Math 120R Workbook

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Section #: Roster #: Name:

Math 120R Class Notes

Math 120R: Pre-calculusUniversity of Arizona

Instructor: Deborah HurSummer Session I 2014


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Contents

Chapter 2: Functions 12.1: What is a Function? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12.2: Graphs of Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52.3: Getting Information from the Graph of a Function . . . . . . . . . . . . . . . 8

2.4: Average Rate of Change of a Function . . . . . . . . . . . . . . . . . . . . . . 122.5: Transformations of Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . 162.6: Combining Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 202.7: One-to-One Functions and Their Inverses . . . . . . . . . . . . . . . . . . . . 24Focus on Modeling: Modeling with Functions . . . . . . . . . . . . . . . . . . . . 28

Chapter 3: Polynomial and Rational Functions 343.1: Quadratic Functions and Models . . . . . . . . . . . . . . . . . . . . . . . . . 343.2: Polynomial Functions and Their Graphs . . . . . . . . . . . . . . . . . . . . . 383.3: Dividing Polynomials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 433.7: Rational Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46

Chapter 4: Exponential and Logarithmic Functions 524.1: Exponential Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 524.2: The Natural Exponential Function . . . . . . . . . . . . . . . . . . . . . . . . 564.3: Logarithmic Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 604.4: Laws of Logarithms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 644.5: Exponential and Logarithmic Equations . . . . . . . . . . . . . . . . . . . . . 68

Chapter 5: Trigonometric Functions: Unit Circle Approach 765.1: The Unit Circle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 765.2: Trigonometric Functions of Real Numbers . . . . . . . . . . . . . . . . . . . . 79

5.3: Trigonometric Graphs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 835.6: Modeling Harmonic Motion . . . . . . . . . . . . . . . . . . . . . . . . . . . . 885.4: More Trigonometric Graphs . . . . . . . . . . . . . . . . . . . . . . . . . . . . 925.5: Inverse Trigonometric Functions and Their Graphs . . . . . . . . . . . . . . . 96

Chapter 6: Trigonometric Functions: Right Triangle Approach 996.1: Angle Measure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 996.2: Trigonometry of Right Triangles . . . . . . . . . . . . . . . . . . . . . . . . . 1056.3: Trigonometry Functions of Angles . . . . . . . . . . . . . . . . . . . . . . . . 1096.4: Inverse Trigonometric Functions and Right Triangles . . . . . . . . . . . . . . 112

Chapter 7: Analytic Trigonometry 1167.1: Trigonometric Identities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1167.2: Addition and Subtraction Formulas . . . . . . . . . . . . . . . . . . . . . . . 1197.3: Double Angle Formulas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1227.4: Basic Trigonometric Equations . . . . . . . . . . . . . . . . . . . . . . . . . . 1267.5: More Trigonometric Equations . . . . . . . . . . . . . . . . . . . . . . . . . . 129


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Appendix: Fundamental Algebra Skills Review 1321.3: Algebraic Expressions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1321.4: Rational Expressions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1371.5: Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1411.6: Modeling with Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1431.7: Inequalities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 147

1.8: Coordinate Geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1491.9: Graphing Calculators; Solving Equations and Inequalities Graphically . . . . 1531.10: Lines . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1551.11: Making Models Using Variation . . . . . . . . . . . . . . . . . . . . . . . . . 159

3


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1

Chapter 2: Functions

2.1: What is a Function?

Topics

1. Definition of Function.

2. Four ways to Represent a Function.

3. Domain of a Function.

4. Piecewise Defined Functions.

Class Notes and Examples

1. Definition of Function.

A function f is a rule that assigns to each element x in a set A exactly one element, calledf(x), in a set B.

Intuitively:

2. Four ways to Represent a Function.

There are four ways in which a function can be represented. In each instance, we shallexplore criteria for checking the function definition quickly.

Example 1. 1. Visual (Graph)

2. Numerical (Table)


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3. Algebraic (Equation)

4. Verbal (Description written in words)

3. Domain of a Function.

The domain of a function is the set of all inputs for the function.

Intuitively:

We shall return to the previous examples to find how to the functions domain quickly.

1. Visual (Graph)

2. Numerical (Table)


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3. Algebraic (Equation)

4. Verbal (Description written in words)

Lets practice finding domains for functions represented algebraically. For each of the fol-lowing, find the domain.

4. Piecewise Defined Functions.

Sometimes the formula/graph that is to be used to find the output of a function depends onthe input. This issue can be compactly described using piecewise defined function notation.

1. The functionf(x) uses the formula x2 for x values less than 2, and uses the formulax + 1 for x values 2 or greater. Write this in piecewise defined function notation:

2. Write the absolute value function in piecewise defined function notation:


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3. GivenFX(x) =

x2 1 x


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2.2: Graphs of Functions

Topics

1. Graphing Functions.

2. Basic Graphs.

3. Graphing Piecewise Defined Functions.

4. Equations that Define Functions.


1. Graphing Functions.

Formally, the graph of the function fis the set of ordered pairs (x, y) that satisfyf(x) =y.In other words, the graph consists of all solutions (x, y) to the equation y = f(x).

We can graph functions in a few ways:

1. Plotting points. (drawback: time consuming and sometimes inaccurate depending onthe number of points plotted and issues with drawing by hand.)

2. With your graphing calculator. (drawback: a good window (often found using logicand tools such as zoom functions and table) is absolutely necessary to get a usefulgraph).

3. Knowing basic graphs and using tools that we cover throughout this course. (drawback:we will be developing these skills as we progress through the course).

2. Basic Graphs.

Students must know the following graphs from memory:

Linear Functions: f(x) =mx + b

Power Functions: f(x) =xn


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Root Functions: f(x) = nx

Reciprocal Functions: f(x) = 1xn

Absolute Value Function: f(x) = |x|

3. Graphing Piecewise Defined Functions.

Using our basic graphs, we can graph simple piecewise defined functions.

1. Draw a sketch ofg(x) = x2 x 1


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4. Equations that Define Functions.

More practice on this topic that we lightly covered in the last section:For all the following, determine ify is a function ofx. For those that do give functions, findthe domain of the function.

1. y+ 2xy= 1

x

2. x2 2y= 12

3. y+ 2xy=

x

4. x2 + y2 =x


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2.3: Getting Information from the Graph of a Function

Topics

1. Domain and Range.

2. Intervals of Positive and Negative.

3. Zeros and Horizontal Intercepts.

4. Increasing and Decreasing Functions.

5. Local Maximum and Minimum Values

of a Function.


1. Domain and Range.

Recall the definitions of domain and range:

The domain is:

The range is:


Definitions:

fis positive on an interval I iff(a)is positive for all a in I.

f is negative on an intervalI iff(a) is negative for all a in I.

3. Zeros / Horizontal Intercepts.

Recall that a horizontal intercept is a point on the graph ofy = f(x) withy coordinate equalto 0.


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A zero is an input value a such that f(a) = 0.

What is the connection between zeros and horizontal intercepts?

What is the connection between zeros/horizontal intercepts and intervals of positiveand negative?


f is increasing on an interval I iff(a)< f(b) whenever a < bin I.

fis decreasing on an interval I iff(a)> f(b) whenever a < bin I.

5. Local Maximum and Minimum Values of a Function.

Definitions:The function value f(a) is a local maximum value off if

f(a) f(x)whenxis near a. In this case, we say that fhas a local maximum at a.

The function value f(a) is a local minimum value off if

f(a) f(x)whenxis near a. In this case, we say that fhas a local minimum at a.


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Local maxima and minima are sometimes refered to as local extrema. In other words, a localextremum is either a local maximum or local minimum.

What is the connection between local extrema and intervals of increasing and decreas-ing?

For each of the graphs, find the following:

-2 -1 0 1 2 3

-

-2

-1

1

2

1. Domain and Range 1. Domain and Range

2. Intervals of Positive and Negative. 2. Intervals of Positive and Negative.

3. Zeros / Horizontal Intercepts. 3. Zeros / Horizontal Intercepts.

4. Increasing and Decreasing Functions. 4. Increasing and Decreasing Functions.

5. Local Maximum and Minimum Values 5. Local Maximum and Minimum Values.


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Calculator example: Consider y = x

6 x. Use your graphing calculator to help yoursketch a complete graph of the function, labeling all intercepts and local extrema. Then findthe following:

1. Domain and Range


3. Zeros / Horizontal Intercepts.


5. Local Maximum and Minimum Values

Highway Engineering Example: A highway engineer wants to estimate the maximumnumber of cars that can safely travel a particular highway at a given speed. Under some

assumptions (see text), she finds that if cars are traveling at a speed s mph, the number Nof cars that can pass a given point per minute is modelled by the function

N(s) = 88s

17 + 17s

20

a) What speed maximizes traffic flow? (round to the nearest mph).

b) Find the intervals on which the function is decreasing(round to the nearest mph). In-terpret the interval in the context of the problem.


