PC - Preparation for Calculus Module Overview

Preparation for Calculus Module Overview

Introduction

Succeeding in calculus requires a solid foundation in algebra, geometry, trigonometry, analytic geometry, and elementary functions. Important prerequisite concepts that span these background areas are reviewed and the use of graphing utilities is introduced. The emphasis is on basic ideas of functions, their graphs, and ways of transforming and combining them. Multiple representations of functions and the use of elementary function types as mathematical models of real-world phenomena are vital to developing an understanding of the significance and impact of calculus in our daily lives.

Essential Questions

What notations are used to express solutions to an equation?
How do you interpret what the solutions to an equation mean and determine if solutions are reasonable?
How do you interpret relations and functions through their graphic, algebraic and numeric representations?
How do you identify intercepts and zeros from a graph, a table, and an equation?
How do you determine if a graph is symmetric with respect to the y-axis, x-axis, origin or none of these?
How do you find the points of intersection of two graphs?
How do you find the slope of a line and use it to write an equation for the line?
How are the slopes of parallel and perpendicular lines related?
What are the important characteristics and representations of a function?
How do you determine whether or not a function is even, odd, or neither? One-to-one?
How do you compose two functions to form a new function?
What is average rate of change and how do you find it from the graph of a function?
How do you write equations and draw graphs for transformations of a parent function?
What are trigonometric reciprocal, quotient, and Pythagorean identities?
How do values and graphs of each of the trigonometric functions illustrate periodicity?
How can technology facilitate the development of mathematical models for real-life data?

Key Terms

The following key terms will help you understand the content in this module.

Interval notation - A notation for representing an interval as a pair of numbers. The numbers are the endpoints of the interval. Parentheses and/or brackets are used to show whether the endpoints are excluded or included. For example, [-2, 6) is the interval of real numbers between -2 and 6, including -2 and excluding 6.

Set-builder notation - A mathematical notation for describing a set, often with an infinite number of elements, by stating the properties that its members must satisfy. For example, the set {x| x > 0} is read, "the set of all x such that x is greater than 0." It is read exactly the same way when the vertical line | is replaced by the colon as in {x x > 0}.

Open interval - (a, b) is interpreted as LaTeX: a<x<b $a<x<b$ where the endpoints are NOT included. (While this notation resembles an ordered pair, in this context it refers to the interval upon which you are working.)

Closed interval - [a, b] is interpreted as $LaTeX: a\le x\le b$ $a\le x\le b$ where the endpoints are included.

Positive Infinity, + ∞ - An idea of something that is endless. It is thought of as greater than any real number and is written as a sideways eight ∞. Infinity is not a number and has nothing to do with numbers the concept of "endlessness" reflects the fact that the number line goes on forever.

Negative Infinity, - ∞ - An idea of something that is endless but less than any real number. It is written as -∞ and like infinity, negative infinity is not a number.

Intercepts - The value where a graph intersects (crosses) a coordinate axis. The y-intercept of a graph is the value of y where it crosses the y-axis, and the x-intercept is the value of x where the curve crosses the x-axis.

Points of intersection - A set of points common to two or more figures.

Symmetry of a graph - Possessing the geometric relationship with pairs of points identically placed with respect to some line, point, or plane. If a graph does not change when reflected over a line or rotated around a point, the graph is symmetric with respect to that line or point.

Zero of a function - A root (solution) of the equation f(x) = 0.

Piecewise function - A function defined by applying different formulas to different parts of its domain.

One-to-one function - A function f that never takes on the same value twice that is, $LaTeX: f\left(x_1\right)\ne f\left(x_2\right)$ $f\left(x_1\right)\ne f\left(x_2\right)$ whenever $LaTeX: x_1\ne x_2$ $x_1\ne x_2$

Slope of a line - A number that measures the steepness of a line. It is the ratio of the change in the vertical coordinate and the change in the horizontal coordinate between any two points on the line (△y/△x).

Average rate of change - A ratio comparing the change in the value of one quantity to the change in value of another quantity over a closed interval.

Even function - A function f(x) that satisfies the property that f(x) = f(-x) for every number x in its domain. For example, f(x)=x²; and g(x)= cos x are both even functions.

Odd function - A function f(x) that satisfies the property that f(-x) =- f(x) for every number x in its domain. For example, $LaTeX: f\left(x\right)=x^3$ $f\left(x\right)=x^3$ and g(x)= sin x are both odd functions.

Scatter plot - A type of graph created by plotting ordered pairs in a coordinate plane that show the relationship between two sets of data.

Linear regression equation - The best-fit line for a set of bivariate (two variable) data.

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