The Handy Math Answer Book

What is the Mean-Value Theorem?

T

he Mean-Value Theorem has nothing to do with crankiness, but it isone of

the most important theoretical tools in the calculus. In written terms, it is

defined as the following: If f(x) is defined and continuous on the interval [a, b],

and differentiable on (a, b), then there is at least one number on the interval (a,

b)—or a< c< b—such that:

When f(a) f(b), this is a special case called Rolle’s Theorem,when we

know that f(c) will equal zero. Interpreting this, we know that there is a point on

(a, b) that has a horizontal tangent.

It is also true that the Mean-Value Theorem can be put in terms of slopes.

Thus, the last part of the above equation (on the right of the equal sign) repre-

sents the slope of a line passing through (a, f(a)) and (b, f(b)). Thus, this theory

states that there is a point c(a, b), such that the tangent line is parallel to a

line passing through the two points.

()

() ()

fc ba

fb fa

= -

• l

where, again, f' is the derivative of f with respect to x.

For a power:

For a chain:

or

where z/x is a partial derivative.

What are the derivativesof trigonometric functions?

There are also derivatives of the six major trigonometric functions—sine, cosine,

cotangent, cosecant, tangent, and secant (for more information on these trigonomet-

ric functions, see “Geometry and Trigonometry”). The following lists the formulas for

those derivatives (in this case, the functions are written with respect to the variable u):

dx

dy

du

dy

dx

= $du

dxd()xnxnn= -^1

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