AP Calculus AB
Class Notes
1.1 Lines
pp 3-11
Objectives:
 Find the x and y increments between two points.
 Find the slope of a line given two points, or, find the location of a point given the slope and
another x or y value.
 Find equations of lines that are perpendicular or parallel to other lines.
 Find the equation of a line given two points or given one point and a slope.
 Identify and convert between point-slope form, slope intercept form, and general form for an
equation of a line.
Omissions:
In AP calculus, we do not do problems involving regression analysis. Omit this material, pp 7-8.
Increments: An increment is the displacement from one place to the other, along the x and y
directions. Increments are found by subtracting the initial position from the final position. Our book
writes this as
Definition: Increments
If a particle moves from the point (x₁, y₁) to the point (𝑥2 , 𝑦2 ), the increments in its coordinates are:
∆𝑥 = 𝑥2 − 𝑥1 and ∆𝑦 = 𝑦2 − 𝑦1
I prefer the notation
x  x f  xo
and
y  y f  yo
which is spoken as, “delta-x equals x-final minus x-naught.” Naught means initial.
Slope of a Line: This is often defined as rise over run, and has the formula and definition
Definition: Slope
Let 𝑃1 (𝑥1 , 𝑦1 ) and 𝑃2 (𝑥2 , 𝑦2 ) be points on a nonvertical line, L. The slope of L is
𝑟𝑖𝑠𝑒
∆𝑦
𝑦2 − 𝑦1
𝑚=
=
=
𝑟𝑢𝑛
∆𝑥
𝑥2 − 𝑥1
Class Notes 1.1
Page 1 of 27
Finding Slope: Find the slope of the line passing through the given points. Then tell whether the line
rises, falls, is horizontal, or is vertical.
1. (3,2), (-4, 3)
2. (1,-4), (2,6)
3. (14, -3), (4,11)
Equations of Lines:
Lines can be written in many forms, and there are three forms that we codify. They are Point-Slope
form, Slope-Intercept form, and General form.
 Slope Intercept form: This formula is most useful if we know the slope and the y-intercept.
 Point Slope form: This is the easiest way to find the equation for a line when you’re given the
slope and one point on the line. This is the most important form and the one we’ll use more
than the others. It is used more because often in calculus, often we will be finding the equation
for a line when we know its slope and one point through which it passes. It’s also worth
mentioning that we can, and should, always leave our answers in point slope form.
Definition: Point-Slope Equation
The equation
𝑦 = 𝑚(𝑥 − 𝑥1 ) + 𝑦1
is the point-slope equation of the line that passes through the point (𝑥1 , 𝑦1 ) with slope m.
Write an equation of the line that passes through the given points.
1. (2,-1), (3,8)
2. (-5,3), (2,2)
Page 2 of 27
3. (1,-6), (1,-2)
Class Notes 1.1
Parallel and Perpendicular Lines: Parallel lines have identical slopes. Perpendicular lines have
slopes that are opposite reciprocals. This can be written
1
m  
m
The only exception is with a horizontal line. Since a horizontal line has a slope of zero, its
perpendicular is a vertical line. However, since we can never have a zero in the denominator, we say
that the vertical line has a slope that is undefined.
The normal: Normal is a synonym for perpendicular.
Write an equation of the line that passes through (6, 10) and is perpendicular to the line that passes
through (4, 6) and (3, 4).
Write an equation of the line that passes through (2, 7) and is parallel to the line x = 5.
Write an equation of the line that passes through (4, 6) and is parallel to the line that passes through
(6, 6) and (10, -4).
Equations of Vertical and Horizontal Lines: Example
What is the equation for the horizontal and vertical lines that pass through the point (3, 2)? Draw out
this example.
General Linear Equation: The general form is useful if you’re wishing to convey the intercepts of
line. It does not show up much otherwise.
Definition: General Linear Equation
The equation
Ax + By = C (A and B both not 0)
is a general linear equation in x and y.
