You’ve now covered the basic skills you need to be able to solve and graph equations. In this lesson, you’ll practice solving equations and inequalities involving quadratics, absolute values, rational expressions, exponents, logarithms, and trigonometric functions.

Why is being able to solve equations so important for Calculus? Oftentimes, we want to know where a function (i.e., at what x-value) is equal to a certain y-value (usually 0). Depending on what this function represents, we can draw a number of conclusions. For example:

• If f(t) represents the distance you are from your home and t represents time, the values of t at which f(t) = 0 represent the time(s) during which you’re at home. The derivative of f(t)—which you recall is denoted f’(t) and gives the slope of f(t) at any point t—represents the velocity. This means that the values of t where f’(t) = 0 represent the time(s) during which you’re not moving (your velocity is 0).
• If f(q) represents the profit a company would make by producing q products (i.e., q represents the quantity), you would want to find the values of q at which f(q) is maximized (i.e., f’(q) = 0).

These examples illustrate real-life situations where you would want to know where a function and its derivative are equal to 0. These are simple functions, and oftentimes functions will be much more complex. The purpose of this lesson is to enable you to solve these complex functions.

Lesson 1: The Basics

Lesson 2: Solving equations and inequalities

Lesson 3: Graphing

Lesson 4: Limits and series