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Chapter Four
APPLICATIONS OF
DERIVATIVES
1. Analysis of Functions: Increase, Decrease and Concavity:
a. Increase, Decrease and Relative Extrema:
DEFINITION: Let f be a function defined on an interval I and let x1 and x2
be any two points in I.
1. If f ( x1 )  f ( x2 ) whenever x1  x2 , then f is said to be increasing on I.
2. If f ( x1 )  f ( x2 ) whenever x1  x2 , then f is said to be decreasing on I.
3. If f ( x1 )  f ( x2 ) for all x1 and x2, then f is said to be constant on I.
y
y
y
decreasing
increasing
f(x1)
x1
f(x1)
f(x2)
x2
f(x1) < f(x2) if x1 < x2
Graph has +ve slope
(1)
constant
x
f(x1)
x1
f(x2)
x2
f(x1) > f(x2) if x1 < x2
Graph has -ve slope
(2)
x
x1
f(x2)
x2
f(x1) = f(x2) if for all x1 and x2
Graph has zero slope
(3)
Theorem 1: Suppose that f is continuous on [a,b] and differentiable on (a,b).
1. If f `(x) > 0 at each point x  (a, b) , then f is increasing on [a,b].
2. If f `(x) < 0 at each point x  (a, b) , then f is decreasing on [a,b].
3. If f `(x) = 0 at each point x  (a, b) , then f is constant on [a,b].
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Note: A function f(x) has critical point like x=c if it is continues there and f`(c) =0
or f`(x) is not found.
Theorem 2: (First Derivative Test for Local Extrema)
Suppose that c is a critical point of a continuous function f, and that f is
differentiable at every point in some interval containing c except possibly
at c itself. Moving across c from left to right,
1. if f ` changes from negative to positive at c, then f has local minimum at c;
2. if f ` changes from positive to negative at c, then f has local maximum at c;
3. if f ` does not change sign at c (that is, f ` is positive on both sides of c or f `
is negative on both sides of c ), then f has no local extremum at c;
Sign of f`
a ------ c ++++ b
Sign of f`
a ++++ c -------- b
Local minimum
(1)
Sign of f`
a ------ c ------ b
Sign of f`
Local maximum
(2)
a ++++ c ++++
b
f has no local extremum
(3)
Theorem 3: (Second Derivative Test for Local Extrema)
Suppose f`` is continuous on an open interval that contains c;
1. If f `` (c) > 0, then f has local minimum at x = c.
2. If f `` (c) < 0, then f has local maximum at x = c.
3. If f `` (c) = 0, then the test fails. The function f may have a local
maximum, a local minimum, or neither.
b. Concavity and Inflection Points (I.P.):
DEFINITION: Concave up, Concave down
The graph of a differentiable function y=f(x) is:
 Concave up in an open interval I if f` is increasing in I.
 Concave down in an open interval I if f` is decreasing in I.
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Theorem 1: The second derivative test for concavity
Let y=f(x) be twice-differentiable on an interval I.
1. If f `` > 0 on I, the graph of f over I is concave up.
2. If f `` < 0 on I, the graph of f over I is concave down.
DEFINITION: Point of Inflection
A point P on a curve y=f(x) is called an inflection point (I.P.) if f is
continuous there and the curve changes from concave upward to concave
downward or from concave downward to concave upward at P.
Theorem 2: Inflection points (I.P.)
Let y=f(x) be continuous and twice-differentiable at x=c then, if f ``(c)=0, the
function f has inflection point (I.P.) at x=c.
c. Horizontal and Vertical Asymptotes:
DEFINITION:
 A line y=b is a horizontal asymptote of the graph of the function y=f(x) if:
f ( x)  b
either xlim

or
lim f ( x)  b
x
 A line x=a is a vertical asymptote of the graph of the function y=f(x) if:
either lim f ( x)   
xa 
or
lim f ( x)   
xa 
d. Oblique Asymptotes:
If the degree of the numerator of a rational function is one greater than
the degree of denominator, the graph has an oblique asymptote, that is, an
asymptote that is neither vertical nor horizontal.
Strategy for Graphing y=f(x):
1. Identify the domain of f.
2. Identify any symmetry the curve may have.
3. Find y` then find the critical points of f, and identify where the curve is
increasing and where it is decreasing.
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4. Find y`` then find the points of inflection, if any occur, and determine the
concavity of the curve.
5. Identify any asymptotes.
6. Plot key points, such as intercepts and the points found in steps 3 and 4,
and sketch the curve.
2. Related Rates of Changes:
Related Rate Problems Strategy:
1. Draw a picture and name the variables and constant. Use t for time.
Assume all variables are differentiable functions of t.
2. Write down the numerical information (in terms of symbols you have
chosen).
3. Write down what you are asked to find (usually a rate, expressed as a
derivative).
4. Write an equation that relates the variables. You may have to combine two
or more equations to get single equation that relates the variable whose rate
you want to the variables whose rates you know.
5. Differentiate with respect to t. Then express the rate you want in terms of
the rate and variables whose values you know.
6. Evaluate. Use known values to find the unknown rate.
3. Optimization:
To optimize something means to make it as useful or effective as
possible. In the mathematical models in which we use differentiable functions to
describe things that interest us, this usually means finding where some function
has its greatest or smallest value. What is the size of the most profitable
production run? What is the best shape for an oil can? What is the stiffest beam
we can cut from a 12-in. log?
In this section we show where such functions come from and how to find
their extreme values.
Critical Points and Endpoints:
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Our basic tool is the observation we made in previous section about local
maxima and minima. There we discovered that the extreme values of any
function f whatever can occur only at:
1. Interior points where f `=0,
named critical points.
2. Interior points where f ` does not exist,
3. Endpoints of the function's domain.
Strategy for Solving Max-Min Problems
1. Draw a picture. Label the parts are important in the problem.
2. Write an equation. Write an equation for the quantity whose maximum or
minimum value you want. If you can, express the quantities as a function
of single variable, say y=f(x). This may require some algebra and use of
information from the statement of the problem. Note the domain in which
the values of x are to be found.
3. Test the critical points and end points. The extreme value of f will be
found among the values f takes on at the endpoints of the domain and at
the points where f` is zero or fails to exist. List the values of f at these
points. If f has an absolute maximum or minimum on its domain, it will
appear on the list. You may have to examine the sign pattern of f ` or the
sign of f `` to decide whether a given value represents a maximum, a
minimum, or neither.
5. The Mean Value Theorem (M. V. T.):
Suppose y=f(x) is continuous on the closed
interval [a,b] and differentiable on the open interval
(a,b), Then there is at least one point c in (a,b) at which
f (b)  f (a )
 f `(c)
ba
min .
f `
f (b)  f (a)
 max . f `
ba
6
1
f (1)  2
1


2
50
1 0
5  12
1
1
 f (1)  2 
5
4
0.2  2  f (1)  0.25  2
2.2  f (1)  2.25
Corollary2: If f` (x) = 0 for all x in an interval (a, b), then
f(x)=C, for all x  (a, b), where C is a constant.
f `(x)=0
f (x)=C
Corollary3: If f`(x) =g`(x) for all in an interval (a, b), then f-g
is constant on (a, b); that is f(x) = g(x)+C, where
C is a constant.
6. L'Hopital's Rule:
Suppose that f(a) = g(a) = 0, and f'(a) and g'(a) exist, and that g'(a)  0, then:
lim
x a
f ( x)
f ' ( x) f ' ( a )
 lim

x

a
g ( x)
g ' ( x) g ' (a)
C