Example: Analyzing and Graphing a General Linear Equation
Find the slope and y-intercept of the line 8x + 5y = 20. Graph the line.
Class Notes 1.1
Page 3 of 27
Twelve Fundamental Functions
You’ll be working with most of the functions that you’ve studied through precalculus, as shown below.
The logistics function is not seen much in AP Calculus AB, but it is a part of most precalc and calc.
Page 4 of 27
Class Notes 1.1
Graphical Transformations
Recall that transformations come in two forms; rigid transformations do not distort the shape and size
of the graph. These include horizontal and vertical shifts and reflections. Non-rigid transformations
distort the size and shape and include horizontal and vertical stretches and shrinks.
Rigid Transformations
vertical shift
horizontal shift
x-axis reflection
y-axis reflection
y  f ( x)  b
y  f ( x  n)
y   f (x)
y  f ( x)
Non Rigid Transformations
vertical stretch/shrink
y  cf (x)
horizontal stretch/shrink
 x
y f 
c
shifts the graph up b units
shifts the graphs n units to the right
reflects the graph in the x-axis
reflects the graph in the y-axis
functions stretches by a factor of c in the vertical
direction if c  1 ; shrinks if 0  c  1 .
functions stretches by a factor of c in the
horizontal direction if c  1 ; shrinks if 0  c  1 .
Example 1:
Example 2:
𝑦 = 𝑥 2 + 5.2
𝑦 = (𝑥 − 3)2
Class Notes 1.1
Page 5 of 27
Example 3: 𝑦 = √𝑥 − 5
Example 4: 𝑦 = √3 − 𝑥
Page 6 of 27
Class Notes 1.1
Example 5:
𝑓(𝑥) = 𝑥 3 − 2
𝑔(𝑥) = (𝑥 + 4)3 − 1
𝑔(𝑥) = 2(𝑥 − 1)3
𝑓(𝑥) = −2|𝑥| − 3
𝑔(𝑥) = 3|𝑥 + 5| + 4
ℎ(𝑥) = |3𝑥|
Class Notes 1.1
Page 7 of 27
AP Calculus AB
Class Notes
1.2 Functions and Graphs pp 12-21
Objectives:
 Find the domain and range of a function.
 State the definition of an even and odd function, and classify a function as even, odd, or neither.
 Graph piecewise functions. Given the graph of a piecewise function, write the function. Graph
piecewise functions on your calculator.
 Graph the absolute value function.
 Transform functions.
 Create composite functions.
Omissions: none
Functions are mathematical machines that spit out an output for each input. The input is called the
domain and is often the x-value. The output is called the range, and is often the y-value.
Domains and Ranges: The domain is the set of all allowable inputs or x-values for a function. If the
domain is not explicitly stated, it is assumed to be all real numbers that are available for the function.
For example, the function f ( x)  x has the domain x  0 because we cannot take the square root of
a negative number.
The range of a function is the set of all outputs or y-values. Recall that since the graph of the sine
function is
The range of the sine function is  1  y  1.
Example 1: Identifying Domain and Range of a function
Identify the domain and range, and then sketch a graph of the function.
1
(a) 𝑦 = 𝑥
(b) 𝑦 = √𝑥
Page 8 of 27
Class Notes 1.2
Example 2: Identifying Domain and Range of a Function
Use a grapher to identify the domain and range, and then draw a graph of the function.
2
(a) 𝑦 = √4 − 𝑥 2
(b) 𝑦 = 𝑥 ⁄3
Even Function and Odd Functions – Symmetry: Functions that are symmetric with respect
to either the y-axis or the origin are said to be even or odd, respectively. There are really three ways to
determine if a function is odd, even, or neither.
 Polynomials are even or odd if all of their exponents are either even or odd respectively.
 Odd functions have graphs that display origin-symmetry. The graphs of even functions have yaxis symmetry.
 Finally, one can determine if the function is even, odd, or neither by the following definition:
Definitions: Even Function, Odd Function
A function 𝑦 = 𝑓(𝑥)is an
even function of x if 𝑓(−𝑥) = 𝑓(𝑥),
odd function of x if 𝑓(−𝑥) = −𝑓(𝑥),
for every x in the function’s domain.
Some rules about odd and even functions:

The only function which is both even and odd is the constant function which is equal to
zero (i.e., f(x) = 0 for all x).

The sum of an even and odd function is neither even nor odd, unless one of the functions
is equal to zero over the given domain.

The sum of two even functions is even, and any constant multiple of an even function is
even.

The sum of two odd functions is odd, and any constant multiple of an odd function is odd.

The product of two even functions is an even function.

The product of two odd functions is an even function.

The product of an even function and an odd function is an odd function.

The quotient of two even functions is an even function.

The quotient of two odd functions is an even function.

The quotient of an even function and an odd function is an odd function.
Class Notes 1.2
Page 9 of 27
State whether the function is odd, even, or neither. Support graphically and confirm algebraically.
1. 𝑓(𝑥) = 2𝑥 4
3. 𝑓(𝑥) = √𝑥 2 + 2
2. 𝑔(𝑥) = 𝑥 3
4. 𝑔(𝑥) =
3
1+ 𝑥 2
End of part one.
Piecewise Functions
The so-called piecewise functions are functions are defined on a particular interval.
Example: Graphing Piecewise-Defined Functions
−𝑥, 𝑥 < 0
Graph 𝑦 = 𝑓(𝑥) = {𝑥 2 , 0 ≤ 𝑥 ≤ 1
1, 𝑥 > 1
Page 10 of 27
Class Notes 1.2
Example: Graph the function 𝑓(𝑥). Let f(x) be defined as follows:
2 𝑖𝑓 𝑥 𝑖𝑠 𝑎𝑛 𝑖𝑛𝑡𝑒𝑔𝑒𝑟
𝑓(𝑥) = {
1 𝑖𝑓 𝑥 𝑖𝑠 𝑛𝑜𝑡 𝑎𝑛 𝑖𝑛𝑡𝑒𝑔𝑒𝑟
a) Find each of the following: f(0), f(1/3), f(1/2), f(√2), f(2), f(π)
b) Sketch the graph of 𝑦 = 𝑓(𝑥)
Suppose m and k are numbers and let g(x) be defined piecewise as follows:
1
𝑥 + 𝑘 𝑖𝑓 𝑥 < 3
𝑔(𝑥) = { 2
𝑚𝑥 + 2 𝑖𝑓 𝑥 ≥ 3
a) Suppose m = 3/2. What must the value of k be in order for the graph of y = g(x) to be connected?
b) Suppose k = 4. What must the value of m be in order for the graph to be connected?
Class Notes 1.2
Page 11 of 27
Example: Writing Formulas for Piecewise Functions
Write a formula for the function y = f(x) whose graph consists of the two line segments in the figure
below.
Composite Functions
When the output of one function becomes the input of a second function, the resulting function is
called a composite function. This concept is used quite frequently in calculus. For example, suppose
we have the following functions:
g ( x)  x  h
and
f ( x)  x 2  3 x
 To form the composite function f ( g ( x)) , we replace every occurrence of x in f(x) with g(x).