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2.4: Average Rate of Change of a Function

Topics

1. Average Rate of Change (ARC).

2. Linear Functions.

3. Other Functions.

4. Interpreting ARC in Application Prob-

lems.


1. Average Rate of Change.

The average rate of change (ARC) of the function y = f(x) betweenx = a andx = b is

average rate of change =change in y

change in x=

The easiest examples of computing ARC are via table and graph:

1. Find the average rate of change over the intervalx = 1 tox = 3.x -1 0 1 2 3f(x) 2 6 9 14 17

2. Find the average rate of change over the interval (10,15).

2. Linear Functions.

1. Find the average rate of change fory = 2x + 1 over the interval (2, 4).


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2. Find the average rate of change fory = 2x + 1 over the interval (x, x + h).

What should be true about ARC for linear functions?

3. Other Functions.

For each function, find the average rate of change over the indicated interval.

1. y= x2 for x = 2 tox = 3.

2. y= 3 + 2x x2 for x = 0 tox = 2.

3. y= 1x2

over (1, 2).


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4. y= x2 for x = a to x = b.

5. y= 3 + 2x x2 over (x, x + h).

6. y=

xover (x, x + h).

7. y= 1x

over (a, b).

4. Interpreting ARC in Application Problems.

Calculate and interpret the ARC in the context of the scenario, including the value and unitsin your response.


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1. The height of a ball in feet aftert seconds is given byy = 40t 16t2. Find the averagerate of change of height over the interval (0 , 1).

2. A soft-drink vendor analysis his sales records and finds that if he sellsx cans of sodain one day, his profit (in dollars) is given by P(x) = 0.001x2 + 3x 1800. Find theaverage rate of change over the interval (1000, 2000).

3. The speed of a car in mph can be expressed in terms of the length of a skid mark in feetwhen the brakes are applied. Suppose for a certain car, this relationship is expressed

as S= 25L. Find the average rate of change from (20, 25).


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2.5: Transformations of Functions

Topics

1. Vertical Transformations

2. Horizontal Transformations

3. Multiple Transformations.

4. Even and Odd Functions


Before we summarize all the transformations, I would like to explore some basic examplesto motivate: Given the following table of values,

x -2 0 2 4 6f(x) 1 -2 3 1 3

Find tables for the following functions. Describe what happens graphically, when comparingthe original table to the new table.

(a)y= f(x) + 1 (b) y = f(x 2)

(c)y= 2f(x) (d) y = f(2x)

(e)y= f(x) (f) y= f (x)


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1. Vertical Transformations.

Vertical Shifts

Vertical Stretch/Shrink

Reflection over x axis.

2. Horizontal Transformations.

Horizontal Shifts

Horizontal Stretch/Shrink

Reflection over y axis.


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3. Multiple Transformations

When there are multiple transformations, it is critical to apply the transformations in anappropriate order to obtain the correct graph. Here are a few examples:

1. Explain how the graph ofg(x) = 2f(x 1) + 4 is obtained from the graph ofy = f(x).

2. Sketch the graph of the function by starting with the graph of a standard function andapplying transformations. f(x) = (x + 2)2 3

3. Sketch the graph of the function by starting with the graph of a standard function andapplying transformations. p(t) = 3

x + 1

4. Find a formula which transforms f(x) =x3 by shifting to the right 2 units, reflectingover the x axis, then shifting up 2 unite.

5. Find a formula which transformsg(x) = |x|by horizontally compressing it by a factorof 1/2, shifting to the left 1, then shifting down 3 units.


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4. Even and Odd Functions.

Recall from section 1.8, that certain graphs have special symmetries. In the case of functions,there are two types of symmetry that are possible:

In function notation we may write these graphical characteristics as:

To Test for Symmetry of a Function Algebraically1. First, evaluatef(x) and fully simplify.2. Next, compare your simplified expression to f(x) andf(x).

Iff(x) simplifies to f(x), the function is even. Iff(x) simplifies tof(x), the function is odd. Iff(x) is neither, then we sayf(x) is neither even nor odd.

Lets practice with a few examples; Determine if the function is even, odd or neither. Identify

symmetry, if any.

1. f(x) =x2 + 1

2. f(x) =x3 + 1

3. f(x) = xx2+2

4. f(x) = |cx| where c is a positive constant.


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2.6: Combining Functions

Topics

1. Sums, Differences, Products, and Quo-tients.

2. Composition of Functions.


1. Sums, Differences, Products, and Quotients.

Given functions f(x) andg(x), we define the following new functions:

Function Domain(f+ g)(x) =

(f g)(x) =

(f g)(x) =

fg

(x) =

1. Suppose f(x) has domain [2, 1] with a zero at x =1 and g(x) has domain [4, 0]with zeros at x = 3,2. Find the domains of

(a) f+ g (b) f g

(c)f g (d) fg

(e) gf.


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2. Let f(x) = xx24 andg(x) =

3 x. Find

Function Domain(f+ g)(x) =

(f g)(x) =

(f g)(x) =

fg (x) =

gf

(x) =

2. Composition of Functions.

Given functions f(x) andg(x), we define the following new functions:(f g)(x) =

(g f)(x) =

To find the domain of a composition, there are two approaches:

1. For algebraic expressions, use the unsimplified composition expression to determinethe domain.

2. Look for x values that are both in the domain of the inside (first) function and give yvalues that in the the domain of the outside (second) function.


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1. Let f(x) = xx24 andg(x) =

3 x. Find

Function Domain(f g)(x) =

(g f)(x) =

(g g)(x) =

(f f)(x) =


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2. Supposefandg are given by the following tables. Find the tables off g,f g,f fand g g. State the domain and range.

x 1 2 3 4 5f(x) 2 3 1 5 7

x 1 3 5 7 9g(x) 2 4 1 6 3

Function Domain(f

g)(x)

(g f)(x)

(g g)(x)

(f f)(x)

3. Given the functionh(x) =f g(x), find possible expressions for f(x) andg(x).(a) h(x) = 1

(x1)2+3

(b) h(x) = (x2 1 + 5)3.


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2.7: One-to-One Functions and Their Inverses

Topics

1. One to One Functions.

2. The Inverse of a Function.

3. Inverse Application Problems.


1. One to One Functions.

Intuitively, a function is called one to one if:

More formally,f(x) is one to one if:

f(x1) =f(x2) whenever x1=x2.In other words, ifx1 and x2 are different input values, the function assigns them todifferent output values.

Equivalently, we may write:Iff(x1) =f(x2), then x1 = x2.In other words, if the output values are the same, the input values must be the same.So each output value comes from exactly one input value.

Now we shall explore via example how to check if a function is one to one or not. For each

of the following, determine if the given function is one to one.

1. x 1 3 4 7 -2

f(x) 3 1 5 -1 1

2. x 1 3 4 7 -2

f(x) 3 1 5 -1 -4

3. The function which assigns to each UA student their Student ID Number.

4. The function which assigns to each item in a grocery store its price.


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5. 6.

7. f(x) =x2 + 1 8. f(x) = 2x+1x3

2. The Inverse of a Function.

The inverse of a function f(x) can be thought of intuitively as:

a function which simply swaps the role of input and output. the function which cancels the original function.

Some easy examples to further explain these two perspectives:

IfV =f(h) gives volume as a function of height, the inverse gives height as a functionof volume.

IfF = T(C) gives Farhenheit temp as a function of Celcius temp, the inverse givesCelcius temp as a function of Farhenheit temp.

Adding by 1 is cancelled by subtracting by 1. Multiplying by 2 is cancelled by dividing by 2.A function f(x) has an inverse (is invertible) if and only if it f(x) is one to one.

We denote the inverse by f1

, where -1 is NOT an exponent.

Important Facts:

f1(y) =x y= f(x)f1(f(x)) =xf(f1(x)) =x


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Next, go back to the earlier examples and find their inverses (if the inverse exists), thensummarize here the procedures used for each type of problem:Tables

Graphs

Functional notation

3. Inverse Application Problems.

For application problems, it is important to NOT switch the variables, as we often choosevariable names based on the application quantities:

1. The volume of liquid in a cylindrical container of height 12 is given by V = 12r2.Find the inverse and explain its practical use.

2. Cassie can complete a task in 4 hours. If she works with a partner that takest hoursto complete the job on their own, we can express their working together rate as

r=1

4+

1

t

Find the inverse and explain its practical use.


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3. The concentration obtained by mixingx ml of 40% solution with 50 ml of 20% solutionis given by

C=0.4x + 10

x + 50

Find the inverse and explain its practical use.


