's question, see if you can write the equations for the other graphs:

B C D

Further Challenge


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C:

D:

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A B

C D



A B

C D


Ans

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D

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From the previous question, see if you can you write the equations for the three other graphs:

A B C

Further Challenge


From the previous question, see if you can you write the equations for the three other graphs:

A B C

Further Challenge


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A:B: C:

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Piecewise Functions


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Piecewise functions can be thought of as several functions at once, 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

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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. This is not always the case.

Piecewise Functions

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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

Discontinuity Notation

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Example: Evaluate the following piecewise function at the given points:

Evaluating a piecewise function is the same as a continuous function, however we must pay close attention to the endpoint definitions.

Evaluating Piecewise Functions


Example: Evaluate the following piecewise function at the given points:

Evaluating a piecewise function is the same as a continuous function, however we must pay close attention to the endpoint definitions.



Ans

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Now we can practice graphing the same piecewise function.

Graphing Piecewise Functions


Now we can practice graphing the same piecewise function.



Ans

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Example: Evaluate the piecewise function at the given values:



Example: Evaluate the piecewise function at the given values:



Ans

wer

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Example: Graph the following piecewise function:



Example: Graph the following piecewise function:


[This object is a pull tab]A

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- Given the following piecewise function, find the value of




Ans

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Ans

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Ans

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Function Composition


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To evaluate composite functions, you must start from the innermost "layer" and work your way out.

For example, if and

To evaluate , first x passes through the function g(x), and that output is then plugged into f(x).

Evaluating Composite Functions


To evaluate composite functions, you must start from the innermost "layer" and work your way out.

For example, if and

To evaluate , first x passes through the function g(x), and that output is then plugged into f(x).



Ans

wer

Students often get confused when they have to substitute an entire

expression into another function, so work slowly and allow time for

discussion.

Slide 70 / 180

Example: Given and find



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22 What is the value of given the following functions:

A

B

C

D

145

26-4



A

B

C

D

145

26-4


Ans

wer

C

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A

B

C

D



A

B

C

D


Ans

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B

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A

B

C

D

15

77

197

152



A

B

C

D

15

77

197

152 [This object is a pull tab]

Ans

wer

D

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25 Find the value of

A

B

C

D

11

9

-3

15


25 Find the value of

A

B

C

D

11

9

-3

15


Ans

wer

B

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26 Given and , find h(x) if

A

B

C

D



A

B

C

D[This object is a pull tab]

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D

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A

B

C

D



A

B

C

D


Ans

wer

D

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A

B

C

D



A

B

C

D


Ans

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C

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Function Roots


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Another important idea to understand regarding functions is the roots of the function.

A root, sometimes called a zero solution of f(x), is the value of x such that f(x)=0. It can also be called the x-intercept.

roots/zeroes/x-intercepts

Roots of a Function

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Example:Find the roots of the following:

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.

Calculating Roots


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Sometimes the equations are not as easily factorable, and the quadratic formula is required.

Recall: ;

Example: Find the roots of the following equation:

Quadratic Formula


Sometimes the equations are not as easily factorable, and the quadratic formula is required.

Recall: ;

Example: Find the roots of the following equation:

Quadratic Formula


Ans

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andso

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Domain and RangeReturn toTable ofContents

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Recall from Algebra II, the Domain of a function is the set of all possible inputs for a function, typically the x-values.

Similarly, the Range of a function is the set of all possible outputs for a function, typically the y-values.

Domain and Range

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Watch for values which may cause: –zero in the denominator –square roots of negative numbers – logs of zero – logs of negative numbers

Certain conditions must be avoided in order for the Domains and Ranges of functions to be real.

Domain and Range

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Example: Find the Domain and Range of the following function:

Domain and Range



Domain and Range


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This is a line with a slope of 3. It will pass through all y-values, so the range is all real numbers.

Since this function is a general polynomial, any value can be plugged in yielding a real number, so the Domain is all real numbers:

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Domain and Range


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33 What is the Domain and Range for the following function:

A

B

C

D



A

B

C


Ans

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B

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34 What is the Domain and Range of the following function:

A

B

C

D


34 What is the Domain and Range of the following function:

A

B

C


Ans

wer

C

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A

B

C

D



A

B

C


Ans

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A

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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 following function:

A More Challenging Example


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 following function:

A More Challenging Example


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We must make sure the denominator is not zero. Since we cannot factor the denominator to cancel out with the numerator, we must find the zeros of the denominator.

