Calculus

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How to Check an Answer to an Algebra Problem

Checking your answers when doing algebra problems is always a good idea — after all, if there's a way to ensure that you have the correct answer, it's worth the time, isn't it? You check your answers in [more…]

How to Find a Normal Line Perpendicular to a Tangent Line

A line normal to a curve at a given point is the line perpendicular to the line that’s tangent at that same point. Find the points of perpendicularity for all normal lines to the parabola [more…]

How to Approximate Area with Left Sums

You can approximate the area under a curve by using left sums. For example, say you want the exact area under a curve between two points, 0 and 3. The shaded area on the left graph in the below figure [more…]

How to Approximate Area with Right Sums

You can approximate the area under a curve by using right sums. This method works just like the left sum method except that each rectangle is drawn so that its right upper corner touches the curve instead [more…]

How to Do Simple Integration by Parts

Integrating by parts is the integration version of the product rule for differentiation. The basic idea of integration by parts is to transform an integral you [more…]

How to Change Unacceptable Forms before Using L'Hôpital’s Rule

You can use L'Hôpital’s rule to find a limit for which direct substitution doesn't work. If substitution produces one of the unacceptable forms, ± ∞ · 0 or ∞ – ∞, you first have to tweak the problem to [more…]

How to Use Tangent Substitution to Integrate

With the trigonometric substitution method, you can do integrals containing radicals of the following forms: [more…]

How to Use Sine Substitution to Integrate

With the trigonometric substitution method, you can do integrals containing radicals of certain forms because they match up with trigonometric functions. A sine can take the place of a radical in a particular [more…]

How to Find the Average Value with the Mean Value Theorem for Integrals

You can find the average value of a function over a closed interval by using the mean value theorem for integrals. The best way to understand the mean value theorem for integrals is with a diagram — look [more…]

How to Find the Volume of a Shape Using the Washer Method

Geometry tells you how to figure the volumes of simple solids. Integration enables you to calculate the volumes of an endless variety of much more complicated shapes. If you have a round shape with a hole [more…]

How to Analyze Arc Length

You can add up minute lengths along a curve, an arc, to get the whole length. When you analyze arc length, you divide a length of curve into small sections, figure the length of each section, and then [more…]

How to Interpret Function Graphs

You’re going to see dozens and dozens of functions in your study of calculus, and the graphs of those functions can visually express such things as inflation, population growth, and radioactive decay. [more…]

How to Horizontally Transform a Function

You can transform any function into a related function by shifting it horizontally or vertically, flipping it over (reflecting it) horizontally or vertically, or stretching or shrinking it horizontally [more…]

How to Vertically Transform a Function

To transform a function vertically, you add a number to or subtract a number from the entire function, or multiply it by a number. To do something to an entire function, say [more…]

How to Solve Limits by Conjugate Multiplication

To solve certain limit problems, you’ll need the conjugate multiplication technique. When substitution doesn’t work in the original function — usually because of a hole in the function — you can use conjugate [more…]

How to Solve Limits at Infinity by Using Horizontal Asymptotes

Horizontal asymptotes and limits at infinity always go hand in hand. You can’t have one without the other. If you’ve got a rational function like [more…]

The Basic Differentiation Rules

Some differentiation rules are a snap to remember and use. These include the constant rule, power rule, constant multiple rule, sum rule, and difference rule. [more…]

How to Work with Lines in Calculus

A line is the simplest function that you can graph on the coordinate plane. (Lines are important in calculus because when you zoom in far enough on a curve, it looks and behaves like a line.) This figure [more…]

How to Recognize Inverse Functions

You can tell that two functions are inverse functions when each one undoes what the other does. When you graph inverse functions, each is a mirror image of the other. Here are some examples of inverse [more…]

How to Use SohCahToa to Find the Trig Functions of a Right Triangle

The study of trigonometry begins with the right triangle. The three main trig functions (sine, cosine, and tangent) and their reciprocals (cosecant, secant, and cotangent) all tell you something about [more…]

How to Graph Sine, Cosine, and Tangent

So, you need to graph a sine, cosine, or tangent function. Sine, cosine, and tangent — and their reciprocals, cosecant, secant, and cotangent — are periodic [more…]

How to Use Limits to Determine Continuity

Here you’ll learn about continuity for a bit, then go on to the connection between continuity and limits, and finally move on to the formal definition of continuity. [more…]

Solving Easy Limits

There are two types of easy limit problems: the ones you should just memorize and the ones where you can plug in the x-number and get the answer in one step. [more…]

How to Solve Limits with a Calculator

You can solve most limit problems by using your calculator. There are two basic methods. For example, say you want to evaluate the following limit: [more…]

How to Solve Limits by Factoring

You can use the algebraic technique of factoring to solve “real” limit problems. All algebraic methods involve the same basic idea. When substitution doesn’t work in the original function — usually because [more…]

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