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…]

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…]

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…]

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…]

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…]

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…]

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…]

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…]

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…]

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…]

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…]