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Focus on Modeling: Modeling with Functions

Topics

1. Making and Using Models.

2. Problems about Geometric Objects.

3. Problems about Distance, Rate, andTime.

4. Problems about Cost, Revenue, and/orProfit.

5. Miscellaneous.



We will follow these guidelines to have an organized approach at handling word problems.

Guidelines for Modeling with Functions

1. Draw a picture (if relevant) and designate variables for unknown quantities.Clearly identify the desired input and output variables.

2. Express the main model in words.

3. Set up the model, using the variables chosen above.

4. If necessary, eliminate any unwanted variables. In this step, a substitution usinga secondary equation might be necessary. Your final expression should give thedesired output as an expression in ONLY the desired input variable.

5. If asked further questions about the model, it is necessary to determine the

domain that makes sense in the context of the problem. This can be logicallydetermined in most cases, with a picture being very helpul.

We shall exemplify these steps in the following problem:Exercise 2 from text:A poster is 10 inches longer than it is wide. Find a function that models its area A in termsof its width w.


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2. Problems about Geometric Objects.

1. Exercise 10 in text.Find a function that models the area Aof a circle in terms of its circumference C.

2. Exercise 17 in text.A rectangle is inscribed in a semicircle of radius 10. Find a function that models theareaA of the rectangle in terms of its height h.


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3. A right triangle has an area of 3000 square centimeters.

(a) Find a formula for the length of the hypotenuse in terms of the length of the base.

(b) Determine the minimum length of hypotenuse, rounded to the nearest 0.001 andinclude units.

3. Problems about Distance, Rate, and Time.

1. Exercise 13 in text.Two ships leave port at the same time. On sails south at 15mph and the other sailseast at 20mph. Find a function that models the distanceD between the ships in termsof the time (in hours) elapsed since their departure.


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2. Exercise 30 in text.A man stands at a point A on the bank of a straight river which is 2 miles wide. Toreach point B, 7 miles downstream on the opposite bank, he first rows his boat to apoint Pon the opposite bank and then walks the remaining distance x to B. He canrow at a speed of 2mph and walk at a speed of 5mph.

(a) Find a function that models the time needed for the trip.

(b) Where should he land so that he reachesB as soon as possible?

4. Problems about Cost, Revenue, and/or Profit.

1. Suppose a square-bottomed box with no top is to be constructed from two differentmaterials; the material for the sides is $0.50 per square inch and the material for the

bottom is $1.25 per square inch. Suppose we are to construct such a box with a volumeof 100 cubic inches.

(a) Find a formula for cost Cas a function of the length of the square base,s.

(b) What is the minimum cost of constructing such a box?


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2. Suppose a manufacturer needs to make a open top cylindrical container. Suppose thematerial for the tube costs $0.10 per square inch and the material for the bottom costs$0.25 per square inch. They have a budget of $5 per box.

(a) Find a formula for the volume of the container as a function of the radiusr .

(b) What is the practical domain of this function?

(c) Find the maximum volume of such a box, rounded to the nearest 0.01.

5. Miscellaneous.

1. Exercise 20 in text.Find two positive numbers whose sum is 100 and the sum of whose square is a minim-ium.


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2. Exercise 31 in text.A bird is released from point A on an island, 5 miles from the nearest point B on astraight shoreline. The bird will travel to his nesting area which is 12 miles downstream,on the shoreline. The bird flies to a point C on the shoreline, then flies along theshoreline to its nesting area D. Suppose the bird requires 10kcal/mile of energy to flyover land and 14 kcal/mile to fly over water.

(a) Draw a diagram to organize the above information. Letxrepresent the distancebetween points B and C.

(b) Find how much energy the total trip takes in terms ofx as defined in part (a).

(c) If the bird instinctively chooses the path that minimizes energy expenditure, towhat point does it fly?


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Chapter 3: Polynomial and Rational Functions

3.1: Quadratic Functions and Models

Topics

1. General form of a quadratic.

2. Zeros of a Quadratic.

3. Maximum or Minimum of a Quadratic.

4. Standard Form of a Quadratic.

5. Factored Form of a Quadratic.

6. Applications.


1. General form of a quadratic.

A quadratic function is a function that may be written in the form

f(x) =ax2 + bx + c

wherea, b, c are real constants with a = 0. We calla the leading coefficient.

2. Zeros of a Quadratic.

The zeros of a quadratic may be found by factoring (if possible) or by using quadraticformula:

x=b b2 4ac

2a

3. Max/min of a Quadratic

Due to the shape of the parabola (graph of a quadratic), a quadratic either has one globalmax or one global min.

What criteria determines whether the vertex is a max or a min?

the vertex gives a minimum if

the vertex gives a maximum if

The vertex is also nice for symmetry reasons, as the vertical line through the vertex also

happens to give the line of symmetry for our parabola.

The text uses completing the square to find the vertex. We shall use a very nice shortcut toavoid completing the square. We derive it here:


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Vertex formula:To find the vertex (h, k) of the quadratic f(x) =ax2 + bx + c

h=

k=

4. Standard Form of a Quadratic.

A particularly convenient way to represent any quadratic is what is referred to as standard

form:

Standard Form of a Quadratic:

f(x) =a(x h)2 + kwhere (h, k) is the vertex and a is the same leading coefficient used in general form.

For the following, a) find the vertex, b) write the quadratic in standard form, c) sketch agraph of the quadratic.

1. f(x) = 2(x 3)2 + 1.

2. f(x) =x2 + 4x + 2.


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6. Applications.

For applications modeled by a quadratic function, we now have exact methods to find theintercepts and extrema.

1. A soft drink vendor at a popular beach analyzes his sales records and finds thatif he sells x cans of sode in one day, his profit (in dollars) is given by P(x) =

0.001x2 + 3x 1800. What is his maximum profit for the day and how many cansmust he sell for maximum profit?

2. A farmer has 2400 feet of fencing and wants to fence off a rectangular area borderinga long straight stream. He does not need fence along the stream. Find the maximumarea that he can enclose.

3. A movie theatre estimates that each $0.50 increase in ticket price, ticket sales decreaseby 40 tickets. If the current price is $10.50 and sales are about 1000 tickets, how muchshould they charge for tickets to maximize revenue?


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3.2: Polynomial Functions and Their Graphs

Topics

1. General Form of a Polynomial.

2. End Behavior of a Polynomial.

3. Zeros of Polynomials.

4. Multiplicity of a Zero.

5. Graphing Polynomials.


1. General Form of a Polynomial.

General Form of a Polynomial:A polynomial function of degree n is a function that may be written in the form:

P(x) =anx2 + an1x

n1 + + a1x + a0wheren is a non-negative integer and an

= 0.

The numbersa0, . . . , an are called the coefficients of the polynomial. The number a0 is the constant coefficient or constant term. The number an is called the leading coefficient. The termanxn is called the leading term.

Lets identify the above defined terms for a concrete example:

f(x) = 2x5

+ x4

6x2

+ x 3

Polynomial Graph Properties:In general all polynomial graphs share the following properties:

The domain is (,). The graph is continuous (no breaks or holes). The graph is smooth (no cusps/pinches/corners).


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2. End Behavior of a Polynomial.

In general, given any function, the end behavior of the function is defined to be what happensto the function values asx gets very large (stated in two cases: xlarge and positive,x largeand negative). For example:

The graph ofy = x2 is a parabola that is concave up. So, as x gets large and positive,y gets large and positive. Similarly, as x gets large but negative, y gets large and positive.

More compactly, we write:

Or equivalently, we could write:

Both notations are acceptable for end behavior.

To find the end behavior in the special case of a polynomial, we will find that the end behav-ior is dominated by the leading term ( a monomial since it is just one term). We summarizethe end behavior of monomials below:

For each of the following, find the end behavior:1. f(x) = 2x3 7x4 + 1 2. f(x) = 2x(x + 1)2(x 3)3


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In general, the multiplicities indicates the shape of the graph near each x-intercept.Shape of the Graph Near a zero of multiplicity m:

5. Graphing Polynomials

We can put all of the above facts together to produce a very good sketch of a polynomial infactored form.

General Method of Graphing a Polynomial:

1. Identify the leading term and end behavior.

2. Factor the polynomial (if necessary). Note the zeros and their correspondingmultiplicities.

3. Draw the ends of the graph.

4. Fill in the middle of the graph starting from left to right, crossing at the x-intercepts with the appropriate multiplicity

Sketch a graph of the following polynomials:a) f(x) =x4 2x3 3x2 b) f(x) = 2x3(x 4)(x + 5)2.


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Now we are ready to try to construct polynomials: For each of the following, find anequation for a polynomial that fits each description/graph:1. P(x) has intercepts (0, 18), (1, 0), (2, 0), (3, 0)

2. 3.

