More Applications of the derivative

The first derivative test:

If c is a critical number and if f’ changes sign at x=c, then

- f has a local minimum at x=c if f’ is negative to the left of c and positive to the right of c;

- f has a local maximum at x=c if f’ is positive to the left of c and negative to the right of c.

A function f has

- a global maximum value f(c) at the input c if f(x) less than or equal f (c) for every x in the domain of f;

- a global minimum value f(c) at the input c if f(x) larger than or equal f(c) for every x in the domain of f.

The extreme Value Theorem

If a function f is continuous on a closed interval (a, b), then f has a global maximum and a global minimum value on this closed interval (a,b).

The Second Derivative Test

- If f’(c ) = 0 and f’’(c )>0 then f has a local minimum at c;

- If f’(c ) = 0 and f’’(c )<0 then f has a local maximun at c.

Find the vertical asymptotoes and horizontal asymptotes of f if f(x)=1/(x^2-1)

X^2-1 = 0

X=1 and x=-1 are vertical asympototes.

Lim(x-->+oo) =0

And

lim(x-->-00) =0

so, y=o is a horizontal asymptote.

Optimization problems

- read the problem. Identify the given quantities and those we must find

- sketch a diagram

- Using available information, express f as a function of just one variable

- Determine the domain of f and draw its graph

- Find the global extreme of f,

- Convert the result

Mean Value Theorem

If the function is continuous on the closed interval (a, b) and differentiable on the open interval (a,b), then there exists at least one number c in the open interval (a, b) such that

F’(c )= ( f(b)-f(a) )/ (b-a)

Ant derivatives:

Given the function f, determine a function F whose derivative is f. the function F is an antiderivative of f

Function -> antiderivative

X^n-------> (X^(n+1) / (n+1) ) +C

Cos X --->Sin X +C

Sin X----> -Cos X +C

(SecX)^2 -> tan X+C

e^X-------> e^X