Slide 1 / 180 Slide 2 / 180 AP Calculus AB Review Unit 2015-10-20 www.njctl.org Slide 3 / 180 Table of Contents Click on the topic to go to that section Slopes Equations of Lines Functions Graphing Functions Piecewise Functions Function Composition Function Roots Domain and Range Inverse Functions Trigonometry Exponents Logs and Exponential Functions Slide 4 / 180 Slopes Return to Table of Contents Slide 5 / 180 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 Slide 6 / 180 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
Transcript
Slide 1 / 180 Slide 2 / 180
AP Calculus AB
Review Unit
2015-10-20
www.njctl.org
Slide 3 / 180
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)
Example: Calculate the slope of the line containing the points
(3,4) and (2,8)
Calculating Slope
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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
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2 What is the slope of the line containing the points:
(2,2) and (8,3) ?
A m= 6
B m= 5
C m= 1/2
D m= 1/6
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3 What is the slope of the line containing the points:
(17,23) and (-6,-18) ?
A m= 41/23
B m= 23/41
C m= -2
D m= -23/41
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Equations of Lines
Return toTable ofContents
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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.
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
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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
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You can see that both examples do not pass the Vertical Line Test:
1. 2.
Failing the Vertical Line Test
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6 Is the following relation a function?
Yes
No
-103
-2-1 0
x y
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7 Is the following relation a function?
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.
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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17 Which of the following is a graph of ?
A B
C 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
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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 NotationSlide 54 / 180
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.
Example: Find the Domain and Range of the following function:
Domain and Range
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Example: Find the Domain and Range of the following function:
Domain and Range
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33 What is the Domain and Range for the following function:
A
B
C
D
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34 What is the Domain and Range of the following function:
A
B
C
D
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35 What is the Domain and Range for the following function:
A
B
C
D
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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
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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
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37 What is the Domain (only) for the following function:
A
B
C
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
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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.
The ranges for these functions can also be determined.What is:
Range of Trig Functions
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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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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
RadiansTe
ache
r Not
es
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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
Teac
her N
otes
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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 TRIANGLESTe
ache
r Not
es
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46 Evaluate
A B C D E
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50 Evaluate
A
B
C
D
E
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51 Evaluate
A
B
C
D
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
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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.
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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.