Similarly, to form the composite function g ( f ( x)) , we replace every occurrence of x in g(x)
with f(x).
 Finally, on the subject of composite functions, there is an alternate notation:
f ( g ( x)) can also be written as f  g
g ( f ( x)) can be written g  f .
Domain of a Composite Function
Finding the domain of a composite function consists of two steps:
Step 1. Find the domain of the "inside" (input) function.
Step 2. Construct the composite function. Find the domain of this composite function. The domain of the
composite is the intersection of these two domains.
Page 12 of 27
Class Notes 1.2
For each of the following, find the composite functions f◦g, g◦f, and the domain of each composite.
1. 𝑓(𝑥) = 𝑥 + 3
3. 𝑓(𝑥) = √𝑥 + 3
𝑔(𝑥) = √9 − 𝑥 2
𝑔(𝑥) = 2𝑥 − 5
2. 𝑓(𝑥) = 𝑥 2 + 2
2
4. 𝑓(𝑥) = 𝑥−3
𝑔(𝑥) = √𝑥 − 5
5
𝑔(𝑥) = 𝑥+2
Composition of Functions Numerically
Using the table below, find the values on the right.
x
f(x)
g(x)
h(x)
Find:
-3
0
3
-2
-2
1
1
-3
0
3
0
-2
1
-3
-2
0
3
-2
-3
1
( f  g )(3)
(h  g )( 2)
( g  g )(1)
( g  f )(1)
( f  f )(3)
( f  h)(1)
( g  h)(3)
( g  f  h)(1)
Class Notes 1.2
Page 13 of 27
AP Calculus AB
Class Notes
1.3 Exponential Functions pp 22-29
Objectives:
x
x
 Graph and identify the domains and ranges for the functions e and e .
 Use the rules for exponential algebra to perform algebraic manipulation of exponential
functions.
 Distinguish between exponential growth and decay.
 Solve exponential grown and decay problems.
Omissions: none
The exponential function is one of the common functions we use in calculus. It’s most familiar
examples are compound interest (exponential growth) and radioactive decay (exponential decay).
Year
1998
1999
2000
2001
2002
2003
Population
(millions)
276.1
279.3
282.4
285.3
288.2
291.0
Ratios
(’98-‘99)
(’99-’00)
(’00-’01)
(’01-’02)
(’02-’03)
279.3/276.1 ≈ 1.0116
282.4/279.3 ≈ 1.0111
285.3/282.4 ≈ 1.0102
288.2/285.3 ≈ 1.0102
291.0/288.2 ≈ 1.0097
To say that something grows exponentially is to say that over time, for equal time intervals, the ratio of
the final to the initial value is constant. For example look at the data showing U.S. population from ‘98
– ’03 at the right. Here, over equal time intervals (one year), the ratio of the population at the end of
the year to that at the beginning of the year is an approximate constant of 1.01. This means that the
U.S. population grew at about 1% per year during this time. While that may not seem like a lot, as you
can see by the table, that is an increase of about 15 million people over a 5 year time period.
Since exponential functions are most often used to describe how the value of something changes over
time, we’re most likely to use the variable t for time, when describing these functions.
Let a be a positive real number. The general form for the exponential function is
f (t )  a t
The a is called the base of the exponent.
Page 14 of 27
Class Notes 1.3
Example 1: Graphing an Exponential Function
Graph the function 𝑦 = 2(3𝑥 ) − 4. State its domain and range.
Example 2: Finding Zeros
Find the zeros of 𝑓(𝑥) = 5 − 2.5𝑥 graphically.
Investigating Graphs: Identify the y-intercept and the asymptote of the graph of the
function.
𝑦 = 5𝑥
𝑦 = −2 ∙ 4𝑥
𝑦 = 4 ∙ 2𝑥
Algebraic Rules for the Manipulation of Exponents: **These must be memorized**
Rules for Exponents
If 𝑎 > 0 and 𝑏 > 0, the following hold for all real numbers x
and y.
1. 𝑎𝑥 ∙ 𝑎𝑦 = 𝑎𝑥+𝑦
4. 𝑎𝑥 ∙ 𝑏𝑥 = (𝑎𝑏)𝑥
2.
𝑎𝑥
𝑎𝑦
𝑎 𝑥
= 𝑎𝑥−𝑦
𝑥
5. (𝑏)
=
𝑎𝑥
𝑏𝑥
3. (𝑎𝑥 )𝑦 = (𝑎𝑦 ) = 𝑎𝑥𝑦
Later in the year, when we are solving differential equations, we’ll have to use these rules to simplify
our functions. For example,
Class Notes 1.3
Page 15 of 27
et c  et e c
Properties of Rational Exponents: Simplify the expression.
1. 35/3 ∙ 31/3
2⁄ 1⁄
3) 2
2. (5
1⁄
4
3. 4
∙ 64
1⁄
4
4.
1
−1
36 ⁄2
Solving Exponential Equations: Solve the equation.
10𝑥−3 = 1004𝑥−5
25𝑥−1 = 1254𝑥
3𝑥−7 = 272𝑥
End of Part I
Definitions: Exponential Growth, Exponential Decay
The function 𝑦 = 𝑦𝑜 ∙ 𝑎 𝑥 , 𝑘 > 0 is a model for exponential growth if 𝑎 > 1, and
a model for exponential decay if 0 < 𝑎 < 1.
Examples of Exponential Growth
Writing Models: For the following two problems, write and solve an exponential growth model that
describes the situation.
Using the data on the first page of this section, predict the U.S. population in 2010. Show all work.
You buy a commemorative coin for $110. Each year, the value of the coin increases by 4%. What is its
value after 10 years?
Page 16 of 27
Class Notes 1.3
Exponential Decay: When the ratio of final to initial values over equal time intervals of a quantity
turns out to be a number between 0 and 1, we say that we have exponential decay.
