Calculus: Hughs-Hallett Chap 4

Joel Baumeyer, FSC

Christian Brothers University

Using the Derivative -- Optimization

The Tangent Line Approximation of a Function

For values of x near a,

We are thinking of a as fixed, so f(a) and f’(a) are constant!

The expression is a linear function which approximates f(x) well near a. It is called the local lineari-zation of f near x = a.

).)((')()( axafafxf

)ax)(a('f)a(f)x(f

Linear Tangent Line Approximation-Suppose f is differentiable at x = a. Then, for

values of x near a, the tangent line approximation to f(x) is:

The expression is called the local linearization of f near x = a. We are thinking of a as fixed, so that f(a) and f’(a) are constant. The error E(x), is defined by:

and

).)((')()( axafafxf ))((')( axafaf

).)((')()()( axafafxfxE .0

)(lim

ax

xEax

l’Hopital’s Rule (from Chapter 4)

If f and g are differentiable and either of the following conditions hold:1. f(a) = g(a) = 0 or

2. or if a =

then:

)x(glimand)x(flimaxax

)x('g

)x('flim

)x(g

)x(flim

axax

Review:If f’ > 0 on an interval, then f is

increasing on that interval.If f’ < 0 on an interval, then f is

decreasing on that interval.If f’’ > 0 on an interval, then the

graph of f is concave up on that interval.

If f’’ < 0 on an interval, then the graph of f is concave down on that interval.

Definition of Maxima and Minima

Suppose p is a point in the domain of f: f has a local (relative) minimum at p if f(p) is less

than or equal to the values of f for points near p. f has a local (relative) maximum at p if f(p) is greater

than or equal to the values of f for points near p.

f has a global minimum at p if f(p) is lessthan or equal to all values of f.

f has a global maximum at p if f(p) is greater than or equal to all values of f.

Definition of a Critical Point

For any function f, a point p in the domain of f is a critical point if: f’(p) = 0, or if f’(p) is undefined

f(p) is then called the critical value of f at the critical point p.

Theorem (Critical Point)

If a continuous function f has a local maximum or minimum at p, and if p is not an endpoint of the domain, then p is a critical point.

The First-Derivative Test for Local Max (M) and Min (m)

The First-Derivative Test for Local Max (M) and Min (m)Suppose p is a critical point of a continuous function f.

If f’ changes from negative to positive at p, then f has a local minimum at p.

If f’ changes from positive to negative at p, then f has a local maximum at p.

Exampleof

First &Second

DerivativeTests

The Second-Derivative Test for Local Max (M) and Min (m)

If f’(p) = 0 and f’’(p) > 0 then f has a local minimum at p.

If f’(p) = 0 and f’’(p) < 0 then f has a local maximum at p.

if f’(p) = 0 and f’’(p) = 0 then the test tells nothing.

Definition of Inflection Point

A point at which the graph of a function changes concavity is called an inflection point.

This may be a point where the second derivative: does not exist, or equals zero.

The Bounds of a FunctionA function is bounded on a interval if there are

numbers L and U such that L f(x) U, where L is the lower bound and U is the upper bound.

The best possible bounds for a function f, over an interval and the numbers A and B such that, for all x in the interval, A f(x) B and where A and B are as close together as possible. A is called the greatest lower bound and B is called the least upper bound.

The Seven Step Paradigm:

1.) I want to and I can2.) Define the situation3.) State the objective4.) Explore the options5.) Plan your method of attack6.) Take action7.) Look back

The Book’s Practical Tips:

1. Make sure that you know what quantity or function is to be optimized.

2. If possible, make several sketches showing how the elements that vary are related. Label your sketches clearly by assigning variables to quantities which change.

3. Try to obtain a formula for the function to be optimized in terms of the variables that you identified in the previous step. If necessary, eliminate from this formula all but one variable. Identify the domain over which this variable varies.

4. Find the critical points and evaluate the function at these points and endpoints to find the global maxima and minima.

Basic Steps in a Word Problem1. Read the problem carefully and completely. Make sure that you know exactly

what is being asked for.

2. Represent the unknown(s) exactly. (Probably the most important step.)

3. Represent all other unknowns in terms of the unknown(s) In (2.) To do this, use a chart, a diagram, a picture; anything that will help. (Read the problem over again!)

4. Look for relationship(s) that exist between known quantities and the unknowns. These relationships must be there or the problem is unworkable. (Read the problem again!) To help do this continue to fill in the chart, diagram, picture with the known values. If there is more than one unknown there will have to be more than one relationship.

One of the best ways to look for relationships in a physical problem is to sketch as accurate picture as possible and label it thoroughly.

5. Translate the relationship(s) in (4.) into algebraic statements; i.e., equations or inequalities.

6. Solve the equations or inequalities in (5.).

7. Check the answer(s) in (6.) for their validity and reasonableness In the problem.

8. Answer the original question(s) asked for!

And now for some very significant theorems!

The Extreme Value TheoremIf f is continuous on the interval [a,b], then f has a

global minimum and a global maximum on that interval.

The Mean Value TheoremIf f is continuous on [a,b] and differentiable on (a,b),

then there exists a number c, with a < c < b, such that

Local Extrema and Critical Points TheoremSuppose f is defined on an interval and has a local max-

imum or minimum at the point x = a, which is not an endpoint if the interval. If f is differentiable at x = a, then f’(a) = 0.

).)((')()()()(

)(' abcfafbfab

afbfcf

or

Constant Function TheoremSuppose that f is continuous on [a,b] and differentiable on (a,b).

If f’(x) = 0 on (a,b), then f is constant on [a,b].

Increasing Function TheoremSuppose that f is continuous on [a,b] and differentiable on (a,b).

• If f’(x) > 0 on (a,b), then f is increasing on [a,b].

• If f’(x) 0 on (a,b), then f is nondecreasing on [a,b]

The Racetrack PrincipleSuppose that g and h are continuous on [a,b] and differentiable

on (a,b), and that g’(x) h’(x) for a < x < b.

• If g(a) = h(a), then g(x) h(x) for a x b.

• If g(b) = h(b), then g(x) h(x) for a x b.