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…]
How to Find a Function's Derivative by Using the Chain Rule
The chain rule is by far the trickiest derivative rule, but it’s not really that bad if you carefully focus on a few important points. [more…]
How the Area Function Works
The area function is a bit weird. Brace yourself. Say you’ve got any old function, f(t). Imagine that at some t-value, call it s, you draw a fixed vertical line. [more…]
The Fundamental Theorem of Calculus: What It Is and How It Works
There are some who say that the Fundamental Theorem of Calculus is one of the most important theorems in the history of mathematics. Here it is: [more…]
How to Find Area with the Shortcut Version of the Fundamental Theorem of Calculus
The Fundamental Theorem of Calculus has a shortcut version that makes finding the area under a curve a snap. Here it is. Let F be any antiderivative of the function [more…]
Integration by Parts Problems where You Go around in Circles
Sometimes if you use integration by parts twice, you get back to where you started from — which, unlike getting lost, is not a waste of time. [more…]
How to Integrate Sine/Cosine Problems with an Odd, Positive Power of Sine
Here’s how you integrate a trig integral that contains sines and cosines where the power of sine is odd and positive. You lop off one sine factor and put it to the right of the rest of the expression, [more…]
How to Integrate Sine/Cosine Problems with Even, Nonnegative Powers of Both Sine and Cosine
Here’s how you integrate a trig integral that contains sines and cosines where the powers of both sine and cosine are even and nonnegative (in other words, zero or positive). You first convert the integrand [more…]
How to Find the Area between Two Curves
To find the area between two curves, you need to come up with an expression for a narrow rectangle that sits on one curve and goes up to another. [more…]
How to Find the Area of a Surface of Revolution
A surface of revolution is a three-dimensional surface with circular cross sections, like a vase or a bell or a wine bottle. For these problems, you divide the surface into narrow circular bands, figure [more…]
How to Use L'Hôpital's Rule to Solve Limit Problems
L’Hôpital’s rule is a great shortcut for doing some limit problems. (And you may need it someday to solve some improper integral problems, and also for some infinite series problems.) [more…]
How to Decompose Partial Fractions
A process called partial fractions takes one fraction and expresses it as the sum or difference of two other fractions. In calculus, this process is useful before you integrate a function. Because integration [more…]
How to Factor a Polynomial Expression
In mathematics, factorization or factoring is the breaking apart of a polynomial into a product of other smaller polynomials. If you choose, you could then multiply these factors together, and you should [more…]
How to Graph a Circle
The first thing you need to know in order to graph the equation of a circle is where on a plane the center is located. The equation of a circle appears as [more…]
How to Graph an Ellipse
An ellipse is a set of points on a plane, creating an oval, curved shape, such that the sum of the distances from any point on the curve to two fixed points [more…]
How to Graph a Hyperbola
Think of a hyperbola as a mix of two parabolas — each one a perfect mirror image of the other, each opening away from one another. The vertices of these parabolas are a given distance apart, and they open [more…]
How to Solve Linear Systems
When you solve systems with two variables and therefore two equations, the equations can be linear or nonlinear. Linear systems are usually expressed in the form Ax + By [more…]
