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,
f ( x)  f (a)  f ' (a)( x  a).
We are thinking of a as fixed, so f(a) and f’(a)
are constant!
The expression f ( x )  f (a )  f ' (a )( x  a ) is a
linear function which approximates f(x)
well near a. It is called the local linearization of f near x = a.
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:
f ( x)  f (a)  f ' (a)( x  a).
 The expression f (a )  f ' (a )( x  a ) 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:
E ( x)  f ( x)  f (a)  f ' (a)( x  a).
and
lim
xa
E ( x)
 0.
xa
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. lim f ( x )   and lim g( x )  
x a
x a
f (x)
f ' (x)
 lim
then: lim
x a g ( x )
x a g ' ( x )
or if a = 
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 less
than 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.
Example
of
First &
Second
Derivative
Tests
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 Function
 A 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.)
2.)
3.)
4.)
5.)
6.)
7.)
I want to and I can
Define the situation
State the objective
Explore the options
Plan your method of attack
Take action
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 Problem
1. 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 Theorem
If f is continuous on the interval [a,b], then f has a
global minimum and a global maximum on that
interval.
 The Mean Value Theorem
If f is continuous on [a,b] and differentiable on (a,b),
then there exists a number c, with a < c < b, such that
f ' (c ) 
f (b)  f (a)
or f (b)  f (a)  f ' (c)(b  a).
ba
 Local Extrema and Critical Points Theorem
Suppose f is defined on an interval and has a local maximum 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.
 Constant Function Theorem
Suppose 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 Theorem
Suppose 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 Principle
Suppose 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.