Writing Models: For the following two problems, write an exponential decay model that describes the
situation.
You buy a stereo system for $780. Each year t, the value V of the stereo system decreases by 5%. What
is the value after 5 years?
You drink a beverage with 120 milligrams of caffeine. Each hour h, the amount c of caffeine in your
system decreases by about 12%. How much caffeine is in your system after 3 hours?
Radioactive Decay and Half Life
When the initial value of a substance is Ao and the substance decays with a half life of h, The amount
of a substance, A(t ) after a time t is given by the function:
A(t )  Ao  12 
t
h
Example: Modeling Radioactive Decay
Suppose the half-life a certain radioactive substance is 20 days and that there are 5 grams present
initially. When will there be only 1 gram of the substance remaining?
Class Notes 1.3
Page 17 of 27
Thallium-208 has a half-life of 3.053 min. How long will it take for 120.0 g to decay to 7.50 g?
If 100.0 g of carbon-14 decays until only 25.0 g of carbon is left after 11 460 y, what is the half-life of
carbon-14?
Page 18 of 27
Class Notes 1.3
AP Calculus AB
Class Notes
1.5 Functions and Logarithms
pp 37-45
Objectives:
 Define one-to-one, and graphically determine whether a function is or is not one-to-one.
Describe the relation of one-to-one and the horizontal line test.
 Draw the inverse of a function. Algebraically determine the inverse of a function.
 Explain the meaning of the logarithmic function. Graph a log function. Use the properties of
log functions to manipulate equations.
Omissions: parametric equations
This section has two parts: inverse functions, and the logarithmic function. This is probably set up like
this because the logarithmic function is often introduced and taught as the inverse to the most
accessible exponential function.
Definition: One-to-One Function
A function f(x) is one-to-one on a domain D if 𝑓(𝑎) ≠ 𝑓(𝑏) whenever 𝑎 ≠ 𝑏.
(A one-to-one function is also called monotonic; it is either always
decreasing, or always increasing.)
Graphically, one-to-one means that the
graph passes the horizontal line test: no
horizontal line crosses the function more
than once.
Graphing Inverse Functions: To graph
the inverse of a function, exchange each
points x and y values. This is the same as
a reflection in the line y = x.
Example: The inverse graphically. Draw the inverse of each of the following graphs.
Class Notes 1.5
Page 19 of 27
The Inverse Numerically
Don’t forget what an inverse is: An inverse is the function that undoes its parent function.
For example,
if the function f has the
then for f 1 , we
f
f 1
following map
x
y
know…
x
y
1
3
2
8
3
11
a
b
Using the table below, find the values on the right.
Find: f 1 (3)
x
-3 -2 0 1
h 1 (3)
g 1 (1)
f 1 (3)
f(x)
g(x)
h(x)
f 1 (h(3))
( f  h 1 )(1)
f 1 ( f 1 (3))
0
3
-2
1
1
-3
3
0
-2
-3
-2
0
3
-2
-3
1
( g 1  h)(2)
Determining the Inverse Function Analytically:
Writing 𝒇−𝟏 as a Function of x
1. Solve the equation 𝑦 = 𝑓(𝑥) for x in terms of y.
2. Interchange x and y. The resulting formula will be 𝑦 = 𝑓 −1 (𝑥).
3. Interchange the domain and range.
This means, switch x with y and solve for y.
Inverses of Power Functions: Find the inverse power function.
1. 𝑓(𝑥) = 𝑥 7
2. 𝑓(𝑥) = −𝑥 6 , 𝑥 ≥ 0
3. 𝑓(𝑥) = 3𝑥 4 , 𝑥 ≤ 0
Page 20 of 27
Class Notes 1.5
Inverses of Nonlinear Functions: Find the inverse function.
1
𝑓(𝑥) = 𝑥 3 + 2
𝑓(𝑥) = −2𝑥 5 + 3
𝑓(𝑥) = 2 − 2𝑥 2 , 𝑥 ≤ 0
End of Part One
Logarithmic Functions
Logarithmic functions are inverse functions of exponential functions. I think that the best way to think
about a logarithmic function is to first look at the exponential. For example:
2 3 asks the question, “what number is the result
of multiplying 2 by itself 3 times?”
23  8
Practice:
problem
log 5 125
translation
log 2 8 is read log base 2 of 8. It asks the question,
“how many times do I multiply 2 by itself to get
8?_____
answer
log 6 36
log 8 1
log 1 / 4
log 4
1
4
1
2
log 4 4 0.38
log 64 64
log  1
Class Notes 1.5
Page 21 of 27
The case of e: e is called the natural base, and gives rise to the natural logarithm.
log e x is written as ln x .
Properties of Logarithms: Memorize the following Inverse Properties and the Algebraic Properties.
Inverse Properties for 𝒂𝒙 and 𝒍𝒐𝒈𝒂 𝒙
1. Base a: 𝑎𝑙𝑜𝑔𝑎 𝑥 = 𝑥, 𝑙𝑜𝑔𝑎 𝑎𝑥 = 𝑥, 𝑎 > 1, 𝑥 > 0
2. Base e: 𝑒ln 𝑥 = 𝑥, ln 𝑒𝑥 = 𝑥, 𝑥 > 0
The function f ( x)  ln x has a domain (0, ) and a range (, ) . Its
graph is on the right. It’s important to note that one can never take the
logarithm of a negative number or zero, and that the log does not
converge. It grows without bounds, and it is one of the slowest growing
functions.
The x-intercept of all functions in the form y  log b x is at x  1.
y




