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How to Solve Improper Integrals that Have One or Two Infinite Limits of Integration

One of the ways in which definite integrals can be improper is when one or both of the limits of integration are infinite. You solve this type of improper integral by turning it into a limit problem where [more…]

How to Find the Volume and Surface Area of Gabriel's Horn

Finding the volume and surface area of this horn problem may blow your mind. Gabriel’s horn is the solid generated by revolving about the x-axis the unbounded region between [more…]

How to Do a Related Rate Problem Involving a Moving Baseball

You can use calculus to determine a rate that’s related to the speed of a moving object. For example, say a pitcher delivers a fastball, which the batter pops up — it goes straight up above home plate. [more…]

How to Determine Limits of Sequences with L'Hôpital's Rule

You can use L’Hôpital’s rule to find limits of sequences. L'Hôpital's rule is a great shortcut for when you do limit problems. Here it is: [more…]

How to Use the nth Term Test to Determine Whether a Series Converges

If the individual terms of a series (in other words, the terms of the series’ underlying sequence) do not converge to zero, then the series must diverge. This is the [more…]

How to Work with Geometric Series

Geometric series are relatively simple but important series that you can use as benchmarks when determining the convergence or divergence of more complicated series. A geometric series is a series of the [more…]

How to Analyze a p-Series

So-called p-series are relatively simple but important series that you can use as benchmarks when determining the convergence or divergence of more complicated series. A [more…]

How to Analyze a Telescoping Series

You don’t see many telescoping series, but the telescoping series rule is a good one to keep in your bag of tricks — you never know when it might come in handy. Consider the following series: [more…]

Using the Direct Comparison Test to Determine If a Series Converges

The direct comparison test is a simple, common-sense rule: If you’ve got a series that’s smaller than a convergent benchmark series, then your series must also converge. And if your series is larger than [more…]

How to Use the Limit Comparison Test to Determine Whether a Series Converges

The idea behind the limit comparison test is that if you take a known convergent series and multiply each of its terms by some number, then that new series also converges. And it doesn’t matter whether [more…]

Determining If a Series Converges Using the Integral Comparison Test

The integral comparison test involves comparing the series you’re investigating to its companion improper integral. If the integral converges, your series converges; and if the integral diverges, so does [more…]

Using the Ratio Test to Determine Whether a Series Converges

The ratio test looks at the ratio of a general term of a series to the immediately preceding term. The ratio test works by looking only at the nature of the series you’re trying to figure out [more…]

How to Use the Root Test to Determine Whether a Series Converges

The root test doesn’t compare a new series to a known benchmark series. It works by looking only at the nature of the series you’re trying to figure out. You use the root test to investigate the limit [more…]

How to Analyze Absolute and Conditional Convergence

Many divergent series of positive terms converge if you change the signs of their terms so they alternate between positive and negative. For example, you know that the harmonic series diverges: [more…]

How to Determine Whether an Alternating Series Converges or Diverges

An alternating series is a series where the terms alternate between positive and negative. You can say that an alternating series converges if two conditions are met: [more…]

How to Find Area with the u-Substitution Method

You can use the Fundamental Theorem to calculate the area under a function (or just to do any old definite integral) that you integrate with the substitution method. What you want to do is to change the [more…]

How to Determine the Dimensions for the Least Expensive Window Frame

You can use calculus to solve practical problems, such as determining the correct size for a home-improvement project. Here’s an example. A Norman window has the shape of a semicircle above a rectangle [more…]

The Difference Quotient: The Bridge between Algebra (Slope) and Calculus (the Derivative)

One of the cornerstones of calculus is the difference quotient. The difference quotient — along with limits — allows you to take the regular old slope formula that you used to compute the slope of lines [more…]

The Definition of the Definite Integral and How it Works

You can approximate the area under a curve by adding up right, left, or midpoint rectangles. To find an exact area, you need to use a definite integral. [more…]

Recognize Even and Odd Numbers, and Multiples of 3, 4, and 5

A number sequence is an arrangement of numbers according to a rule. For example, you can have a sequence of numbers that are odd or even, or multiples of 3, 4, 5, and so on. [more…]

How to Find Square Numbers

You get a square number by multiplying a number by itself, so knowing the square numbers is a handy way to remember part of the multiplication table. Although you probably remember without help that 2 [more…]

How to Identify Prime Numbers

A prime number can be divided by only itself and 1. This makes it stubborn. Like certain people you may know, a prime number resists being placed in any sort of a box. Look at how the number 13 is depicted [more…]

How to Add and Subtract on the Number Line

You can use the number line to demonstrate simple addition and subtraction. These first steps in math become a lot more concrete with a visual aid. Here’s the main thing to remember: [more…]

Use the Number Line to Find Zero

An important addition to the number line is the number 0, which means nothing, zilch, nada. Step back a moment and consider the bizarre concept of nothing. For one thing — as more than one philosopher [more…]

How to Multiply with the Number Line

Suppose you start at 0 and circle every other number on a number line, as shown in the below figure. As you can see, all the even numbers are now circled. In other words, you’ve circled all the multiples [more…]

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