Calculus

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How to Solve a Quadratic Equation by Completing the Square

You can solve quadratic equations by completing the square. Completing the square involves creating a perfect square trinomial from the quadratic equation, and then solving that trinomial by taking its [more…]

How to Use Sigma Notation

For adding up long series of numbers like the rectangle areas in a left, right, or midpoint sum, sigma notation comes in handy. Here’s how it works. Say you wanted to add up the first 100 multiples of [more…]

How to Do Integration by Parts More than Once

Sometimes you have to use the integration-by-parts method more than once because the first run through the method takes you only part way to the answer. [more…]

How to Calculate Instantaneous Speed with Limits

You can calculate the instantaneous speed of an object using limits. Say that you and your calculus-loving cat are hanging out one day, and you decide to drop a ball out of your second-story window. Here’s [more…]

How to Know When a Derivative Doesn't Exist

There are three situations where a derivative fails to exist. The derivative of a function at a given point is the slope of the tangent line at that point. So, if you can’t draw a tangent line, there’s [more…]

How to Use the Chain Rule to Find the Derivative of Nested Functions

Sometimes, when you need to find the derivative of a nested function with the chain rule, figuring out which function is inside which can be a bit tricky — especially when a function is nested inside another [more…]

How to Find Local Extrema with the First Derivative Test

All local maximums and minimums on a function's graph — called local extrema — occur at critical points of the function (where the derivative is zero or undefined). [more…]

How to Find Absolute Extrema on a Closed Interval

Every function that’s continuous on a closed interval has an absolute maximum value and an absolute minimum value (the absolute extrema) in that interval — in other words, a highest and lowest point — [more…]

How to Locate Intervals of Concavity and Inflection Points

You can locate a function's concavity (where a function is concave up or down) and inflection points (where the concavity switches from positive to negative or vice versa) in a few simple steps. The following [more…]

How to Make Linear Approximations

Because ordinary functions are locally linear (that means straight) — and the further you zoom in on them, the straighter they look—a line tangent to a function is a good approximation of the function [more…]

How Integration Works: It’s Just Fancy Addition

The most fundamental meaning of integration is to add up. And when you depict integration on a graph, you can see the adding up process as a summing up of thin rectangular strips of area to arrive at the [more…]

How to Approximate Area with Right Rectangles

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

How to Find Antiderivatives by Guessing and Checking

The guess-and-check method works when the integrand— that’s the thing you want to antidifferentiate (the expression after the integral symbol, not counting the [more…]

How to Find Antiderivatives with the Substitution Method

When a function’s argument (that’s the function’s input) is more complicated than something like 3x + 2 (a linear function of x — that is, a function where [more…]

How to Solve (and Factor) a Quadratic Equation with the Quadratic Formula

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. The solution or solutions of a quadratic equation, [more…]

How to Write Riemann Sums with Sigma Notation

You can use sigma notation to write out the right-rectangle sum for a function. For example, say you’ve got f(x) = x2 + 1.

By the way, you don’t need sigma notation for the math that follows. It’s just [more…]

How to Integrate Sine/Cosine Problems with an Odd, Positive Power of Cosine

When you integrate a trig integral that includes cosine, and if the power of cosine is odd and positive, you can convert and then use substitution to integrate. To make this conversion, you need to know [more…]

How to Integrate Tangent/Secant Problems with an Odd, Positive Power of Tangent

Here’s how you integrate a trig integral that contains tangents and secants where the tangent power is odd and positive. You’ll need the tangent-secant version of the Pythagorean identity: [more…]

How to Integrate Tangent/Secant Problems with an Even, Positive Power of Secant

Here’s how you integrate a trig integral that contains tangents and secants where the secant power is even and positive. Like with all tangent/secant integrals, you use the tangent-secant version of the [more…]

How to Integrate Problems with an Even, Positive Power of Tangent

Here’s how you integrate a trig integral that contains tangents (and no secant factors) where the tangent power is even and positive. [more…]

How to Use Trig Substitution to Integrate

With the trigonometric substitution method, you can do integrals containing radicals of the following forms (given a is a constant and u is an expression containing [more…]

How to Integrate by Using Partial Fractions when the Denominator Contains Only Linear Factors

You can use the partial fractions method to integrate rational functions (Recall that a rational function is one polynomial divided by another.) The basic idea behind the partial fraction approach is “unadding” [more…]

Integrating Using Partial Fractions when the Denominator Contains Irreducible Quadratic Factors

You can use the partial fractions method to integrate rational functions, including functions with denominators that contain irreducible quadratic factors [more…]

How to Find a Function’s 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 is with a diagram — check it out below [more…]

How to Find the Volume of a Complicated Shape with the Meat-Slicer Method of Integration

In geometry, you learned how to figure the volumes of simple solids like boxes, cylinders, and spheres. Integration enables you to calculate the volumes of an endless variety of much more complicated shapes [more…]

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