Post #26

Sorry for posting this late but the parade yesterday wiped me out.

Implicit Derivatives

The only difference between implicit derivatives and regular derivatives is that implicit derivatives include dy or y', the actual derivative of y.

y=x+2 y'=1

In an implicit derivative, you are always asked to solve for y'.

Example:

x^2+2y=0

1. Take derivative of both sides first.

2x+2y'=0

2. Then solve for y'.

y'=(-2x)/2

Some examples include:

4x+13y^2=4 y'=(-4/26y)

cos(x)=y y'=-sin(x)

y^3+y^2-5y-x^2=4 y'=2x/((3y+5)(y-1))

Rolle's Theorem

To understand Rolle's Theorem, you must first understand the Extreme Value Theorem.  The Extreme Value Theorem states that a continuous function on the closed interval [a,b] must have both a maximum and a minimum.  They can be on the endpoint a and b though.

So Rolle's Theorem gives the conditions that guarantee the existence of an extrema in the interior of a closed interval.

Rolle's Theorem-If a function is continuous on the closed interval [a,b] and differentiable on the open interval (a,b) and f(a)=f(b), then there is at least one point, designated "c", where the derivative of f(c)=0.

Example:  Find all points "c" where the derivative of f(c)=0  f(x)=x^4-2x^2   [-2,2]

1.  I first found that f(a) equals f(b).

(-2)^4-(-2)^2=8
(2)^4-2(2)^2=8

2.  Once finding that f(a)=f(b), I set the derivative equal to 0 and solved for x.

4x^3-4x=0
4x(x^2-1)=0
4x=0  x^2-1=0
x=-1,0,1

There are three point of extrema within the [-2,2] of the function x^4-2x^2.  One at x=-1  another at x=0 and the last at x=1.


Mean Value Theorem

The Mean Value Theorem states that If the function f(x) is continuous on the closed interval [a,b] and differentiable on the open interval (a,b), then there exists a number "c" in (a,b) such that f'(c)=(f(b)-f(a))/(b-a).

This is essentially the same thing as average speed, which finds the slope of the equation.