maxima and minima

SKETCHING THE GRAPH USINGTHE FIRST DERIVATIVE TEST

Standard of Competence: To use The concept of Function Limit and

Function deferential in problem solving

Basic Competence: To use The derived to find the caracteristic of

functions and to solve the problems

Indicator:•To find the function increases and the function decreases by first derivative concept•To sketch the function graph by the propertis of the Derived Functions•To find end points of function graph

Definitions of Increasing and Decreasing Functions

A function is increasing when its graph rises as it goes from left to right. A function is decreasing when its graph falls as it goes from left to right. inc inc

dec

The increasing/decreasing concept can be associated with the slope of the tangent line. The slope of the tangent line is positive when the function is increasing and negative when decreasing

Test for Increasing and Decreasing Functions

Theorem 3.6 The First Derivative Test

Using First Derivatives to Find Maximum and Minimum Values and Sketch Graphs

• Example 1: Graph the function f given by

• and find the relative extremes.• Suppose that we are trying to graph this function but • do not know any calculus. What can we do? We can • plot a few points to determine in which direction the • graph seems to be turning. Let’s pick some x-values• and see what happens.

f (x)2x3 3x2 12x 12.

Using First Derivatives to Find Maximum and Minimum Values and Sketch Graphs

• Example 1 (continued):

Using First Derivatives to Find Maximum and Minimum Values and Sketch Graphs

• Example 1 (continued): • We can see some features of the graph from the sketch. • Now we will calculate the coordinates of these features • precisely.

• 1st find a general expression for the derivative.

• 2nd determine where f (x) does not exist or where • f (x) = 0. (Since f (x) is a polynomial, there is no • value where f (x) does not exist. So, the only • possibilities for critical values are where f (x) = 0.)

f (x)6x2 6x 12

Optimizing an Open Box• An open box with a square base

is to be constructed from 108 square inches of material.

• What dimensions will produce a box that yields the maximum possible volume?

Which basic shape would yield the maximum

volume?• Should it be tall?• Should it be square?• Should it be more cubical?• Perhaps we could try calculating a few volumes and get lucky.

Is this the maximum volume?

Is this the maximum volume?

Is this the maximum volume?

Is this the maximum volume?

How are we doing?

• Guess and Check really is not a very efficient way to approach this problem.

• Lets use Calculus and get directly to the solution of this problem.

• We can apply the maxima theory for a derivative to resolve this problem.

Working rule for finding points of maxima and minima

• Let f be a function such that f’ (x) exists.• 1) if f ’ (C) = 0 and f(C) ’’ > 0

then f has local minima• 2) if f ’(C) = 0 and f ’’(C) < 0 then f

has local maxima