L ectur e1 1| 1 Chapter 5
Applications of Differentiation
Outline 1. Extreme values
2. Curvature and inflection points 3. Curve sketching
4. Related rate
5. Indeterminate forms and L’Hopital’s rule
L ectur e1 1| 2 Extreme Values
Optimization problems:
For a given quantity
where varies over some real numbers, what are
That is find the extreme values.
L ectur e1 1| 3 EX What are the highest/lowest points of the graph
when varies over .
L ectur e1 1| 4 EX Find the max and min acceleration of a space shuttle.
L ectur e1 1| 5 EX Find the optimal branching angle so as to minimize the resistance of the blood in the blood vessels.
L ectur e1 1| 6 Consider the graph
is highest for the whole graph is lowest for the whole graph is highest on interval is lowest on interval is lowest on interval
L ectur e1 1| 7
Def Let be a function with domain and . Then is the
Absolute maximum of if
Absolute minimum of if
These values are called extreme values of the function .
L ectur e1 1| 8
Definition The number is a
Local maximum value of if
for all in some open interval about
Local minimum value of if
for all in some open interval about . These values are called local extreme values of .
L ectur e1 1| 9
L ectur e 1 1| 10 EX The function
takes on infinitely many a-max and a- min .
L ectur e 1 1| 11 EX has a-min at but has no a-max.
has no a-max, a-min, l-max, l- min.
L ectur e 1 1| 12 EX Consider
with .
a-max
a-min . It is also l-min.
l-min l-max
L ectur e 1 1| 13
Extreme value theorem
If is a continuous function and is a closed interval, then attains both a- max and a-min at some points.
IMPORTANT The theorem cannot be
applied if is discontinuous at some point or is NOT a closed interval!
L ectur e 1 1| 14
Rule
If has a local extreme at , then either
(i) does not exist, or (ii)
L ectur e 1 1| 15 Warning
That does not exist or does not guarantee to be a local extreme value of .
EX has , but is neither a l-max nor l-min.
EX , does not exist, but is neither l-max nor l-min.
L ectur e 1 1| 16
Critical numbers
A critical number of a function is in the domain of such that either
1. does not exist, or 2. .
So we find that every l-extreme of is c-number. But some c-numbers of may not be l-extreme. We will have a method to check.
L ectur e 1 1| 17 EX Find the critical numbers of
L ectur e 1 1| 18 EX Find the critical numbers of
L ectur e 1 1| 19
Closed interval method
To find the a-max/a-min of a continuous function :
1. Find at all c-numbers.
2. Find the values of .
3. The largest of the values from Steps 1 and Step 2 is a-max; the smallest is the a-min.
L ectur e 1 1| 20 EX Find the extreme values of
L ectur e 1 1| 21 EX Find the a-max/a-min of the function
