Understanding Calculus: Problems, Solutions, and Tips

Calculus is the mathematics of change, a field with important applications in science, engineering, medicine and business; surveying the goals of the series; investigating the classic tangent line problem, which illustrates the concept of limits.

In the first of two review episodes on precalculus, examining graphs of equations and properties such as symmetry and intercepts; the use of equations to model real life; study of functions, which Edwards calls an important concept in mathematics.

Different types of functions and how they can be identified by their distinctive shapes when graphed; reviewing trigonometric functions, using both the right triangle definition as well as the unit circle definition, which measures angles in radians.

Jumping into real calculus by going deeper into the concept of limits introduced in the first episode; the working definition of limits and how to determine a limit in three distinct ways; how to recognize when a given function does not have a limit.

A function is continuous if there is no interruption in the curve when its graph is drawn; the three conditions that must be met for continuity; applications of associated ideas; the greatest integer function and the intermediate value theorem.

Infinite limits describe the behavior of functions that increase or decrease without bound, in which the asymptote is the specific value that the function approaches without ever reaching it; some examples from relativity theory and biology.