1 sec 4.3 curve sketching. 2 curve sketching problems given: a function y = f(x). objective: to...
TRANSCRIPT
1
Sec 4.3
Curve Sketching
2
Curve Sketching Problems
Given: A function y = f(x).
Objective: To sketch its graph.
3
Steps
(1) Find a “Frame” for the graph Domain Asymptotes – Horizontal, Vertical, Slant
(2) Find out how the graph “wiggles” Derivative – intervals of increase/decrease;
max/min Second derivative – intervals for concave
up/down; point(s) of inflection
(3) Sketch
4
Example (1)
12
x
xxfSketch
Frame:
Domain:
Asymptotes:
Starts here Ends here
Next Question: How does the graph wiggle between the two ends ?
5
11
0
Wiggle:Derivative:
2nd derivative:
Final Step: Put the wiggly graph onto the Frame.
22
2
1
1'
x
xxf
32 1
332''
x
xxxxf
xf
22 1
11
x
xx
xf
xf
01 1
33
+
++
– –
––
33
6
11 0
xf01 1
33
33
Starts here
Decreasing; Concave down
Decreasing; Concave up
Increasing; Concave up
Increasing; Concave down
Decreasing; Concave down
Decreasing; Concave up
Ends here
A “twist” : Concavity changes – a point of inflection
Graph rebounds after a dip – a
local min
A “twist” : Concavity changes – a point of inflection
Local max
A “twist” : Concavity changes – a point of inflection
7
Example (2)
12
x
xxfSketch
Frame:
Domain:
Asymptotes:
Starts here Ends here
Next Question: How does the graph wiggle within each of the three sections ?
?
?
?
?
?
?
?
?
?
?
?
8
Wiggle:Derivative:
2nd derivative:
22
2
1
1'
x
xxf
32
2
1
32''
x
xxxf
1 0 1 xf
9
Example (3)
4
92
2
x
xxfSketch
Frame:
Domain:
Asymptotes:
Starts here Ends here
Next Question: How does the graph wiggle within each of the three sections ?
?
?
?
?
?
?
?
?
?
?
?
10
Wiggle:Derivative:
2nd derivative:
22 4
10'
x
xxf
32
2
4
4310''
x
xxf
2 0 2 xf
11
Example (4)
523/2 xxxfSketch
Frame:
Domain:
Asymptotes:
Starts here
Ends here
Next Question: How does the graph wiggle between the two ends ?
?
?
?
12
Wiggle:Derivative:
2nd derivative:
13
10' 3/1 xxxf
129
10'' 3/4 xxxf
1 0 21
xf
13
Example (5)
3
22
x
xxxfSketch
Frame:
Domain:
Asymptotes:
Starts here
Ends here
Next Question: How does the graph wiggle within the two regions ?
?
?
?
?
?
?
14
Wiggle:Derivative:
2nd derivative:
23
51'
x
xxxf
33
8''
x
xf
1 3 5 xf
15
Example (6)
Sketch
Frame:
Domain:
Asymptotes:
Repeat here
Next Question: How does the graph wiggle in one of the regions ?
?
?
?
Periodicity:
x
xxf
sin1
cos
?
Repeat here
16
Wiggle:Derivative:
2nd derivative:
x
xfsin1
1'
2sin1
cos''
x
xxf
2
2
23
xf