4. 5.


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3.3: Dividing Polynomials

Topics

1. Long Division of Polynomials.

2. Synthetic Division.

3. The Remainder and Factor Theorem.



1. Long Division of Polynomials.

Division Algorithm:IfP(x) and D(x) are polynomials, with D(x)= 0, then there exist unique polynomials Q(x) andR(x), where R(x) is either 0 or of degree less than the degree ofD(x) such that

P(x) =D(x) Q(x) + R(x)

Or we may write this as:

P(x)

D(x)=Q(x) +

R(x)

D(x)

Long division is one method that always works in finding the quotient and remainder asstated in the division algorithm. We shall look at a few examples to

1. DivideP(x) = 6x2 2x+ 1 by D(x) = x 1. Write your final answer in the formP(x) =D(x) Q(x) + R(x)

2. DivideP(x) = x2 2x3 + 2 by D(x) = x2 + 1. Write your final answer in the formP(x) =D(x) Q(x) + R(x)


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2. Synthetic Division.

In the very special case that the divisor can be written as (x c), or as a product of factorsof that form, there is a short cut to polynomial division called synthetic division. We willexplore this process via example below.

1. DivideP(x) = 6x2 2x+ 1 by D(x) = x 1. Write your final answer in the formP(x) =D(x) Q(x) + R(x)

2. DivideP(x) = 4x3 3x+ 5 by D(x) = x+ 2. Write your final answer in the formP(x) =D(x) Q(x) + R(x)

3. The Remainder and Factor Theorem.

If you only need to determine the remainder after dividing by a factor (xc), there is a veryeasy way to do so, referred to as the remainder theorem.

Remainder Theorem:If the polynomial P(x) is divided by (x c), then the remainder is the value P(c).

1. Use synthetic division and the Remainder Theormem to evaluateP(c), whenP(x) = 2x2 + 9x + 1 and c = 1.


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Notice that when the remainder is zero, that means that (x c) is a factor; this gives (inpart) the factor theorem introduced in 3.2 class notes:

Factor Theorem:IfP(x) is a polynomial, then the following are equivalent:

c is a zero ofP(x) (x c) is a factor ofP(x).


Now we shall combine our newly reviewed techniques of polynomial division and our 3.2concepts to graph polynomials of higher degree.

Sketch a graph of each of the following polynomials:

1. y=

x3

x2 + 8x + 12. Hint:

2 is a zero.

2. y= x4 2x2 + x Hint: 1 is a zero.


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3.7: Rational Functions

Topics

1. Rational Functions.

2. Domain, Vertical Asymptotes, and

Holes.3. Zeros.

4. End Behavior Asymptotes: Horizontal,Slant, or other.

5. Graphing Rational Functions.6. Applications.


1. Rational Functions.

A rational function r(x) is a function that may be written in the form r(x) = P(x)Q(x)

, where

P(x) andQ(x) are polynomials.

Which of the following are rational functions?

y= x1/2 y= x1 y= x2 + 2

y = 2x y=1 + x

x y= 3

2. Domain, Vertical Asymptotes, and Holes.

We shall study a few key examples first before summarizing results on the next page.

For each of the following, determine the domain. Graph the function in your calculator andanalyze what happens at the domain exclusions.

1. y= x+1(x)(x2)

2. y= x(x+1)(x+1)


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3. y= x22x3x29

Domain:

Vertical Asymptotes

Holes

3. Zeros.For each of the following, determine the zeros.

1. y= x+1(x)(x2)

2. y= x(x+1)(x+1)

3. y= x22x3x29


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4. End Behavior Asymptotes: Horizontal, Slant, or other.

To best understand the end behavior, it is convenient to rewrite the rational function inlong-divided form (as seen in the division algorithm in section 3.3).

P(x)

D(x)=Q(x) +

R(x)

D(x)

We shall study a few key examples first before summarizing results on the next page.

For each of the following, long divide and rewrite in the above form. Using logic or yourgraphing calculator, determine what happens to the graph for extreme values ofx (i.e. findend behavior).

1. y= 4x+8x1

2. y= x+5x21

3. y= 8x2+2x1x+3

The most general tool in determining the end behavior and any (end behavior) asymptoteis by considering the quotient from long division.

For extreme values ofx, P(x)D(x) Q(x)


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Lets try all of the above steps to generate good graphs of the following:

1. 2x(x+2)(x1)(x4)

2. 2x2+x1x3x2

3.

x2+5x+4

x3

4. 2x3+2xx21


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The next pattern we will consider is:This leads us naturally to exponential functions. Lets look at a specific example via table:

x y-2 1/9-1 1/30 11 323

In general we have the following definition:Exponential Functions:The exponential function with base a is defined for all real numbers x by

y= ax

whereais a positive constant not equal to 1.

2. Graphs of Exponential Functions.

The graphs of exponential functions break into two general shapes, which have some of thesame key features:

Increasing Exponential: Decreasing Exponential:

Domain: Domain:

Range: Range::

Asymptote: Asymptote:


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1. A sum of $5000 is invested at an interest rate of 12% per year. Find the amounts inthe account after 10 years if interest is compounded as follows:

(a) Annually.

(b) Quarterly.

(c) Monthly.

Annual Percentage Yield:

The annual percentage yield (APY, also called effective growth rate) can be inter-preted as follows:

The actual amount of growth observed in a year.

The simple interest rate (n=1) that yields the same amount at the end of theyear.

2. Find the APY for an account that is compounded monthly at a rate of 4%.


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4.2: The Natural Exponential Function

Topics

1. The Number e.

2. The Natural Exponential Function.

3. Continuously Compounded Interest.


1. The Number e.

We begin by considering a silly example. Suppose you invest $1 in an account bearing 100%interest for one year, compounded ntimes per year. Using a graph or table, explain brieflyhow the accumulated amount changes as the number of compoundings increases.

We shall define the number e to be:

2. The Natural Exponential Function.

The Natural Exponential Function:The natural exponential function is defined for all real numbers x by

y= ex

wheree is the constant defined above.Graph:


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2. The balance in an account is $7361 after 3% interest is compounded continuously for5 years. What was the initial amount of the investment?

3. Find the APY for an account that is compounded continuously at a rate of 4%. Roundto the nearest 0.01%.

Another important model that will prove useful is called the logistic growth model.This model has some key features that we will explore via example.

4. An infectious disease begins to spread in a small city of population 10,000. Aftertdays,the number of people who have succumbed to the virus is modeled by the function

v(t) = 10, 000

5 + 1245e0.97t

(a) How many infected people are there initially?

(b) Find the number of infected people after one day, approximated appropriately.

(c) Graph the functionv and describe its behavior.


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Logistic Growth Model:For quantities Pthat grow nearly exponentially for small values of t then approacha horizontal asymptote, the logistic model may fit the data well:

P(t) = d

1 + kect

wherec, d, k are positive constants.

5. For the logistic growth model, determine the following:

(a) What is the y intercept?

(b) What happens as t

?


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4.3: Logarithmic Functions

Topics

1. Review from 2.7: Inverse Functions.

2. Logarithmic Functions.

3. Common Logarithms.

4. Natural Logarithms.


1. Review from 2.7: Inverse Functions.

Recall, the inverse of a functionf(x) can be thought of intuitively as:

a function which simply swaps the role of input and output. the function which cancels the original function.

To motivate, lets explore some exponential equations:

a) 2x = 8 b) 212x = 1/4

c) 2x = 2 d) 2x = 5

So in the context of exponentials, we are currently at a standstill for solving certain typesof equations that have irrational solutions. This is where logarithms will come in!

2. Logarithmic Functions.

In order to cancel the exponential base a we shall define the base a logarithm, denoted byloga as follows:

Definition of Logarithm base a

x= loga(y) is the solution to the equation y = ax, where y is any positive number.

In other words, the following two equations are equivalent: y= ax loga(y) =x

In particular, a logarithm and an exponential of the same base are INVERSE functions.


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Exponential(Original Function) Logarithm (Inverse Function)

input: input:

output: input:

domain: domain:

range: range:

asymptote: asymptote:

Graph: Graph:

Exponential(Original Function) Logarithm (Inverse Function)

input: input:

output: input:

domain: domain:

range: range:

asymptote: asymptote:

Graph: Graph:


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In addition to the graphical results obtained above from noting the inverse relationship, weshall find a lot of nice algebraic result too (we shall investigate this more deeply in the nextsection):

1. Solve the equation: 2x = 5

2. Solve the equation: 2 (3)x

1

= 8

3. Simplify each expression:

(a) log2(4)

(b) log1/3(9)

3. Transformation of Logarithms.

Describe each transformation in an appropriate order, identifying an appropriate base graph.Determine the domain, range, and asymptote.

