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3.1 Functions, 3.1 Relations and Functions, Difference Quotient

Difference Quotient: Rational Function

June 6, 2017 admin

Example: Find the difference quotient for f(x)=1/x^2

The Difference Quotient: {f(x+h)-f(x)}/h

Solution:

The Difference Quotient Formula
Write the difference quotient for the given function	= ${1/(x+h)^2-1/x^2}/h$
Simplify the complex fraction by multiplying the numerator and denominator by the common denominator	= ${1/(x+h)^2-1/x^2}/h {{x^2(x+h)^2}/1}/{{x^2(x+h)^2}/1}$
Distribute the common denominator to each fraction in the numerator.	= ${1/(x+h)^2-1/x^2}/h {{x^2(x+h)^2}/1}/{{x^2(x+h)^2}/1}$ = ${{x^2(x+h)^2}/(x+h)^2-{x^2(x+h)^2}/x^2}/{h{x^2(x+h)^2}}$
Cancel the common factor	= ${{x^2(x+h)^2}/(x+h)^2-{x^2(x+h)^2}/x^2}/{h{x^2(x+h)^2}}$ = ${x^2-(x+h)^2}/{h{x^2(x+h)^2}}$
Simplify by squaring the binomial	= ${x^2-(x+h)^2}/{h{x^2(x+h)^2}}$ = ${x^2-(x+h)(x+h)}/{h{x^2(x+h)^2}}$ = ${x^2-(x^2+xh+xh+h^2)}/{h{x^2(x+h)^2}}$
Simplify by combining like terms and distributing the negative	= ${x^2-(x^2+2xh+h^2)}/{hx^2(x+h)^2}$ = ${x^2-x^2-2xh-h^2}/{hx^2(x+h)^2}$
Combine like terms	= ${-2xh-h^2}/{hx^2(x+h)^2}$
Cancel a common h from the numerator and denominator	= ${-2x-h}/{x^2(x+h)^2}$

The difference quotient for f(x)=1/x^2 is ${-2x-h}/{x^2(x+h)^2}$

1.5 Applications of Quadratic Equations, Applications of Quadratic Equations

Application: Quadratic Equation (Position Equation)

June 6, 2017 admin

1.5 Applications of Quadratic Equations, Applications of Quadratic Equations

Application: Quadratic Equation (Pythagorean Theorem)

June 6, 2017 admin

3.1 Functions, 3.5 Combination of Functions; Composition of Functions

Evaluating Combined Functions Using a Graph

June 6, 2017 admin

1.3 Complex Numbers, Complex Numbers

Powers of i

June 5, 2017 admin

Example: Simplify the power of i.

i^-59

Solution:

i^-59

Rewrite the exponential expression with negative exponents by using the reciprocal property.

= $1/{i^59}$

Simplify the i^59 by dividing 59 by 4. The remainder will tell you which position in the pattern you are in. 59 divided by 4 is 14 with a reminder of 3.

= $1/{i^3}$

= 1/{-i}

= 1/{-i} {i /i}

= $i/{-i^2}$

= i/{-(-1)}

= i/{1}

Binomial Distribution

Binomial Distribution Probability Formula

June 5, 2017 admin

Example: There is a quiz consisting of four multiple choice questions. Each questions has three answer choices.

binomial probabilities

Integration

Integration by Trig Substitution

June 2, 2017 admin

1.2 Applications of Linear Equations, Applications of Linear Equations

Consecutive Numbers

May 23, 2017 admin

Here is a handout about consecutive numbers.

MAT0024_2.5CW Consecutive

1.2 Applications of Linear Equations, Translating Phrases

Translating Phrases

May 23, 2017 admin

Here is a handout on translating phrases from words into mathematical expressions and equations. Solutions are provided on last page.

MAT0024_1.1CW Translating Phrases

3.5 Graphing Techiques: Transformations, 5.1 Exponential Functions, 6.3 Exponential Functions, Graphing Exponential Functions

Graphing an Exponential Equation by Transformations

April 29, 2017 admin

Example: For the function below. Graph using transformations. Find the y-intercept. State the horizontal asymptote and the domain and range.

$f(x)=-2^{x+2}+2$

First we must examine the base function y=2^x

Graph using plotting points. We can use the standard set of x-values to find ordered pairs.

x	y
-2	2^(-2)=1/4
-1	2^(-1)=1/2
0	2^0=1
1	2^1=2
2	2^2=4

The graph below shows the points plotted and the line that connects them. This graph has a horizontal asymptote at y=0. The domain is (- infty, infty) and the range is (0, infty)