Curves & Lines in Calculus

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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

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

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

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.

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

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

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).

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 —

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

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

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

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

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

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

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

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