1. y= log2(2x + 5) + 1.

2. y= log2(x 1).


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4. Common and Natural Logarithms.

Since y = 10x and y = ex are widely used exponentials in real-life applications, we havespecial notation for the corresponding base logarithms.Common and Natural Logarithms

Define the common logarithmas the base 10 logarithm, denoted simply as log(x).

log(x) is just a shortened way to write log10(x).

Define the natural logarithm as the base e logarithm, denoted simply as ln(x).

ln(x) is just a shortened way to write loge(x).

Logarithms are particularly useful when values are extremely small, extremely large, andspan a wide range. Here we shall consider one application example in which logarithms areused in the model:

1. The age of an ancient artifact can be determined by the amount of radioactive carbon-14 remaining in it. It D0 is the original amount of carbon-14 and D is the amountremaining, then the artifacts age A in years is given by

A= 8267 ln

D

D0

Find the age of an object if the amount D of carbon-14 that remains in the object is73% of the original amount.


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4.4: Laws of Logarithms

Topics

1. Review from 1.2: Exponents Proper-ties.

2. Laws of Logarithms.

3. Expanding and Combining LogarithmicExpressions.

4. Change of Base Formula.


1. Review from 1.2: Exponents Properties.

Properties of Exponents

Sum Property : bm+n =

Subtraction Property : bmn

=

Multiplication Property : bmn =

Recall that logarithms were constructed as inverses to exponential functions:Exponential(Original Function) Logarithm (Inverse Function)

input: input:

output: input:

This relationship gives rise to equivalent properties in the language of logarithms:

2. Laws of Logarithms.

Properties of Logarithms

Sum Property :

Subtraction Property :

Multiplication Property :


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3. Expanding and Combining Logarithmic Expressions.

1. Expand each logarithm completely:

(a) log2(4yz1/3)

(b) ln

S5T2

Q3

(c) log

x2y3

zw4

(d) ln

x

y

x

(e) log(10(x

1)2(x + 10)3)


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2. Rewrite as a single logarithm.

(a) 2ln x + 3ln y 4 ln z

(b) ln(6x) + 12ln x ln(2x)

(c) log(5x) log(x) 3 log(3y) + log(t)

(d) ln 5x

3

yx

(e) log(10(x 1)2(x + 10)3)


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4. Change of Base Formula.

In instances where we would like to change from base b logarithms to base a logarithms (inpractice, we often use a=10 or e), there is a simple formula which helps us to do so. Sinceits easy to derive, we shall do that first:

In summary, we have just proven the change of base formula:Change of Base Formula

When is this formula useful? Here we shall see two distinct examples of its usage:

1. Evaluate log3(6) accurate to 4 decimals.

2. Simplify the expression log3(x) log5(3).

3. Simplify the expression log100(x) + log(x).


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4.5: Exponential and Logarithmic Equations

Topics

1. Exponential Equations.

2. Logarithmic Equations.

3. Application Problems.


1. Exponential Equations.

Main focus: Apply algebra skills such as isolating an exponential expression, using the oneto one property of exponentials, substitution, and factorization to one main end:

Using algebra, reduce our equation to one or more simple equations in the form

y= asome variable expression.

Only when we reach this simple form will log base a be able to cancel the exponential base a.

The next page lists several exponential examples, which we will turn to now, before movingon to log equations.

2. Logarithmic Equations.

Main focus: Apply algebra skills such as isolating a logarithmic expression, combining log-arithms, using the one to one property of logarithms, substitution, and factorization to onemain end:

Using algebra, reduce our equation to one or more simple equations in the form

y= loga(some variable expression).

Only when we reach this simple form will exponential base a be able to cancel the log base a.

The next page lists several logarithmic examples, which we will turn to now.


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15. log9(x 5) + log9(x + 3) = 1.

16. log2(log4(x)) = 1.

17. log2(2x 3) = 1


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3. Application Problems.

1. Find the time required for an investment of $5000 to grow to $8000 at an interest rateof 7.5% per year, compounded quarterly.

2. Find the time required for a student loan to triple in balance, if the interest rate is 9%per year, compounded continuously. (Assume no other fees are assessed for simplicity).

3. A small lake is stocked with a certain species of fish. The fish population is modeledby the function

P = 10

1 + 4e0.8t

(a) Find the fish population after 3 years. Approximate to the nearest fish.

(b) After a very long time, what happens to the population of the fish, based on thismodel?

(c) After how many years will the fish population reach 5000 fish?


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More Practice:

Find the solution(s) exactly.

1. 10x = 5

2. 21x = 3

3. 4 + 35x = 8

4. 2e2x+1 = 200

5. 10

1 + ex = 4

6. x22x 2x = 0

7. 4x3e3x 3x4e3x = 0


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8. e2x 3ex = 2

9. ex 12ex 1 = 0

10. ln(x) = 10

11. log(x

2) = 1

12. log2(x2 x 2) = 2

13. log(x) + log(x 1) = log(4x)

14. log5(x + 1) log5(x 1) = log5(20).


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15. log9(x 5) + log9(x + 3) = 1.

16. log2(log4(x)) = 1.

17. log2(2x 3) = 1


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Chapter 5: Trigonometric Functions: Unit Circle Ap-proach

5.1: The Unit Circle

Topics

1. Unit Circle.

2. Terminal Points on the Unit Circle.

3. The reference number.


1. Unit Circle.

The unit circle

The unit circle (Ms. Hurs shorthand is S1) is the circle of radius 1 centered at the origin (0,0). Its

equation is

x2 + y2 = 1.

We shall use the equation of the unit circle to either verify that a point is on the unitcircle, or perhaps to find the x or y coordinate on the unit circle given one of the twocoordinates and a quadrant.

1. Show that

35 ,

45

is a point on the unit circle.

2. Find the point on the unit circle withx coordinate 13

in quadrant IV.

2. Terminal Points on the Unit Circle S1.

Given a real number t (any decimal), we can imagine t as defining a distance along the cir-cumference of a unit circle. positive meaning counterclockwise motion and negative meaningclockwise motion.

There are some particularly nice values for t which give us easy to calculate terminalpoints. Lets list some of those out:


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3. The Reference Number

The reference number tLet t be any real number. The reference number, denoted by t associated with t is the shortestdistance along the unit cicle between the terminal point and the x axis

1. For each of the following, determine the reference number:

(a) t= 3/2

(b) t= 11/3

(c) t= 8

(d) t= 3

(e) t= 31/52. Ift corresponds to a terminal point (a, b) in quadrant I, find the terminal points for:

(a) + t

(b)t(c) 2 + t


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5.2: Trigonometric Functions of Real Numbers

Topics

1. The Trigonometric Functions.

2. Values of the Trigonometric Functions.

3. Fundamental Identities.


1. Unit Circle.

Definition of the Trigonometric FunctionsLett be any real number andP(x, y) be the terminal point on the unit circle determined by t. Wedefine:

sin(t) =y cos(t) =x tan(t) = yx

csc(t) = 1y sec(t) = 1x cot(t) =

xy

In particular for sin(t), cos(t), and tan(t), we have really nice geometric interpretations ofthe outputs of those functions:

For each of the following special angles, find the values of all the trig functions:

1. t= 0

2. t=


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3. t= /2

4. t= /4

5. t= /3

6. t= /6

2. Values of the Trigonometric Functions.

To evaluate the trig functions at other values oft, drawing the unit circle, finding the point,and using reference numbers can be very helpful. For each of the following angles, find thevalues of all the trig functions:

1. t= 200

2. t= 10/4

3. t= 2/3

4. t= 5/6


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3. Fundamental Identities.

Even-Odd Properties:

Fundamental Identities:

1. Given that cos(t) = 2/3 andt is in QIII, find the values of all the trig functions at t.


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2. Given that tan(t) = 2 and t is in QIV, find the values of all the trig functions at t.

3. Write tan(t) in terms of cos(t) where t is in QII.


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Now, we shall draw sketches of each graph.

2. Graphs of Transformations of Sine and Cosine.

To aid us in drawing the transformation of sine and cosine graphs (sometimes called sinu-soidal graphs or harmonic motion), it helps to keep track of characteristics that are horizontal(about the inputs) and those that are vertical (about the outputs).

Horizontal Characteristics:

Vertical Characteristics:

Base Graphs:


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Lets explore a few simple examples before diving into more complex ones: For each of thefollowing, determine the period, phaseshift, average (midline), amplitude, max, and min.Then draw a sketch of two cycles (two periods) of the graph.

