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### 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 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 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…]

### 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 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 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 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 Calculate Arc Length with Integration

When you use integration to calculate arc length, what you’re doing (sort of) is dividing a length of curve into infinitesimally small sections, figuring the length of each small section, and then adding [more…]

### Finding the Key Parts of All Hyperbolas

A *hyperbola* is the set of all points in the plane such that the difference of the distances from two fixed points (the *foci*) is a positive constant. Hyperbolas always come in two parts, and each one is [more…]

### How to Use Sigma Notation to Find the Area Under a Curve

You can use sigma notation to write out the Riemann sum for a curve. This is useful when you want to derive the formula for the approximate area under the curve. For example, say that you want to find [more…]

### How to Find Absolute Extrema over a Function's Entire Domain

A function's *absolute max* and *absolute min* over its *entire domain* are the highest and lowest values (heights) of the function anywhere it's defined. When you consider a function's entire domain, a function [more…]

### How to Measure the Changing Area Under a Curve

You can use an area function to measure the area under a curve, even as the area changes. For example, say you've got any old function, *f* (*t*). Imagine that at some [more…]