Therefore, the domain is all real numbers except x=-1 and x=3 written as:

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36 What is the Domain (only) for the following function:

A

B

C

Domain: All real numbers

Domain: All real numbers except x=-3, x=2 and x=-5

Domain: All real numbers except x=-3 and x=-5

Domain: All real numbers except x=3 and x=5D



A

B

C

Domain: All real numbers

Domain: All real numbers except x=-3, x=2 and x=-5

Domain: All real numbers except x=-3 and x=-5

Domain: All real numbers except x=3 and x=5D


Ans

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C

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A

B

C

D



A

B

C

D


Ans

wer

Note, this answer shows the Domain has to avoid division by zero, and square root of negative numbers.

D

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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.

One-to-One Functions


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.

One-to-One Functions

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[This object is a teacher notes pull tab]

Note: Even if the function is not one-to-one, it can still have an inverse, but the inverse is not a function.

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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 y-values.

If given an equation that is easy to plot, you can use the Horizontal Line Test. This states that 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.

Horizontal Line Test

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Example:

Notice: The line crosses the graph twice and fails the Horizontal Line Test. Therefore, it is not a One-to-One function.

Failing the Horizontal Line Test

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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.

Passing the Horizontal Line Test

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38 Is the following graph a One-to-One function?

Yes

No



Yes

No


Ans

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No

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Yes

No



Yes

No [This object is a pull tab]

Ans

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Yes

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Example: Find the inverse of f(x), given:

Finding the Inverse


Example: Find the inverse of f(x), given:

Finding the Inverse


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Step 5 involves the previously discussed Function Composition. (click for link)

Inverse Function can be defined as:

andGiven two One-to-One Functions

if: and

then and are Inverses of each other.

Inverse Definition

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Slide 110 / 180

Example: Given:

Are these two functions inverses of each other? Check to make sure it follows the definition.

Inverses


Example: Given:

Are these two functions inverses of each other? Check to make sure it follows the definition.

Inverses


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and

inverses

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The Inverse of is

and the Inverse of is

Terminology


The Inverse of is

and the Inverse of is

Terminology

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The inverse notation can cause confusion, the "-1" superscript is NOT an exponent:

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40 Which of the following is the correct notation for the Inverse Function of ?

A

B

C

D

E


40 Which of the following is the correct notation for the Inverse Function of ?

A

B

C

D

E


Ans

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D

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41 Given the following function, which is its inverse function?

A

B

C

E Not Invertable

D


41 Given the following function, which is its inverse function?

A

B

C

E Not Invertable

D


Ans

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D

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42 Given , Find

A

B

C

E Not Invertable

D


42 Given , Find

A

B

C

E Not Invertable

D


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A

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43 Given , Find

A

B

C

E Not Invertable

D


43 Given , Find

A

B

C

E Not Invertable


Ans

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C

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Another special relationship that you may recall about functions and their inverses is that their graphs are a reflection across the line y=x.

Graphs of Inverses

Slide 117 / 180

Trigonometry


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Trig FunctionsThese are the six trig functions you are familiar with from Geometry and Precalculus.


Trig FunctionsThese are the six trig functions you are familiar with from Geometry and Precalculus.

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Review these relationships with students:

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All these trig functions are defined in terms of a right triangle:

OppositeHypotenuse

Adjacent

Trig - Right Triangles

Slide 120 / 180

The graphs of these functions should be easily recognizable:

Slide 121 / 180

The range of sin and cos ?

The range of csc and sec ?

The range of tan and cot ?

The ranges for these functions can also be determined.What is:

Range of Trig Functions


Slide 122 / 180

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.

A-S-T-C


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.

A-S-T-C

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Most students try to remember the ASTC order with a catchy phrase like: All Students Take Calculus

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In Calculus class almost all problems are in radians, not in degrees. This table shows the "special" angles, in both, that you should be familiar with.

Degrees 0 30 45 60 90 180 270 360

Radians 0

Radians

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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.

Trigonometry

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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.

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The Unit Circle is divided into 4 quadrants. They are listed below.

III

III IV

The Unit Circle

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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.

Special Angles in the II, III, and IV Quadrants

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This method requires you memorize values from the table and remember:

2. THE TRIG TABLE

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This method requires you to draw any of the above triangles on a set of axes depending on given angle, and remember:

3. SPECIAL RIGHT TRIANGLES

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46 Evaluate

A B C D E


46 Evaluate

A B C D E[This object is a pull tab]

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E

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50 Evaluate

A

B

C

D

E


50 Evaluate

A

B

C

D

E


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C

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51 Evaluate

A

B

C

D

E


51 Evaluate

A

B

C

D

E


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E

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Double Angle Formulas

The following Trig Identities are some of the more common ones, you may recall from Pre-calculus.

