University of Akron

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.

Just as definite integrals can be used to find the area under a curve, they can also be used to find the area between two curves. In this section, we use a familiar concept from Calc I (that is, the Fundamental Theorem of Calculus) to determine the area enclosed by two or more curves after identifying points of intersection.

Functions are a fundamental concept in mathematics. In this section, we discuss the notion of a function and the relationship between two important pieces of a function:domain and range. These notes include examples for identifying functions given a specific domain and range, as well as a variety of examples practicing how to evaluate a function with unique inputs. We then examine the visual graphs of some functions and use the Vertical Line Test (VLT) in determining whether the graph represents...

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