x











Properties of Logarithms

For any real numbers 𝑥 > 0 and 𝑦 > 0,
1. Product Rule: 𝑙𝑜𝑔𝑎 𝑥𝑦 = 𝑙𝑜𝑔𝑎 𝑥 + 𝑙𝑜𝑔𝑎 𝑦
𝑥
2. Quotient Rule: 𝑙𝑜𝑔𝑎 𝑦 = 𝑙𝑜𝑔𝑎 𝑥 − 𝑙𝑜𝑔𝑎 𝑦
3. Power Rule: 𝑙𝑜𝑔𝑎 𝑥 𝑦 = 𝑦𝑙𝑜𝑔𝑎 𝑥
Solving Logarithmic Functions:
−5 + 2 ln 3𝑥 = 5
ln(𝑥 + 5) = ln(𝑥 − 1) − ln(𝑥 + 1)
Page 22 of 27
log(5 − 3𝑥) = log(4𝑥 − 9)
ln(5.6 − 𝑥) = ln(18.4 − 2.6𝑥)
6.5log 5 3𝑥 = 20
10 ∙ ln 100𝑥 − 3 = 117
Class Notes 1.5
AP Calculus AB
Class Notes
1.6 Trigonometric Functions
pp 46-55
Objectives:
 Graph the six trigonometric functions and their inverses.
 Find the values of each of the six trig functions along the unit circle.
 Use the fundamental identities to simplify trig expressions.
 Use the double angle formula to reduce the order of trig expressions.
Omissions: regression problems.
A good starting point for our review of trigonometric
functions is the arc lengths along the unit circle. The
following diagram shows that the distance traveled, along
the circle of radius one, from the positive x-axis in a
counterclockwise direction are show on the right.
Graphs of Trigonometric Functions:
Students must know the graphs of all trig functions as well
as their domains and ranges.
sine graph
domain:

range:
cosine graph domain:

y








x












tangent graph domain:
range:

Class Notes 1.6






range:
y




secant graph domain:
y


x



range:
y

x













x




Page 23 of 27
cosecant graph
domain:

range:
cotangent graph
domain:
y









range:
y

x













x

Students also must know the sine and cosine values for all of the whole multiples of 



6 .and
 .
4
and should therefore be able to determine the values of all of the six trig functions at these multiples.
r




2
3
5
sin r
cos r
tan r
sec r
csc r
cot r
6
4
3
2
3
4
6
Page 24 of 27
Class Notes 1.6

7
5
4
3
5
7
11
6
4
3
2
3
4
6
Fundamental Identities
Fundamental identities are those that are based on the sine and cosine functions.
cos x
sin x
tan x 
cot x 
cos x
sin x
1
1
sec x 
csc x 
cos x
sin x
We use them to simplify other expressions.
1. cot 𝜃 sec 𝜃
2. sin ∅ (csc ∅ − sin ∅)
cot 𝑥
3. csc 𝑥
4.
1−𝑠𝑖𝑛2 𝑥
𝑐𝑠𝑐 2 𝑥−1
sin 𝛼
5. sec 𝑎 ∙ tan 𝛼
End of part 1
Class Notes 1.6
Page 25 of 27
Double Angle Formulas
You must memorize these:
sin 2 x  2 sin x cos x
cos 2 x  sin 2 x

cos 2 x   1  2 sin 2 x
 2 cos 2 x  1

The Double Angle formulas are used often in calculus. One reason for this is to reduce the order of a
problem. For example, we will not learn how to solve the integral:
2
 sin xdx
However, if we know the formula:
cos 2 x  2 sin 2 x  1 ,
2
We can write sin x as sin 2 x  12 (cos 2 x  1) . From this,
 sin
2
xdx   12 (cos 2 x  1)dx ,
and this second integral is one that we can do.
Use the figure to find the exact value of the trigonometric function.
1.
2.
3.
4.
sin𝜃
tan𝜃
cos2𝜃
sin2𝜃
Page 26 of 27
Class Notes 1.6
Inverse Trigonometric Functions
The inverse trigonometric functions arcsin, arccos and arctan are part of the AP Calculus AB
Curriculum,
y  cos 1 x or y  arccos x
y  sin 1 x or y  arcsin x
y  tan 1 x or y  arctan x
Class Notes 1.6
Page 27 of 27