Pythagorean Identity

Half Angle Formulas

Sum Identities

Trig Identities

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52 Evaluate

A

B

C

D


52 Evaluate

A

B

C

D


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B

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Inverse Trig FunctionsInverse Trig Functions follow the same rules as other Inverse Functions we learned earlier. (Click here)

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.

page22svg


Inverse Trig FunctionsInverse Trig Functions follow the same rules as other Inverse Functions we learned earlier. (Click here)

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.Te

ache

r Not

es


Remind students that the "-1" is not an exponent.

page22svg

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For sinx:

For cosx:

For tanx:

Remember that Inverse Functions must be One-to-One. Recalling our basic trig graphs, we can see that none of them are One-to-One. Therefore, we must restrict the range.

(Click here)

Inverse Functions

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Example: Evaluate

In other words, we must find what angles have sin values of , remembering our range restrictions.

Evaluating


Example: Evaluate

In other words, we must find what angles have sin values of , remembering our range restrictions.

Evaluating


Ans

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This value can be easily found using the Unit Circle or trig table.

Answer:

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Slide 147 / 180

56 Evaluate

A

B

C

D

E


56 Evaluate

A

B

C

D

E


Ans

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C

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57 Evaluate

A

B

C

D

E


57 Evaluate

A

B

C

D

E


Ans

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B

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ExponentsReturn toTable ofContents

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Slide 150 / 180

Properties of Exponents

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Simplify each of the following expressions.

Practice

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Slide 153 / 180

59 Simplify.

A

B

C

D

E None of the above


59 Simplify.

A

B

C

D

E None of the above


Ans

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D

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60 Simplify:

A

B

C

D

E None of the above


60 Simplify:

A

B

C

D

E None of the above


Ans

wer E

correct answer = 1

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Slide 157 / 180

63 Simplify:

A

B

C

D

E None of the above


63 Simplify:

A

B

C

D

E None of the above


Ans

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B

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Slide 159 / 180

65 Simplify:

A

B

C

D

E None of the above


65 Simplify:

A

B

C

D

E None of the above


Ans

wer

A

Slide 160 / 180

Logs and Exponential Functions

Return toTable of

Contents

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Slide 161 / 180

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Slide 166 / 180

69 Find

A

B

C

D

E


69 Find

A

B

C

D

E


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A

Slide 167 / 180

Log Properties:

Change of Base formula:


Log Properties:

Change of Base formula:Teac

her N

otes


The following are log properties that are calculated, but might help students if they knew them without having to stop and obtain them:

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Example: Find

Logarithms


Slide 169 / 180

70 Find

A

B

C

D

E


70 Find

A

B

C

D

E


Ans

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C

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71 Find

A

B

C

D

E


71 Find

A

B

C

D

E


Ans

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D

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72 Find

A

B

C

D

E Undetermined


72 Find

A

B

C

D

E Undetermined


Ans

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D

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73 Find

A

B

C

D

E


73 Find

A

B

C

D

E


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Hint: Use Change of Base formula with a new base of 6.

Answer: E

Slide 173 / 180

Special Case of Log

This is called the natural log, and it has a base of . follows the same rules and has the same properties as .

Note that:


Special Case of Log

This is called the natural log, and it has a base of . follows the same rules and has the same properties as .

Note that: Te

ache

r Not

es


Students should be familiar with , it has a value of:

Slide 174 / 180

Using what we learned about the relationships between logs and exponents, we can now solve equations containing them.

Exponential and Logarithm Equations

Slide 175 / 180

Example: Solve for x: (remember domain requirements for log)



Example: Solve for x: (remember domain requirements for log)



Ans

wer potential answers are:

each answer must be checked with the original problem, to make sure that we are not taking the log of a negative number...

Answer:

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Slide 177 / 180


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Slide 180 / 180

78 Solve for x:

A

B

C

D

E None of the above


78 Solve for x:

A

B

C

D

E None of the above [This object is a pull tab]

Ans

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Answer: C

Hint: Isolate the exponential term, then take of both sides.

Documents

Slide 1 / 180content.njctl.org/.../review-2-2015-10-20-1-slide-per-page-w-answers.p… · 20/10/2015 · Slide 12 / 180 Once you have the slope of a line, it is important to be able