Transcript Document

Sketching the Graph of a Function Using Its 1st and 2nd Derivatives
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What does the 1st derivative of a function tell us about the graph of the function?
It tells us __________________________________________________________
What does the 2nd derivative of a function tell us about the graph of the function?
It tells us __________________________________________________________
Therefore, to sketch the graph of a function, f(x), we should,
For where the function f(x) is increasing/decreasing and attains its local extrema:
1. Find the 1st derivative, f (x).
2. Find the x-values such that f (x) = 0, and sometimes, also find the x-values such that f (x) doesn’t exist.*
3. Determine the intervals for which f is increasing and decreasing, determine the locations of its local extrema, if any.
For where the function f(x) is concave up/concave down and attains its inflection points:
1. Find the 2nd derivative, f (x).
2. Find the x-values such that f (x) = 0, and sometimes, also find the x-values such that f (x) doesn’t exist.
3. Determine the intervals for which f is _______________________________________________________________
Example: f(x) = 1/3 x3 + x2 – 8x + 5
*Definition:
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) doesn’t exist.
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Intervals
( , )
( , )
( , )
Intervals
x
x
f (x)
f (x)
Inc/Dec?
CU/CD?
Graph
Graph
( , )
( , )
( , )
15
10
5
-4
-2
2
-5
-10
-15
-20
4
6
Sketching the Graph of a Function (cont’d)
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If a function is not a constant function, then it will increase and/or _____________. If it is not a linear function, it will be
concave up and/or __________. If so, the graph of the function can only consist of one or more of the following 4 pieces:
Inc and CU
Dec and ____
____ and ____
Dec and CD
Example: f(x) = x4 – x2 – 2x – 1 (Note: The only critical number from f (x) is x = 1.)
Intervals
(
x
f (x)
Inc/Dec?
4
f (x)
3
2
CU/CD?
1
Graph
-3
-2
-1
1
2
3
-1
-2
f at Key Numbers
-3
-4
Max/Min/Inf
)
(
)
(
)
(
)
(
)
General Techniques/Considerations When Sketching a Function
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When sketching the graph of a function, f(x), besides considering increasing/decreasing and concavity (i.e., concave
up/concave down), we also need to considering following:
A. Domain: determine all possible values of x
B. Intercepts: y-intercept (by plug __ into f(x)) and x-intercept(s) (by setting f(x) = __ and solve for x)*
C. Symmetry: determine whether it is symmetric with respect to (wrt) the ______ or wrt the ______ (see below.)
D. Asymptotes: determine whether there is any _______ and/or ______ asymptotes (see below.)
* Only when the x-intercepts are manageable to find.
Of course, for where f(x) is increasing/decreasing (incl. local extrema) and concavity (incl. inflection points), we have to
find the following:
E. Intervals of Increase/Decrease: Use the I/D Test: f (x) > 0  increasing and _________  __________
F. Local Maximum/Minimum: Find the x-values where f (x) = 0 or f (x) doesn’t exist. f will likely have local extrema
at these x-values (but not a must).
G. Concavity and Inflection Points: Use the Concavity Test: f (x) > 0  concave up and ______  _____. Find the
x-values where f (x) = 0 or f (x) doesn’t exist. f will likely have inflection points at these x-values (but not a must).
When you have all these components,
H. Sketch the function.
C. Symmetry
D. Asymptotes
Sketch a Function Using A-H (from the Previous Page)
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Ex 1. f(x) = (x2 – 4)/(x2 + 1)
A. Domain:
B. Intercepts:
C. Symmetry:
D. Asymptotes:
E. Intervals of Increase/Decrease:
Intervals
F. Local Maximum/Minimum:
(
x
G. Concavity and Inflection Points:
f (x)
H. Sketch the function
Inc/Dec?
f (x)
5
4
CU/CD?
3
Graph
2
1
f at Key Numbers
-5
-4
-3
-2
-1
1
-1
-2
-3
-4
-5
2
3
4
5
Max/Min/Inf/VA
)
(
)
(
)
(
)
(
)
Sketch a Function Using A-H
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Ex 1. f(x) = x/(x2 – 4)
A. Domain:
B. Intercepts:
C. Symmetry:
D. Asymptotes:
E. Intervals of Increase/Decrease:
F. Local Maximum/Minimum:
G. Concavity and Inflection Points:
H. Sketch the function
Intervals
5
x
4
f (x)
(
3
Inc/Dec?
2
f (x)
1
-5
-4
-3
-2
-1
1
-1
2
3
4
5
CU/CD?
Graph
-2
-3
-4
-5
f at Key Numbers
Max/Min/Inf/VA
)
(
)
(
)
(
)
(
)