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Calculus - Review Topics

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    Study Guide Calculus – Review Topics 1. Absolute Value 1. What Is Absolute Value? The idea of absolute value comes up often in algebra and calculus. The absolute value of a number ( x ) , written as ( |x| ) , tells us how far the number is from zero , without caring about direction. So: • Distance is always positive or zero • That means absolute value is never negative In symbols: 2. Absolute Value on the Number Line Think about a number line : • The number 5 is 5 units to the right of zero • The number – 5 is 5 units to the left of zero Even though they are on opposite sides, their distances from zero are the same. So: This is why absolute value ignores direction and focuses only on distance . 3. Algebraic (Piecewise) Definition

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    Study Guide One common way to define absolute value in algebra is by breaking it into cases: What this means: • If ( x ) is positive , absolute value leaves it unchanged • If ( x ) is zero , the absolute value is zero • If ( x ) is negative , absolute value removes the negative sign This definition guarantees that the result is always nonnegative . 4. Absolute Value Using Square Roots Another definition that is sometimes useful in calculus is: Why does this work? • Squaring ( x ) makes it nonnegative • Taking the principal square root gives a nonnegative result For example: 5. Key Idea to Remember No matter which definition you use: • Absolute value measures distance from zero • Absolute value is never negative • Different definitions describe the same concept

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    Study Guide Key Takeaways • Absolute value shows how 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| = √ x 2 ) is another valid definition 2. Functions 1. What Is a Function? A function is a special kind of relationship between two variables, usually written as an ordered pair ((x, y)). In a function: • Each input value (x) is paired with one and only one output value (y) This rule is very important. If even one (x) - value is matched with more than one (y) - value, the relation is not a function. 2. Domain and Range • The domain is the set of all possible input values ((x) - values) • The range is the set of all possible output values ((y) - values) Because of this: • (x) is called the independent variable • (y) is called the dependent variable , since its value depends on (x)

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    Study Guide 3. Function Notation Instead of writing (y), we often use function 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 Test Geometrically, a graph represents a function if and only if it passes the vertical line test . Vertical Line Test: • Draw any vertical line on the graph • If the line intersects the graph at most one point , the graph represents a function • If it intersects the graph at two or more points , the relation is not a function Why? • One vertical line means one (x) - value • Multiple intersection points mean that one (x) has multiple (y) - values • This violates the definition of a function Many important ideas in calculus are based on functions, which is why this concept is so important. Example 1: Equations That Are Functions Paste the first example image (functions) here , under this subheading. Each of the following equations defines exactly one (y) - value for every allowed (x) - value , so they are functions:

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    Study Guide Even though some of these functions have restrictions on their domains, each valid (x) still produces only one output. Example 2: Equations That Are NOT Functions Paste the second example image (not functions) here , under this subheading. The following equations are not functions because at least one (x) - value corresponds to more than one (y) - value : Each of these fails the vertical line test.

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    Study Guide Key Takeaways • A function assigns one output to each 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 the vertical line test • If one (x) has multiple (y) - values, it is not a function 3. Linear Equations 1. What Is a Linear Equation? A linear equation is any equation that can be written in the form where (a) and (b) are not both zero . Even if an equation is not written in this form at first, it can usually be rearranged algebraically into this standard form. The graph of a linear equation is always a straight line . 2. Understanding Slope The slope of 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 ((x 1 , y 1 )) and ((x 2 , y 2 )) lie on a line, the slope can be calculated using:

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