## ESSENTIAL CALCULUS SKILLS

You learn a lot during a calculus course.

How much of this is useful?

Of course, it depends on what you do after calculus.

For example, much of the material is needed in higher-level math courses.

But how about for physics, engineering, and other applications of calculus?

- derivatives of polynomials and trig functions are absolutely essential. You need to practice these until you’re fluent.
- for integration, sometimes you can get by with a table of integrals, but to be successful with this, you need to understand the ideas behind integration, and you need to have some experience doing integrals by hand.
- but what if you take a course that applies calculus where the professor doesn’t allow a table of integrals? In that case, it may be handy to be familiar with u-sub, trig-sub, and integration by parts. (Familiarity with these techniques may also help you better understand how to use tables of integrals, and especially what to do when you need to integrate an expression that you can’t quite find in a standard table.)
- derivatives and anti-derivatives of exponentials and logarithms show up in some common differential equations in science and mathematics.
- double and triple integrals naturally come about in some applications of physics and engineering.
- optimization and extreme value problems are very common in many important applications of calculus. It’s worth learning these techniques very well.
- it’s very common to make a graph when you have data in math or science, so it would be helpful to understand conceptually how derivatives and integrals are related to interpreting a graph.
- most real-world problems require numerical solutions, so any chance to apply calculus numerically is helpful. Example: Simpson’s rule.
- when you derive an equation in physics or another field, it’s helpful if you can verify that it exhibits appropriate behavior for various limiting cases. Taylor series expansions are often helpful with this.
- multi-variable calculus is handy because the real-world seldom involves relations with just two independent variables, x and y. Vector calculus is essential for going beyond the first year of physics.

## CHRIS MCMULLEN, PH.D.

Copyright © 2018 Chris McMullen, author of the *Improve Your Math Fluency* series of math workbooks

Newest releases (in math):

**Essential Calculus Skills Practice Workbook (with Full Solutions)**- 50 Challenging Algebra Problems
**50 Challenging Calculus Problems**(available in September, 2018)- Fractions Essentials Workbook with Answers