Chapter 4 – Applications of
Differentiation
4.1 – Maximum and Minimum Values
Optimization – the ‘best’ way
Absolute maximum/minimum: c is an absolute maximum if f (c)  f ( x)  x  D
Local maximum/minimum: c is a local maximum if f (c)  f ( x)  x some open
interval around 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: If f has a local minimum (or maximum) at c , and if f '(c ) exists,
then f '(c)  0 .
(Proof on page 282)
 NOTE: Just because f '(c)  0 does NOT mean that a local minimum or maximum
occurs at c . Examples?
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.
 If f has a local maximum or minimum at c , then c is a critical number of f .
The closed interval method: To find the absolute max and min values of a continuous
function f on a closed interval [a, b] :
a) Find the values of f at the critical number of f in (a,b)
b) Find the values of f at the endpoints of the interval.
c) The largest value from 1 and 2 is the max, the smallest is the min.
4.2 – The Mean Value Theorem
Rolle’s Theorem: Let f be a function that satisfies the following three hypotheses:
a) f is continuous on the closed interval [a, b]
b) f is differentiable on the open interval (a, b)
c) f (a )  f (b)
Then there is a number c  (a, b) such that f '(c)  0 .
What does this look like?
 Prove that the equation x3  x  1  0 has exactly one real root.
The Mean Value Theorem: Let f be a function that satisfies the following hypotheses:
a) f is continuous on the closed interval [a, b]
b) f is differentiable on the open interval (a, b)
Then there is a number c  (a, b) such that
f (b)  f (a)
f '(c) 
ba
or, equivalently,
f (b)  f (a)  f '(c) b  a 
What does this look like? (Proof on page 292)
If an object moves in a straight line with the position function s  f (t ) , then the average
velocity between t  a and t  b is
and the velocity at t  c is f '(c ) .
f (b)  f (a )
ba
Thus, the MVT tells us that at some time c between a and b , the instantaneous velocity
f '(c ) is equal to the average velocity.
 In a two hour, 120 mile car ride, the car had to have gone 60mph at least once.
Thm: If f '( x)  0 for all x  (a, b) , the f is constant on (a, b)
Thm: If f '( x)  g '( x) x  (a, b) , then f  g is constant on (a, b) : that is
f ( x)  g ( x)  c where c is constant.
 If one person starts a race in front of another racer and they both travel with the same
velocity at every moment, the distance between them remains constant for the entire race.
4.3 – How Derivatives Affect the Shape of a Graph
Increasing/Decreasing:
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.
Example: Find where the function f ( x)  3x 4  4 x3  12 x 2  5 is increasing and where is
it decreasing. (Make a chart, remember critical points)
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 max at c .
b) If f ' changes from negative to positive at c , then f has a local min at c .
c) If f ' does not change sign at c (stays positive or stays negative), then f ' has no
local max or local min at c .
If the graph of f lies above all of tangents on an interval I, then it is concave upward.
If the graph of f lies below all of tangents on an interval I, then it is concave downward.
What does this look like?
Concavity Test:
a) If f ''( x)  0 x  I , then the graph of f is concave upward on I.
b) If f ''( x)  0 x  I , then the graph of f is concave downward on I.
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.
Smiles and frowns!
Example: Discuss y  x 4  4 x3 WRT concavity, points of inflection, and local minima
and maxima.
Do:
1) Pages 286 – 287; #5, 11, 21, 27, 51, 57, 62
2) Pages 295 – 296; #2, 7, 14, 17, 34
3) Pages 304 – 306; #1, 8, 21, 38, 42, 64
4.4 – Indeterminate Forms and L’Hospital’s Rule
ln x
?
x 1
a) Can’t do the limits of the numerator and denominator separately. Why?
b) No common terms to cancel
What happens when we try to evaluate lim
x 1
 The limit may or may not exist and is called an indeterminate form of type  / 
There is also an indeterminate form of type 0/ 0 .
Ex: lim
x 0
x
x2
L’Hospital’s rule:
Suppose f and g are differentiable and g '( x)  0 near a (except possibly at a ). Suppose
that:
lim f ( x)  0 and
lim g ( x)  0
OR
lim f ( x)   and
lim g ( x)  
xa
xa
x a
x a
(In other words we have an indeterminate form of type 0/ 0 or  /  .) Then:
lim
x a
f ( x)
f '( x)
 lim
g ( x) xa g '( x)
If the limit on the right side exists (or is  ).
So, what about lim
x 1
ex
x  x 2
Ex: lim
ln x
?
x 1
ln x
x  3 x
Ex: lim
tan x  x
x 0
x3
Ex: lim
Ex: lim
x 
sin x
1  cos x
(What’s the issue here?)
What about indeterminate products? Turn them into quotients!
Ex: lim  x ln x 
(Can rewrite two ways?)
x0
Same idea for differences…
Ex:
lim
x  / 2

sec x  tan x 
Indeterminate powers: 00 , 0 ,1
a) take the natural logarithm
b) write the function as an exponential
If y  lim  f ( x)
g ( x)
then
ln y  g ( x) ln f ( x)
x a
 f ( x)
g ( x)
 e g ( x )ln f ( x )
Ex: lim 1  sin 4 x 
cot x
x 0
(Why?)
(Use a)
Do: Page 313; #1, 5 – 16, 22
4.5 – Summary of curve sketching
Guidelines for sketching a curve, pages 317 – 318
4.6 – Graphing with calculus and calculators
Improving upon your graphs from 4.5
4.7 – Optimization problems
These are great! 
1) Understand
2) Picture
3) Notation
4) Equation
5) Simplify
6) Calculus
UPN ESC
Page 337; #9, 19
4.8 – Applications to Business
Skipping – might come back to it
4.9 – Newton’s Method
Cool, helpful, kinda annoying
y  f ( x1 )  f '( x1 )  x  x1 
x2 is the x intercept ( x2 ,0) and is on that line, so:
0  f ( x1 )  f '( x1 )  x2  x1 
 f ( x1 )
 x2  x1
f '( x1 )
f ( x1 )
f '( x1 )
Keep on doing the same thing and you end up with….
x2  x1 
xn 1  xn 
f ( xn )
f '( xn )
lim xn  r
n 
Ex: Starting with x1  2 find the third approximation x3 to the root of the equation
x3  2 x  5  0
Ex: Use Newton’s method to find
6
2 correct to 8 decimal places.
Note: Newton’s method doesn’t always work – can go ‘outwards’ or ‘get stuck’
Examples?
4.10 – Antiderivatives
A physicist who knows the velocity of a particle might wish to know its position at a
given time.
An engineer who can measure the variable rate at which water is leaking from a tank
wants to know the amount leaked over a certain time period.
A biologist who knows the rate at which a bacteria population is increasing might want to
deduce what the size of the population will be at some future time.
In each case we’re looking for a function F whose derivative is a known function f .
A function F is called an antiderivative of f on an interval I if F '( x)  f ( x) for all
xI
What is the antiderivative of f ( x)  x 2 ? What about g ( x)  x 4 ? Are they unique?
If F is an antiderivative of f on an open interval I , then the most general
antiderivative of f on I is F ( x)  C where C is an arbitrary constant.
Hungh?
So how many antiderivatives might a function have?
Ex: Find the most general antiderivative of:
a) f ( x)  sin x
b) f ( x)  1/ x (careful)
c) f ( x)  x n n  1
d) f ( x) 
2 x5  x
x
Ex: Find f if f ''( x)  12 x 2  6 x  4 , f (0)  4 and f (1)  1