1. y= cos(x) + 2

2. y= 3cos(x)

3. y= sin(2x)

4. y= sin(x 1)


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In general, to graph transformations of sin(t) or cos(t), we have the following procedure:

1. Identify the base graph.

2. Identify the period, phaseshift, average (midline), amplitude, max, and min.

3. Locate the starting point of the base graph on the transformed graph. Use the pattern ofthe base graph to determine max/min/average points.

4. Smoothly connect the points plotted in step 3.

Particularly, for graphs of the form y = A cos(B(tC)) + Dor y = A sin(B(tC)) + D, we have:period phaseshift

average (midline) amplitude

max min

For each of the following, determine the period, phaseshift, average (midline), amplitude,max, and min. Then draw a sketch of two cycles (two periods) of the graph.

1. y= 3 cos(2x) 1

2. y= 0.5cos(/3(x 1)) + 1.5


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3. y= 2 sin(2x + /3) + 4

For each of the following, determine a formula in terms of sine or cosine for the given graph.

1.

2.


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5.6: Modeling Harmonic Motion

Topics

1. Simple Harmonic Motion. 2. Damped Harmonic Motion.


1. Simple Harmonic Motion.

Simple Harmonic Motion

If the equation describing the displacement y of an object at time t is

y= a sin(t) or y= a cos(t)

then the object is in simple harmonic motion. In this case,

amplitude= |a| (maximum displacement of the object)period= 2 (time required to complete one cycle)frequency=

2(number of cycles per unit of time)

More generally, we can define harmonic motion in which the function may beshifted vertically or horizontally.

y= a sin((t c)) + bor y= a cos((t c)) + b

averagey = b (average value)phase shift=c (horizontal shift from base to final graph)

1. Suppose the displacement of a mass suspended by a spring is modeled by the function

y= 10 sin(4t)

Wherey is measured in inches and tis in seconds.

(a) Find the amplitude, period, and frequency of the motion.

(b) Sketch a graph of the displacement of the mass.


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2. A mass is suspended from a spring. The spring is compressed at a distance of 4 cm andthen released. It is observed that the mass returns to the compressed position after 13second.

(a) Find a function that models the displacement of the mass.

(b) Sketch the graph of the displacement of the mass.

3. Suppose a ferris wheel has a radius of 10 m and the bottom of the wheel passes 2 mabove the ground.

If the ferris wheel makes one complete revolution every 20 seconds, find an equationthat gives the height above the ground of a person on the ferris wheel as a function oftime.


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2. Damped Harmonic Motion.

To sketch transformations of Tangent, Cotangent, Secant,and Cosecant, we use our knowl-edge of the features and shapes of the base graphs.

Damped Harmonic Motion

If the equation describing the displacement y of an object at time t is

y= kect sin(t) or y = kect cos(t)

then the object is in damped harmonic motion. In this case,

damping constant= c (maximum displacement of the object)initial amplitude=kperiod (aka quasi-period)= 2

(time required to complete one cycle)

1. Two mass spring systems are experiencing damped harmonic motion, both at 0.5 cyclesper second with an initial maximum displacement of 10 cm. Suppose the dampingconstant is 0.5 for the first spring and 0.1 for the second.

(a) Find functions of the formg(t) =kect cos(t) to model the displacement at timet for each spring.

(b) Graph the two functions together for 0 < t


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2. A tuning fork is struck and oscillates in damped harmonic motion. The amplitude ofthat motion is measured and 3 seconds later it is found that the amplitude has droppedto 1/4 of this value. Find the damping constant cfor the tuning fork.


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5.4: More Trigonometric Graphs

Topics

1. Graphs of Tangent, Cotangent, Se-cant,and Cosecant.

2. Graphs of Transformations of Tangent,Cotangent, Secant,and Cosecant.


1. Tangent, Cotangent, Secant,and Cosecant.

Using the fact that tan(t) is just the slope of the terminal side of t, it is fairly simple tosketch a good graph of tangent:Graph of Tangent and Summary of Features

Based on the above graph, we can also determine the graph of Cotangent.

Graph of Cotangent and Summary of Features


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To obtain graphs of Secant and Cosecant, it is often helpful to refer to graphs of Cosine andSine:

Graph of Secant and Summary of Features

Graph of Cosecant and Summary of Features


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2. Graphs of Transformations of Tangent, Cotangent, Secant,and Cosecant.

To sketch transformations of Tangent, Cotangent, Secant,and Cosecant, we use our knowl-edge of the features and shapes of the base graphs.

Examples: Determine all features of the transformations. Sketch 2 cycles of each graph.

1. y= tan(x) + 1

2. y= cot(x).


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3. y= 2 sec(2x)

4. y= 1/2 csc(x /4)


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2. Inverse Cosine

We can consider the appropriate restriction by looking at either the graph ofy = cos(t) orconsidering the unit circle.

Inverse Cosine (aka Arccosine)

3. Inverse Tangent

We can consider the appropriate restriction by looking at the graph ofy= tan(t) .

Inverse Tangent (aka Arctan)

4. Calculations with inverse trig functions

Find the exact value or expression, if it is defined.

1. sin1(1/2) 2. arccos(3/2)


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3. tan1(1)

4. sin(sin1(5))

5. cos1(cos(5))

6. sin(cos1(0))

7. tan(sin1(0))

8. sec(+ tan1(

3)


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1. Draw each angle in standard position:(a) /3 (b) 120 (c)45


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Coterminal Angles:Two answers in standard position are coterminal if they have the SAME terminal side.

2. Find two angles that are coterminal to the given angle:(a) /3 (b) 120 (c)45

3. Length of a Circular Arc.

Length of a Circular Arc:In a circle of radius r, the length s of an arc that subtends a central angle of radians is

s= r

3. Find the length of an arc that subtends a central angle of on a circle of radius r,given:

(a) = 5/6 andr= 2 inches.

(b) = 120 andr = 3 inches.


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6. The top and bottom ends of a windshield wiper blade are 34 inches and 14 inches,respectively, from the pivot point. While in operation, the wiper sweeps through 135.Find the area swept by the blade.

5. Circular Motion.

If an object moves along a circular wheel, there is a connection to the speed of the objectand how fast the wheel is rotating.

Linear and Angular:

Thelinear speed v is the rate at which the distance traveled is changing.

v=d

t

If an object is traveling on wheels, the distance is travelled on the outside of the wheels. In otherwords, d= r, so we have

v=r

t

Theangular speed is the rate at which the central angle is changing.

= t

Finally, the connection betweenv and :

7. A ceiline fan with 16 inch blades rotates at 45 rpm.(a) Find the angular speed of the fan in rad/min.

(b) Find the linear speed of the tips of the blades in inches/minute.


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6.2: Trigonometry of Right Triangles

Topics

1. Trigonometric Ratios.

2. Special Triangles.

3. Applications of Trigonometry of RightTriangles.


1. Trigonometric Ratios.

The Trigonometric Ratios:

sin() = oppositehypotenuse

cos() = adjacenthypotenuse

tan() = oppositeadjacent

csc() = hypotenuseopposite sec() = hypotenuse

adjacent cot() = adjacentopposite

For each of the following, find all 6 trig ratios:

1. The triangle with opposite side length 4 and hypotenuse length 5.

sin() = cos() =

tan() = csc() =

sec() = cot() =

2. The triangle with opposite side length 1 and adjacent side length 3.

sin() = cos() =

tan() = csc() =

sec() = cot() =


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2. Special Triangles.

The Special Triangles:

A triangle has six parts (3 angles and 3 sides). To solve a triangle means to determine all ofits parts (all angles and all sides).

For the following, solve each triangle exactly.

1. The right triangle ABC, withA= 90, B = 45, AB = 2.

2. The right triangle ABC, withA= 30, B = 60, AB = 5.


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3. The right triangle ABC, withA= 20, B = 90, AC= 3.

4. The right triangle ABC, withA= 90, B =,AC=h.

3. Applications of Trigonometry of Right Triangles.

Angle of Elevation vs. Angle of Depression:

1. A water tower is located 325 feet from a building. From a window in the building, anobserver notes that the angle of elevation to the top of the tower is 40 and the angleof depression to the bottom of the tower is 25.

(a) Draw a diagram that shows this scenario.

(b) How tall is the tower? (c) How high is the window?


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2. A hot air balloon is floating above a straight road. To estimate their height abovethe ground, the balloonists simultaneously measure the angle of depression to twoconsecutive mileposts on the road on the same side of the balloon. The angles ofdepression are found to be 20 and 22.

(a) Draw a diagram that shows this scenario.

(b) How high is the balloon?

3. (p 451, Written Homework Problem # 62)When the moon is seen at its zenith at a point A on the earth, it is observed to be atthe horizon from point B . PointsA and B are 6155 miles apart, and the radius of theearth is approximately 3960 miles.

(a) Find the angle in degrees.

(b) Estimate the distance from pointA to the moon.


