Chapter 5/unit 4/5 notes

Absolute extreme values

Let f be a function with domain d, then f© is the

(a) absolute maximum value on d if and only if f(x) <_ f© for all the x in D

(b) absolute minimum value on d if and only if f(x)>_ f© for all x in d

The extreme value theorem

If f is continuous on a closed interval [a,b] then f has both a maximum value and a minimum value on the interval. Could be interior points or end points.

Local extreme values

Let x be an interior points of the domain function f. The f© is a

(a) local maximum value at c if and only if f(x)<_f© for all x in some open interval containing c

(b) local minimum value at c if and only if f(x)>_f© for all x in some open interval containing c

Local extreme value theorem

If a function f has a local maximum it a local value minimum at an interior points d of its domain and if f’ exists at c then

F’(C)=0

Critical point

A point in the interior of the domain of a function f at which f’=0 or f’ does not exist at a critical point of f

Stationary point

A point in the interior of the domain of a function at which f’ =0 is this point of f

Mean value theorem for derivatives

If y=f(x) is continuous at every poly of the closed interval [a,b] and differentiable at every point of its interior (a,b) then there is at least one pint c in (a,b) at which

f’(x)= f(b)-f(a)/ b-a

1st derivative test

It must be continuous.

Critical point at x=c

F’ changes from positive to negative it is going to be a maximum

F’ changes from negative to positive, it is a minimum

F’ first change sign, no extreme value or max/min

Concavity test

f’>0 is increasing

f’<0 is decreasing

f’’> 0 concave up

f’’< 0 concave down

Second derivative test

Test the concavity at values where f’=0

Find critical points where f’=0

if f’©= 0 and f’’© <0 then f has a local max at x=c

If f’©=0 and f’’©>0 the f has a local minimum at x=c

Steps for solving max-min word problems

1) understand the problem- read the problem carefully. Identify the into you need to solve the problem.

2) develop a mathematical model of the problem- draw pictures and label parts that are important to the problem. Introduce the variable to represent the quantity to be maximized or minimized. Using that variable, write a function verbose extreme value gives the information sought.

3) graph the function- find the domain of the function. Determine what values of the variable make sense in the problem.

4)identify the critical points and end points- find where the derivative is zero or fails to exist

5) solve the mathematical model- if unsure of the result, support or confirm you solution with another method

6) interpret the solution- translate your mathematical results into the problem setting and decide whether the result makes sense

Linearization

If f is differentiable at x=a, then the equation of the tangent line,

L(x)=f(a)+f’(a)(x-a)

defines that of f at a.

The approximation of f(x)~ L(x) is the standard linear approximation of f at a. The point x=a is the center of the approximation.

Newtons Method

Guess a first approximation to a solution of the equation f(x)=0. a graph helps

Use the first approximation to get a second, the second to get a third and so on using the formula

Xn+1= Xn- (f(Xn)/f’(xn))