University of Akron

In this section, we learn how to solve for x given two types of equations: rational equations and radical equations. We learn about what's called an "extraneous root" and how to properly check our final solutions. Additionally, we learn how to factor a difference of two cubes or a sum of two cubes to help us identify our roots.

In this section, we further examine quadratic functions by analyzing their U-shaped graphs. Key features to identify on a quadratic's graph include the vertex, axis of symmetry, end behavior, x- and y-intercepts, and concavity. Given a quadratic equation in standard form, these notes guide you through the proper steps on how to properly sketch the graph of the function, thoroughly identify each key feature, and rewrite the equation in shifted/vertex form as well.

These notes explore the properties of quadratic functions and how quadratics can be classified by the number of possible solutions each one can have. To find the solution to a quadratic function/equation, we look at its x-intercepts and where they occur on a graph. For complex situations, we choose from a variety of algebraic techniques to determine the solutions. These techniques, as outlined in the notes, are as follows: Principle of Zero Products, Principle of Square Roots, Quadratic Formula,...

The combination of a real number with an imaginary number creates a single entity known as a complex number. In this section, we become familiar with the notion of complex numbers and how to perform arithmetic operations on them.

A "composition of functions" is an operation where two functions are combined by using the output of one function as the input for the other, thus creating a new function. Essentially, we are substituting one function as the input for the other function. These notes provide a few brief examples on composing new functions given preexisting functions, as well as how to find the domains of these new functions.

The "algebra of functions" refers to performing basic arithmetic operations such as addition, subtraction, multiplication, and division on different functions. Essentially, we are combining two functions to create one brand new function to study. These new functions are required to be equipped with compatible domains in order for these operations to be valid. We take a look at how to evaluate these new functions as well as how to determine their new domains.

The average of some finite set of values is a familiar concept/computation. We generalize this idea to calculus where we are now asked to find the average value of a function using integration techniques. To do so, we fall back on the idea of approximation from Calc I.

We've been using integration since Calculus I, but several physical applications of the definite integral are common outside of math class, specifically in the departments of engineering and physics. In this section, we solve numerous problems dealing with work calculations in a variety of settings, with an emphasis on spring problems.

Definite integrals can be used to find the volume of solids. In calculus, the "slicing method" of finding a solid's volume involves dividing that solid into many thin slices, estimating the volume of each slice, and then summing up those volumes through integration. In this section, we learn a method called the "shell method" that involves slicing a solid into smaller sections to find its overall volume. We also discuss the differences between the shell method, disc method, and wa...

Definite integrals can be used to find the volume of solids. In calculus, the "slicing method" of finding a solid's volume involves dividing that solid into many thin slices, estimating the volume of each slice, and then summing up those volumes through integration. In this section, we learn two methods, namely the "disc method" and the "washer method," that involve slicing a solid into smaller sections to find its overall volume.