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6.3: Trigonometry Functions of Angles

Topics

1. Trigonometric Functions of Angles.

2. Trigonometric Identities.

3. Areas of Triangles.


1. Trigonometric Functions of Angles.

The Trigonometric Functions:Let be an angle in standard position and let P(x, y) be any point on the terminal side. Ifr=

x2 + y2 is the distance from the origin to the point P(x, y), then

sin() = yr cos() = xr tan() =

yx

csc() = ry sec() = rx cot() = xy

As with unit circle trig, it is convenient to determine the reference number, and use the factthat the associated terminal points are the same up tosign.

Reference Angle:

Let be an angle in standard position. The reference angle associated with is the acute angleformed by the terminal side of and the x axis.

1. Determine the reference angle (in the same units as the original angle). Evaluate thefollowing exactly:

(a) cos(135) (b) cot(390)

(c) sec(10/3) (d) csc(11/4)


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2. Trigonometric Identities.


Quotient Identity

Reciprocal Identities

Pythagorean Identities

Negative Angle Identities (aka Even-Odd Identities)

2. Express csc() in terms of cos(), given that terminates in QII.


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3. If csc() = 1/4 and terminates in QIII, find all 6 trig values:

sin() = cos() =

tan() = csc() =

sec() = cot() =

4. If tan() = 2 and terminates in QIV, find all 6 trig values:

sin() = cos() =

tan() = csc() =

sec() = cot() =

3. Areas of Triangles.

The area of a triangle with side lengths a and b with included angle is

5. Find the area of a triangle ABC that has side lengths AB = 10cm, BC = 4cm, andincluded angle 120.


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6.4: Inverse Trigonometric Functions and Right Triangles

Topics

1. The Inverse Since, Inverse Cosin, andInverse Tangent Functions.

2. Solving for Angles in Right Triangles.

3. Evaluating Expressions Involving In-verse Trigonometric Functions.

4. Basic Solving Equations.


1. The Inverse Sine, Inverse Cosins, and Inverse Tangent Functions.

Inverse Trig Functions:

2. Solving for Angles in Right Triangles.

1. Solve the triangle:


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2. Find all angles between 0 and 180 with sin() = 1/2.

3. Find all angles between 0 and 180 with sin() = 0.3.

4. An observer views the space shuttle from a distance of 2 miles from the launch pad.

(a) Express the height of the space shuttle as a function of the angle of elevation .

(b) Express the angle of elevation as a function of the heighth of the space shuttle.


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3. Evaluating Expressions Involving Inverse Trigonometric Functions.

1. Evaluate cos(sin1(4/5)) exactly.

2. Evaluate sec(sin1(12/13)) exactly.

3. Rewrite the expression as an algebraic expression in x.cos(sin1(x))

4. Rewrite the expression as an algebraic expression in x.sec(tan1(x))


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4. Basic Solving Equations.

1. Find all solutions between 0 and 360.sin(x) =

3

2

2. Find all solutions between 0 and 360.cos(x) =

2

2

3. Find all solutions between 0 and 360.tan(x) = 1

4. Find all solutions between 0 and 360.sin(x) = 1/4

5. Find all solutions between 0 and 360.tan(x) = 2


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Chapter 7: Analytic Trigonometry

7.1: Trigonometric Identities

Topics

1. Building our toolbox of identities.

2. Simplifying Trig Expressions.

3. Proving Identities.


1. Building Our Toolbox of Identities.

We introduce here another set of useful identities below, called the Co-function identities.The Co in co-function can be thought of as COmplementary. Here we motivate the identitiesby focusing on right triangles; however, these identities hold for ANY angle (radian, degree,big, small, positive or negative).


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Quotient Identity

Reciprocal Identities

Pythagorean Identities

Negative Angle Identities (aka Even-Odd Identities)

Cofunction Identities

2. Simplifying Trig Expressions.

Simplify the following completely:

1. cos(t) + tan(t) sin(t) 2. sin()cos()

+ cos()1+sin()


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7.2: Addition and Subtraction Formulas

Topics

1. Addition and Subtraction Formulas.

2. Evaluating Expressions Involving Inverse Trig Functions.

3. Expressions of the formA sin x + B cos x


1. Addition and Subtraction Formulas.

Addition and Subtraction Formulas:

Evaluate the following EXACTLY:(a) cos(75) (b) sin(/12)

(c) sin(20) cos(40) + cos(20) sin(40)


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Prove the identity:

1 + tan(x)

1 tan(x) = tan

4+ x


Simplify the following completely:

1. sin(cos1(1/3) + tan1(2))

2. sin(cos1(1/3) tan1(2))

3. cos(sin1(1/x) + tan1(x))


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4. cos(sin1(1/x) tan1(x))

3. Expressions of the form A sin x + B cos x.

Sums of Sines and CosinesIfAand B are real numbers, then

A sin x + B cos x= k sin(x + )

wherek=

A2 + B2 and satisfies

cos() = AA2 + B2

and sin() = BA2 + B2

Express each of the following in the form k sin(x + )

1. y= sin(x) + 3cos(x)

2. sin(x) + cos(x)


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7.3: Double Angle Formulas

Topics

1. Double Angle Formulas.


3. Power Reduction Formulas.


In our course, we are covering a very specific subset of these topics from this section. WeWILL cover double angle and power reduction formulas. We will not cover other topics (likehalf angle and product to sum).

1. Double Angle Formulas.

We can derive these from the sum and difference formulas from last section:

Double Angle Formulas:

Now lets practice putting these identities to use:


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1. Given that sin(x) = 45

and cos(x) = 35

, find

(a) sin(2x) (b)cos(2x) (c) tan(2x)

2. Given that cot(x) = 1/3 andx terminates in QII, find(a) sin(2x) (b)cos(2x) (c) tan(2x)

3. Given that csc(x) =c andx terminates in QII, find(a) sin(2x) (b)cos(2x) (c) tan(2x)

4. Prove the following identity:

2tan(x)

1 + tan2(x)= sin(2x)


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5. Prove the following identity:

cos4 x sin4 x= cos(2x)


Just as in the previous section, drawing a diagram makes these problems much simpler thanthey appear!

1. sin(2cos1(2/3))

2. cos(2 tan1(x))

3. cos(2 sin1(1/x))


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3. Power Reduction Formulas.

The power reduction formulas are derived from the double angle identities for cosine:

Power Reduction Formulas:

Use the formulas for lowering powers to rewrite the expression in terms of the first power ofcosine.

(a) sin4(x) (b) cos2(x)sin2(x)


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7.4: Basic Trigonometric Equations

Topics

1. Basic Trigonometric Equations.

2. Solving Trigonometric Equations By Factoring.


1. Basic Trigonometric Equations.

Find all solutions in [0, 2) exactly.(a) cos() = 1/2 (b) sin() = 2/2

(c) tan() = 1 (d) sin() = 1/5

(e) 3 cos() = 1 (f) cot() = 2


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Find all solutions exactly.(a) cos() = 1/2 (b) sin() = 2/2

(c) tan() = 1 (d) sin() = 1/5

(e) 3 cos() = 1 (f) cot() = 2

In general, find all solutions in one period. Then find all remaining solutions using the period.

2. Solving Trigonometric Equations By Factoring.

Find all solutions exactly.

1. 2cos2() 1 = 0


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2. 3tan()sin() 2tan() = 0.

3. 2cos2() 7cos() + 3 = 0.

4. 2sin2() sin() 1 = 0.


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7.5: More Trigonometric Equations

Topics

1. Solving Trig Equations By Using Identities.

2. Equations with Trig Functions of Multiple Angles.

3. Summary of all Trig Identities (only Sum/Diff given on the test; others must be mem-orized!)


1. Solving Trig Equations By Using Identities.

(i) Find all solutions. (ii) Find the solutions in the interval in [0, 2).(a) 2 cos2(x) + sin(x) = 1 (b) csc2() = cot() + 3

(c) cos(2) = 3 sin() 1 (d) 2 sin() tan() tan() = 1 2sin()


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2. Equations with Trig Functions of Multiple Angles.

(i) Find all solutions. (ii) Find the solutions in the interval in [0, 2).(a) 2 cos(2) =

2 (b)

3 tan(4) = 1

(c) sec(3) 2 = 0 (d) sin(2) = 1/3


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3. Summary of all Trig Identities used in Math 120R

Quotient Reciprocal

Pythagorean Negative Angle

Cofunction

Sum and Difference

Double Angle

Power Reduction


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Appendix: Fundamental Algebra Skills Review

1.3: Algebraic Expressions

Topics

1. Adding and Subtracting Polynomials.

2. Multiplying Algebraic Expressions.

3. Special Product Formulas.

4. Factoring Common Factors.

5. Factoring Trinomials.

6. Special Factoring Formulas.

7. Factoring by Grouping.

Warm-Up

Given f(x) = 2x3 4x + 1 and g(x) =x 1, expand and simplify the following:1. f(x) + g(x) =

2. g(x) f(x) =

3. f(x)g(x) =


1. Adding and Subtracting Polynomials

How does one add or subtract polynomials?

