APPLICATIONS OF DIFFERENTIATION
Maximum and Minimum Values
· A
function f has an absolute maximum (or global maximum) at c if f (c)
≥ f (x)
for
all x in D, where D is the domain of f . The number f (c)
is called the maximum
value of f
on D.
· Similarly,
f has an absolute
minimum at c if f (c)
≤ f (x)
for all x
in D and number
f (c)
is called the minimum value of f on D.
· The
maximum and minimum values of f are called the extreme values of f .
· A
function f
has a local maximum (or relative maximum) at c if f (c)
≥ f (x)
when x
is near c . [This means that f (c)
≥ f (x)
for all x
in some open interval containing c .]
· Similarly,
f has a local minimum at c if f (c)
≤ f (x)
when x is near c .
The Extreme Value Theorem
If
f is continuous on a closed interval[a,b], then f attains an absolute maximum value
f
(c)
and an absolute minimum value f (d) at some numbers c and d in [a,b].
Fermat’s Theorem
Critical Number
A
critical number of a function f is a number c in the domain of f such that either
f
' (c) = 0 or f
' (c) does not exist.
The Closed Interval Method
To
find the absolute maximum and minimum values of a continuous function f on a
closed
interval[a,b]:
1.
Find the values of f
at the critical numbers of f in (a,b).
2.
Find the values of f
at the endpoints of the interval.
3.
The largest of the values from Steps 1 and 2 is the absolute maximum value; the
smallest
of these values is the absolute minimum value.
How Derivatives Affect the Shape of a
Graph
Increasing/Decreasing Test
a)
If f ' (x)> 0 on an interval, then f is increasing on that interval.
b)
If f ' (x)< 0 on an interval, then f is decreasing on that interval.
The First Derivative Test
Suppose
that c is a critical number of a continuous
function f
.
a)
If f ' changes from positive to negative at c , then f has a local maximum at c .
b)
If f ' changes from negative to positive at c , then f has a local minimum at c .
c)
If f ' does not change sign at c , then f has no local maximum or minimum at c .
1. To
use the I/D test we have to know where f '
(x)>0
and where f
' (x)< 0. This depends
on
the signs of the three factors of f '
(x)
, namely, 12x
, x − 2 and x +1. We divide the
real
line into intervals whose endpoints are the critical numbers -1, 0 and 2.
2. The
sign of f
' (x) is represented on the number line.
3. We
see that f
' (x) changes from negative to positive at
-1, so f
(−1) = 0 is a local
minimum
value by the First Derivative Test. Similarly, f ' changes from negative to
positive
at 2, so f
(2) = −27 is also a local minimum value. f (0) = 5 is a local maximum
value
because f
' (x) changes from positive to negative at
0.
Definition
If
the graph of f
lies above all of its tangents on an
interval I, then it is called concave
upward on I. If the graph of f lies below all of
its tangents on I, it is called concave
downward on I.
Concavity Test
a)
If f " (x) > 0 for all x in I, then the graph of f is concave upward on I.
b)
If f " (x) < 0 for all x in I, then the graph of f is concave downward on I
Definition
A
point P on a curve y
= f (x)
is called an inflection point if f is continuous there and
the
curve changes from concave upward to concave downward or from concave
downward
to concave upward at P.
The Second Derivative Test
Suppose
f " is continuous near c .
a)
If f ' (c) = 0 and f
" (c) > 0 , then f
has a local minimum at c .
b)
If f ' (c) = 0 and f
" (c) < 0 , then f
has a local maximum at c .
c)
If f ' (c) = 0 and f
" (c) = 0 , then the test is inconclusive.
Optimization Problems
Steps
in Solving Optimization Problems
1.
Understand the problem.
2.
Draw a Diagram.
3.
Introduce Notation. Assign a symbol to the quantity that is to be maximized or
minimized
(say Q). Also select symbols (a, b, c, …, x, y) for other unknown
quantities
and label the diagram with these symbols.
4.
Express Q in terms of some of the other symbols from Step 3.
5.
Find the relationship between Q and the unknown quantities.
6.
Find the absolute maximum or minimum value of f .
Reference:
Stewart,
James. Calculus
5th edition
No comments:
Post a Comment