Calculus Workbook For Dummies Cheat Sheet
To effectively work through calculus problems, you have to understand a number of topics: the process of evaluating limits, methods of solving various differentiation and integration problems, and the [more…]
Classifying Differential Equations by Order
The most common classification of differential equations is based on order. The order of a differential equation simply is the order of its highest derivative. You can have first-, second-, and higher-order [more…]
Distinguishing among Linear, Separable, and Exact Differential Equations
You can distinguish among linear, separable, and exact differential equations if you know what to look for. Keep in mind that you may need to reshuffle an equation to identify it. [more…]
Defining Homogeneous and Nonhomogeneous Differential Equations
In order to identify a nonhomogeneous differential equation, you first need to know what a homogeneous differential equation looks like. You also often need to solve one before you can solve the other. [more…]
Using the Method of Undetermined Coefficients
If you need to find particular solutions to nonhomogeneous differential equations, then you can start with the method of undetermined coefficients. Suppose you face the following nonhomogeneous differential [more…]
Differential Equations For Dummies Cheat Sheet
To confidently solve differential equations, you need to understand how the equations are classified by order, how to distinguish between linear, separable, and exact equations, and how to identify homogenous [more…]
How to Do an Area Approximation Using Sigma Notation
Sigma notation comes in handy when you’re approximating the area under a curve. For example, express an 8-right-rectangle approximation of the area under [more…]
How to Use Trig Substitution to Integrate radicals of the sine form
Before reading this article, you should check out the discussion of trig substitution in the companion article, “How to Use Trig Substitution to Integrate.” [more…]
How to Analyze Position, Velocity, and Acceleration with Differentiation
Every time you get in your car, you witness differentiation first hand. Your speed is the first derivative of your position. And when you step on the accelerator or the brake — accelerating or decelerating [more…]
How to Express Exponential Relationships with Logarithms
You may come across logarithms in your calculus work. A logarithm is just a different way of expressing an exponential relationship between numbers. For instance, [more…]
How to Simplify Roots
When you work with roots in an equation, you often need to simplify them. There are two methods: the quick, sort of intuitive method, and a slightly longer method. The quick method of simplification works [more…]
How to Factor Mathematical Expressions
You often need to factor expressions (break those expressions into their simpler components, or factors) for calculus. Factoring means “unmultiplying,” like rewriting 12 as [more…]
How to Use Math Root Rules
When using math root rules, first note that you can’t have a negative number under a square root or any other evennumber root — at least, not in basic calculus. Here are a couple of easy rules to begin [more…]
How to Solve a Quadratic Equation by Factoring
A quadratic equation is any second degree polynomial equation — that’s when the highest power of x, or whatever other variable is used, is 2. You can solve quadratic equations by factoring. [more…]
How to Find Local Extrema with the Second Derivative Test
The Second Derivative Test is based on two prize-winning ideas: First, that at the crest of a hill, a road has a hump shape — in other words, it’s curving down or concave down. And second, at the bottom [more…]
Solving Differential Equations Using an Integrating Factor
A clever method for solving differential equations (DEs) is in the form of a linear first-order equation. This method involves multiplying the entire equation by an [more…]
How to Antidifferentiate Any Polynomial Using the Sum, Constant Multiple, and Power Rules
The anti-differentiation rules for integrating greatly limit how many integrals you can compute easily. In many cases, however, you can integrate any polynomial in three steps by using the Sum Rule, Constant [more…]
How to Use Identities to Integrate Trigonometry Functions
You’ll be surprised how much headway you can often make when you integrate an unfamiliar trigonometry function by first tweaking it using the Basic Five trig identities: [more…]
Computing Integrals and Representing Integrals as Functions
In trying to understand what makes a function integrable, you first need to understand two related issues: difficulties in computing integrals and representing integrals as functions. [more…]
Understanding What Makes a Function Integrable
When mathematicians discuss whether a function is integrable, they aren’t talking about the difficulty of computing that integral — or even whether a method has been discovered. Each year, mathematicians [more…]
Use a Shortcut for Integrating Compositions of Functions
You can use a shortcut to integrate compositions of functions — that is, nested functions of the form f(g(x)). Technically, you’re using the variable substitution [more…]
Use a Shortcut for Integrating Compositions of Functions
You can use a shortcut to integrate compositions of functions — that is, nested functions of the form f(g(x)). Technically, you’re using the variable substitution [more…]
Using the Product Rule to Integrate the Product of Two Functions
The Product Rule enables you to integrate the product of two functions. For example, through a series of mathematical somersaults, you can turn the following equation into a formula that’s useful for integrating [more…]
How to Integrate Even Powers of Secants with Tangents
You can integrate even powers of secants with tangents. If you wanted to integrate tanm xsecn x when n is even — for example, tan8 x sec6 x — you would follow these steps: [more…]
How to Integrate Odd Powers of Tangents with Secants
You can integrate odd powers of tangents with secants. To integrate tanm x secnx when m is odd — for example, tan7x sec9 x — you would follow these steps: [more…]