2. Multiplying Polynomials

How does one multiply two polynomials?


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3. Special Product Formulas:

Use the scratch work space below to complete the below list of special product formulas.

IfA andB are any numbers or algebraic expressions, then

1. (A + B)(AB) =2. (A + B)2 =

3. (AB)2 =4. (A + B)3 =

5. (AB)3 =

Scratch work:

4. Factoring Common Factors:

If all terms of a sum have a common factor (multiple), then one may pull out the commonfactor. For example, factor the following completely:

1. 3x2

6x=


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2. 2x3 2x=

3. 8x4y2 + 6x2y3 2xy4 =

5. Factoring Trinomials:

To factor a trinomial of the form ax2 +bx+c (if a factorization exists), we generally do soby trial and error. The key in rewriting has to do with checking that when foiled out, ourfactorization must give us ax2 + bx + c . This is best seen by example:For each of the following, factor completely:

1. x2 + 7x + 12

2. x2 3x 10

3. 6x2 + 7x 5

4. 8x2 13x 15


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6. Special Factoring Formulas:

1. A2 B2 = (AB)(A + B)2. A2 + 2AB+ B2 = (A + B)2

3. A2

2AB+ B2

= (AB)2

4. A3 B3 = (AB)(A2 + AB+ B2)5. A3 + B3 = (A + B)(A2 AB+ B2)

For each of the following, factor completely:

1. x2 + 6x + 9

2. 4x2 4xy+ y2

3. 2x4 8x2

4. x5y2 xy6

7. Factoring by Grouping:

We shall explore this tool via example. Factor the following completely.

1. x3 + x2 + 4x + 4 2. x3 2x2 3x + 6


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8. Factoring by least powers

Factor by first factoring out the lowest power of each common factor

1. 2x2(x + 1) 6x(x + 1)2

2. x1/2 x5/2

3. x3/2 + 2x1/2 + x1/2

4. x1/2(x + 1)1/2 + x1/2(x + 1)1/2


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1.4: Rational Expressions

Topics

1. Domain of an Algebraic Expression.

2. Simplifying Rational Expressions.

3. Multiplying and Dividing Rational Expressions.

4. Adding and Subtracting Rational Expressions.

5. Compound Fractions.

6. Rationalizing the Denominator or Numerator.

7. Avoiding Common Errors.

Warm-Up

When dividing, what is the only number that we cannot divide by?

When square-rooting, what types of numbers are not square-rootable?


1. Domain of an Algebraic Expression

For each of the following expressions, find the domain.

1. x2x

cx2

+x

2.

x 5

3. 3

1

x

4.x

x3x


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2. Simplifying Rational Expressions.

To simplify rational expressions, one often uses methods like factoring and common denom-inators. We shall review by example:

Simplify the following expressions completely.

1. 3x2x2

14x21

2. z2+z6

4(z+2)z2(z+2)

3. x+2x1+ 7

4. 6n7 +

2nn+1

3. Multiplying and Dividing Rational Expressions.

Some basic rules to remember about multiplying and dividing rational expressions:

To multiply rational expressions, we simply .

To divide rational expressions, we simply .

Given f(x) = xx+1

andg(x) = x21x

, complete the following operations and simplify.

1. (f g) (x) 2.fg

(x)


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4. Adding and Subtracting Rational Expressions.

To add or subtract rational expressions, we simply .

Given f(x) = x+1x22x and g(x) = 3x2x2 , complete the following operations and simplify.

1. (f+ g) (x) 2. (f

g) (x))

5. Compound Fractions.

Write each expression in simplified form.

1.3

4+h 3

4

h

2. k2

k1

k3

3. (x1

+ y1

)2


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6. Rationalizing the Denominator or Numerator.

We use a trick involving what is referred to as the conjugate radical. A + B

C is conjugateto A BCand vice versa.

1. Rationalize the denominator:

1

1 +

2

2. Rationalize the numerator:

4 + h 2

h

7. Avoiding Common Errors.

See the box of common errors on pg 41 of the text. Short story: only multiplication/divisiondistribute over addition and only exponentiation/radicals distribute over products.


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1.5: Equations

Topics

1. Solving Linear Equations.

2. Solving Quadratic Equations.

3. Solving other Types of Equations.


1. Solving Linear Equations.

Solve the following linear equations:

1. 3x + 2 = 12 x 2. 13 x + 16 =x + 112

Briefly summarize your technique for solving linear equations here:

2. Solving Quadratic Equations.

Solve the following quadratic equations exactly:

1. x2 2x= 8

2. 2x2 x 1 = 0

3. (x 1)(x + 2) = 10

4. x2 + x 1 = 0

Briefly summarize your techniques for solving quadratic equations here (there are two par-ticularly useful methods):


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3. Solving other Types of Equations.

Solve the following equations exactly, below each solution state a short description of yourtactic:

1. 3

x+

5

x + 2= 2

2. 2x= 1

2

x

3. x4 8x2 + 8 = 0

4.

|2x

5

|= 3

More Practice

1. Solve exactly: 5y2(4y 7)(y+ 2) = 0

2. Solve exactly: t3 = 7t

3. Solve exactly: 2y=

3y 2 + 1

4. Solve forr : V = 43 r3

5. Solve forS: 1R = 1S+

1T

6. Solve exactly:|x + 2| = 1.


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1.6: Modeling with Equations

Topics


2. Problems about Area or Length.

3. Problems about Mixtures.

4. Problems about the Time Needed to Doa Job.

5. Problems about Distance, Rate, andTime.



We will follow these guidelines to have an organized approach at handling word problems.

Guidelines for Modeling with Equations

1. Identify the variable (the quantity we are trying to solve for!).

2. Translate from words into algebra. Often a diagram or table can help us withthis step.

3. Set up the Model. Find the simple formula for the given quantity or quanti-ties.

4. Solve the equation and check your answer. Be sure to include units in your finalanswer!

2. Problems about Area or Length.1. A square garden has a walkway 3 feet wide around its outer edge. If the area of the

entire garden, including the walkway, is 18,000 square feet, what are the dimensions ofthe planted area?


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2. A rectangular lot is 8 feet longer than it is wide and has an area of 2900 square feet.Find the dimensions of the lot.

3. A 20 foot ladder leans against a building. If the base of the ladder is 8 feet away fromthe building, how high up on the building does the ladder reach? Give an exact answer.

3. Problems about Mixtures.

1. What quantity of a 60% acid solution must be mixed with a 30% solution to produce300 mL of a 50% solution?


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2. A 100 gallon container is full with a 2% concentration of bleach. How much should bedrained and replaced with pure bleach in order to strengthen the contents to 5%?

4. Problems about the Time Needed to Do a Job.

1. Hank and Sally can mow their property in 60 minutes when they work together. Since

Hank uses a riding mower while Sally uses a push mower, Hank works twice as fast.How long does it take Hank to mow the lawn by himself?

2. Betty and Karen have been hired to paint the houses in a new development. Workingtogether, the women can paint a house in two thirds the time that it takes Karenworking along. Betty takes 6 hours to paint a house alone. How long does it takeKaren to paint a house working alone?


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5. Problems about Distance, Rate, and Time.

1. Amy travels 450 miles in her car at a certain speed. If the car had gone 15 mph faster,the trip would have taken 1 hour less. Determine the speed of her car.

2. A boat travels down a river with a current. Travelling with the current, a trip of 58miles takes 3 hours while the return trip travelling against the current takes 4 hours.How fast is the current?


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1.7: Inequalities

Topics

1. Solving Linear Inequalities.

2. Absolute Value Inequalities.

3. Solving Non-linear Inequalities.

4. Modeling with Inequalities.


1. Solving Linear Inequalities.

Linear inequalities are the simplest to work with. We may solve these in the same way wewould an equality with one big caveat:

Now lets go through some examples from the accompanying worksheet of this type (2 a,b,c).

2. Absolute Value Inequalities.

We use the definition of absolute value to handle these types of inequalities. Intuitively, letsthink about what the absolute value function does:

For POSITIVE numbers x, how can we simplify|x|?

For NEGATIVE numbers x, how can we simplify|x|?

We can write this more compactly as follows:

Now lets go through some examples from the accompanying worksheet of this type (2 d,e,f).

3. Solving Non-linear Inequalities.

Since there are so many types of non-linear functions, it is difficult to set one metho

Documents

Math 120R Workbook