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Transcript 
Wednesday, March 29, 2017
MAT 145
Wednesday, March 29, 2017
MAT 145
Wednesday, March 29, 2017
MAT 145
Wednesday, March 29, 2017
MAT 145
f (x) = x
3
Wednesday, March 29, 2017
g(x) = x -1
MAT 145
x -3
h(x) =
x+2
To determine the absolute maximum and absolute
minimum values of a continuous function f on a closed
interval [a,b], carry out these steps.
(1) Determine all critical numbers of the function f
on a < x < b.
(2) Determine the value of the function f at each
critical number.
(3) Determine the value of f at each endpoint of the
closed interval [a,b].
(4) Now compare outputs: The largest of the values
calculated in steps (2) and (3) is the absolute
maximum value; the smallest of these values in the
absolute minimum.
Wednesday, March 29, 2017
MAT 145
Wednesday, March 29, 2017
MAT 145
Wednesday, March 29, 2017
MAT 145
h
m
Wednesday, March 29, 2017
MAT 145
a
t
Wednesday, March 29, 2017
MAT 145
Wednesday, March 29, 2017
MAT 145
Concavity Animations
More Concavity Animations
Wednesday, March 29, 2017
MAT 145
Concavity Animations
More Concavity Animations
Wednesday, March 29, 2017
MAT 145
Wednesday, March 29, 2017
MAT 145
To determine the points of inflection for a function y = f(x),
carry out these steps.
(1) From the original function y = f(x), calculate the
function’s second derivative, y = f ’’(x).
(2) Determine every x-axis location (a) at which f ’’ = 0 or
(b) at which f ’’ is undefined yet f is defined.
(3) Now, look for a sign change in f ’’ at each point you
determined in Step (2).
There is a point of inflection at every x-axis location
at which both Step (2) and Step (3) are true.
Wednesday, March 29, 2017
MAT 145
Wednesday, March 29, 2017
MAT 145
Here’s a graph of
g’(x). Determine all
intervals over which g
is increasing and over
which g is decreasing.
Identify and justify
where all local
extremes occur.
Wednesday, March 29, 2017
MAT 145
Here’s a graph of
h”(x). Determine all
intervals over which h
is concave up and over
which h is concave
down. Identify and
justify where all points
of inflection occur.
Wednesday, March 29, 2017
MAT 145
For f(x) shown below, use calculus to determine and justify:
• All x-axis intervals for which f is increasing
• All x-axis intervals for which f is decreasing
• The location and value of every local & absolute extreme
• All x-axis intervals for which f is concave up
• All x-axis intervals for which f is concave down
• The location of every point of inflection.
f (x) = 5- 3x + x
2
Wednesday, March 29, 2017
MAT 145
3
Wednesday, March 29, 2017
MAT 145
f (x) = 7 - 2 x
g(x) = 3e
Wednesday, March 29, 2017
-1£ x < 2
x
h(x) = 4 - x
1£ x £ 4
2
MAT 145
-3 < x < 2
Wednesday, March 29, 2017
MAT 145
f (x) = x + 3x -144x
3
g(x) = x
4
5
2
( x - 4)
2
h(t) = 4t - 3 , - 2 £ x £ 3
Wednesday, March 29, 2017
MAT 145
f (x) = x 3 + 3x 2 -144x
4
5
(
)
critical numbers : -8,6
8
g(x) = x x - 4
critical numbers : 0, ,6
7
3
h(t) = 4t - 3 , - 2 £ x £ 3 critical number :
4
Wednesday, March 29, 2017
2
MAT 145
f (x) = x 16 - x
-1£ x £ 4
2
g(x) = 4x - 6x - 72x + 4
[-3,4]
x
h(x) = 2
x - x +1
[0,3]
3
Wednesday, March 29, 2017
2
MAT 145
f (x) = x 16 - x
2
g(x) = 4x - 6x - 72x + 4
3
2
x
h(x) = 2
x - x +1
Wednesday, March 29, 2017
MAT 145
x + 3x - 4
f (x) =
-3 £ x £ 1
x+2
3
Wednesday, March 29, 2017
2
MAT 145
Wednesday, March 29, 2017
MAT 145
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