## Cheat Sheet

# 1,001 Calculus Practice Problems For Dummies

Solving calculus problems is a great way to master the various rules, theorems, and calculations you encounter in a typical Calculus class. This Cheat Sheet provides some basic formulas you can refer to regularly to make solving calculus problems a breeze (well, maybe not a breeze, but definitely easier).

## Useful Calculus Theorems, Formulas, and Definitions

Following are some of the most frequently used theorems, formulas, and definitions that you encounter in a calculus class for a single variable. The list isn’t comprehensive, but it should cover the items you’ll use most often.

## Limit Definition of a Derivative

## Definition: Continuous at a number *a*

## The Intermediate Value Theorem

## Definition of a **Critical Number**

A *critical number* of a function *f* is a number *c* in the domain of *f* such that either *f*^{ }*'*(*c*)* *= 0 or *f*^{ }*'*(*c*) does not exist.

**Rolle’s Theorem**

Let *f* be a function that satisfies the following three hypotheses:

*f*is continuous on the closed interval [*a, b*].*f*is differentiable on the open interval (*a, b*).*f*(^{}*a*)*f*(^{}*b*).

Then there is a number *c* in (*a, b*) such that *f*^{ }*'*(*c*)* *= 0.

**The Mean Value Theorem**

Let *f* be a function that satisfies the following hypotheses:

*f*is continuous on the closed interval [*a, b*].*f*is differentiable on the open interval (*a, b*).

**Newton’s Method Approximation Formula**

Newton’s method is a technique that tries to find a root of an equation. To begin, you try to pick a number that’s “close” to the value of a root and call this value *x*_{1}. Picking *x*_{1} may involve some trial and error; if you’re dealing with a continuous function on some interval (or possibly the entire real line), the intermediate value theorem may narrow down the interval under consideration. After picking *x*_{1}, you use the recursive formula given here to find successive approximations:

A word of caution: Always verify that your final approximation is correct (or close to the value of the root). Newton’s method can fail in some instances, based on the value picked for *x*_{1}. Any calculus text that covers Newton’s method should point out these shortcomings.

**The Fundamental Theorem of Calculus **

Suppose *f* is continuous on [*a, b*]. Then the following statements are true:

**The Trapezoid Rule**

where

**Simpson’s Rule**

where *n* is even and

## Special Limit Formulas in Calculus

Many people first encounter the following limits in a Calculus textbook when trying to prove the derivative formulas for the sine function and the cosine function. These results aren’t immediately obvious and actually take a bit of work to justify. Any calculus text should provide more explanation if you’re interested in seeing it!

## Derivative and Integration Formulas for Hyperbolic Functions

The hyperbolic functions are certain combinations of the exponential functions *e** ^{x}* and

*e*

^{–}*. These functions occur often enough in differential equations and engineering that they’re typically introduced in a Calculus course. Some of the real-life applications of these functions relate to the study of electric transmission and suspension cables.*

^{x}