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### The Mean Value Theorem

You don’t need the mean value theorem for much, but it’s a famous theorem — one of the two or three most important in all of calculus — so you really should learn it. Fortunately, it’s very simple. [more…]

### How to Use Differentiation to Calculate the Maximum Volume of a Box

One of the most practical uses of differentiation is finding the maximum or minimum value of a real-world function. In the following example, you calculate the maximum volume of a box that has no top and [more…]

### How to Use Differentiation to Calculate the Maximum Area of a Corral

Finding the maximum or minimum value of a real-world function is one of the most practical uses of differentiation. For example, you might need to find the maximum area of a corral, given a certain length [more…]

### Related Rates: the Expanding Balloon Problem

Say you’re filling up your swimming pool and you know how fast water is coming out of your hose, and you want to calculate how fast the water level in the pool is rising. You know one rate [more…]

### Related Rates: the Trough of Swill Problem

Say you’re filling up your swimming pool and you know how fast water is coming out of your hose, and you want to calculate how fast the water level in the pool is rising. You know one rate [more…]

### Related Rates: Two Cars at a Crossroads

Say you’re filling up your swimming pool and you know how fast water is coming out of your hose, and you want to calculate how fast the water level in the pool is rising. You know one rate [more…]

### How to Find the Tangent Lines of a Parabola that Pass through a Certain Point

Ever want to determine the location of a line through a given point that’s tangent to a given curve? Of course you have! Here’s how you do it.

Determine the points of tangency of the lines through the point [more…]

### How to Find a Normal Line to a Curve

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 Determine Marginal Cost, Marginal Revenue, and Marginal Profit in Economics

Marginal cost, marginal revenue, and marginal profit all involve how much a function goes up (or down) as you go over 1 to the right — this is very similar to the way linear approximation works. [more…]

### How to Approximate Area with Left Rectangles

You can approximate the area under a curve by summing up “left” rectangles. For example, say you want the area under the curve *f* (*x*) = *x*^{2} + 1 from 0 to 3. The shaded area of the graph on the left side [more…]

### How to Approximate Area with Midpoint Rectangles

A good way to approximate areas with rectangles is to make each rectangle cross the curve at the midpoint of that rectangle's top side*.* A midpoint sum is a much better estimate of area than either a left-rectangle [more…]

### How to Approximate Area with the Trapezoid Rule

With the trapezoid rule, instead of approximating area by using rectangles (as you do with the left, right, and midpoint rectangle methods), you approximate area with — can you guess? — trapezoids. [more…]

### How to Approximate Area with Simpson's Rule

With Simpson’s rule, you approximate the area under a curve with curvy-topped “trapezoids.” The tops of these shapes are sections of parabolas. You can call them “trapezoids” because they play the same [more…]

### How to Do 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 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 Convert Square Roots to Exponents

Finding square roots and converting them to exponents is a relatively common operation in algebra. Square roots, which use the radical symbol, are nonbinary operations — operations which involve just one [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…]