Calculus - Review Topics - Quick Guides

Page 1 of 20

Calculus - Review Topics - Page 1 preview image

Study GuideCalculus–Review Topics1.Absolute Value1. What Is Absolute Value?The idea ofabsolute valuecomes up often in algebra and calculus.The absolute value of a number(x), written as(|x|), tells ushow far the number is from zero,without caring about direction.So:•Distance is alwayspositive or zero•That means absolute value isnever negativeIn symbols:2. Absolute Value on the Number LineThink about anumber line:•The number5is 5 units to the right of zero•The number–5is 5 units to the left of zeroEven though they are on opposite sides, their distances from zero are the same.So:This is why absolute value ignores direction and focuses only ondistance.3. Algebraic (Piecewise) Definition

Page 2 of 20

Calculus - Review Topics - Page 2 preview image

Study GuideOne common way to define absolute value in algebra is by breaking it into cases:What this means:•If(x)ispositive, absolute value leaves it unchanged•If(x)iszero, the absolute value is zero•If(x)isnegative, absolute value removes the negative signThis definition guarantees that the result isalways nonnegative.4. Absolute Value Using Square RootsAnother definition that is sometimes useful in calculus is:Why does this work?•Squaring(x)makes itnonnegative•Taking theprincipal square rootgives a nonnegative resultFor example:5. Key Idea to RememberNo matter which definition you use:•Absolute value measuresdistance from zero•Absolute value isnever negative•Different definitions describe thesame concept

Page 3 of 20

Calculus - Review Topics - Page 3 preview image

Study GuideKey Takeaways•Absolute value showshow far a number is from zero•Direction does not matter•(|x|0)for all real numbers•Piecewise definition handles positive, zero, and negative values•(|x| =√x2) is another valid definition2.Functions1. What Is a Function?Afunctionis a special kind of relationship between two variables, usually written as an ordered pair((x, y)).In a function:•Eachinput value(x) is paired withone and only oneoutput value (y)This rule is very important.If even one (x)-value is matched withmore than one(y)-value, the relation isnota function.2. Domain and Range•Thedomainis the set of all possibleinput values((x)-values)•Therangeis the set of all possibleoutput values((y)-values)Because of this:•(x) is called theindependent variable•(y) is called thedependent variable, since its value depends on (x)

Page 4 of 20

Calculus - Review Topics - Page 4 preview image

Study Guide3. Function NotationInstead of writing (y), we often usefunction notation.•(f(x)) means “the value of the function at (x)”•It is read as“f of x”For example:This means the function takes an input (x) and produces an output based on the formula (3x + 1).4. Graphical View: The Vertical Line TestGeometrically, a graph represents a functionif and only ifit passes thevertical line test.Vertical Line Test:•Draw any vertical line on the graph•If the line intersects the graph atmost one point, the graph represents a function•If it intersects the graph attwo or more points, the relation isnot a functionWhy?•One vertical line means one (x)-value•Multiple intersection points mean that one (x) has multiple (y)-values•This violates the definition of a functionMany important ideas in calculus are based on functions, which is why this concept is so important.Example 1:Equations That Are FunctionsPaste the first example image (functions) here, under this subheading.Each of the following equations definesexactly one (y)-value for every allowed (x)-value, so theyare functions:

Page 5 of 20

Calculus - Review Topics - Page 5 preview image

Study GuideEven though some of these functions haverestrictionson their domains, each valid (x) still producesonly oneoutput.Example 2:Equations That Are NOT FunctionsPaste the second example image (not functions) here, under this subheading.The following equations arenot functionsbecause at least one (x)-value corresponds tomore thanone (y)-value:Each of these fails the vertical line test.

Page 6 of 20

Calculus - Review Topics - Page 6 preview image

Study GuideKey Takeaways•A function assignsone outputtoeach input•Domain = input values•Range = output values•(f(x)) means the value of the function at (x)•A graph represents a function if it passes thevertical line test•If one (x) has multiple (y)-values, it isnot a function3.Linear Equations1. What Is a Linear Equation?Alinear equationis any equation that can be written in the formwhere (a) and (b) arenot both zero.Even if an equation is not written in this form at first, it can usually berearranged algebraicallyintothis standard form.The graph of a linear equation is always astraight line.2. Understanding SlopeTheslopeof a line tells us how the line is tilted—whether it goes up, goes down, or stays flat.Slope is usually represented by the letter(m)and can be defined in several equivalent ways:If two points ((x1, y1)) and ((x2, y2)) lie on a line, the slope can be calculated using:

Preview Mode

This document has 20 pages. Sign in to access the full document!

Report

Learning Tools

Writing Tools

Browse Resources

Calculus - Review Topics - Page 1

Calculus - Review Topics

Study Now!

Document Details

Related Documents

Calculus - The Derivative

Calculus - Limits

Calculus - Integration

Calculus - Applications of the Derivative

Calculus - Applications of the Definite Integral

Precalculus - Polynomial and Rational Functions

Precalculus - Functions

Precalculus - Exponential and Logarithmic Functions

Math Word Problems - Translating Expressions

Math Word Problems - Equations

Company

Explore

Study Tools