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Curve Sketching, Extrema & Inflection Points
No description
f(x) = 2x + 6x  5x +1
f '(x) = 6x + 12x 5
f " (x) = 12x +12
0 = 12x +12
0 = x + 1
1  1
1 = x
f(1) = 2(1) +6(1) 5(1) + 1
= 2 + 6 + 5 + 1
y = 10
= (1, 10)
How to Find Inflection Points
Local: Max/min within an interval (not including actual endpoints of the function)
Absolute: Max/min for entire function (includes actual endpoints)
To find extrema, find where f'(x)=0
Test intervals to the right and left of answers and determine whether f'(x) is greater then or less then 0
If f'(x) is less then 0, it is decreasing. If it is greater then 0, it is increasing.
Extrema
Curve Sketching
How to Find Extrema
Curve Sketching, Extrema & Inflection Points
By : Melina Hernandez, Rickelle Shaw, and Wendolyne Valdez
concave upward
Inflection Points
f(x)=1/3x 1/2x
Sketch the graph using extrema and inflection points
Overview
c
concave downward
inflection points indicate where the concavity changes
to find where f = 0
test intervals to right of answers, determine whether f (x) is > or < 0
f (x)< 0, concave down
f (x)>0, concave up
3
2
f'(x)= x x6
x=2,3
2
2
3
f''(x)=2x1
x=0.5
0.5
practical application of differential calculus
creates fairly accurate sketch of functions.
This method of picturing what the graph of a function actually looks like could potentially involve many different aspects of the graph, such as asymptotes. However, we are only going to be covering curve sketching with regards to extrema and inflection points.
Bibliography
Karassev, Alex. "Increasing and Decreasing Functions." Curve Sketching Tutorial. Nipissing University, n.d. Web. 15 May 2014.
Dawkins, Paul. "Pauls Online Notes : Calculus I  Minimum and Maximum Values." Pauls Online Notes : Calculus I  Minimum and Maximum Values. Paul Dawkins, 2003. Web. 15 May 2014.
"Lecture Notes: Relative Extrema." Hidegkuti, Powell, 2010. Web. 15 May 2014.
Husch, Lawrence S. "Visual Calculus  Drill  Concavity." Visual Calculus  Drill  Concavity. University of Tennessee, 1995. Web. 16 May 2014.
"Calculus AB: Applications of the Derivative." SparkNotes. SparkNotes, 2014. Web. 16 May 2014.
3
2
2
12
inflection point
Inflection Point
Plug in 1 into x:
3
2
f (x) = x 3x + 2 on [ 3, 2]
f '(x) = 3x 3
3(x 1) = 0
3(x1) (x+1) = 0
x= 1, x= 1
# (x1) (x+ 2) f(x)
2
0
2


+

+

+
+
+
1
1
3
max @ x= 1
min @ x= 1
3
2
2Full transcript
by
TweetWendolyne Valdez
on 27 May 2014Transcript of Curve Sketching, Extrema & Inflection Points
f(x) = 2x + 6x  5x +1
f '(x) = 6x + 12x 5
f " (x) = 12x +12
0 = 12x +12
0 = x + 1
1  1
1 = x
f(1) = 2(1) +6(1) 5(1) + 1
= 2 + 6 + 5 + 1
y = 10
= (1, 10)
How to Find Inflection Points
Local: Max/min within an interval (not including actual endpoints of the function)
Absolute: Max/min for entire function (includes actual endpoints)
To find extrema, find where f'(x)=0
Test intervals to the right and left of answers and determine whether f'(x) is greater then or less then 0
If f'(x) is less then 0, it is decreasing. If it is greater then 0, it is increasing.
Extrema
Curve Sketching
How to Find Extrema
Curve Sketching, Extrema & Inflection Points
By : Melina Hernandez, Rickelle Shaw, and Wendolyne Valdez
concave upward
Inflection Points
f(x)=1/3x 1/2x
Sketch the graph using extrema and inflection points
Overview
c
concave downward
inflection points indicate where the concavity changes
to find where f = 0
test intervals to right of answers, determine whether f (x) is > or < 0
f (x)< 0, concave down
f (x)>0, concave up
3
2
f'(x)= x x6
x=2,3
2
2
3
f''(x)=2x1
x=0.5
0.5
practical application of differential calculus
creates fairly accurate sketch of functions.
This method of picturing what the graph of a function actually looks like could potentially involve many different aspects of the graph, such as asymptotes. However, we are only going to be covering curve sketching with regards to extrema and inflection points.
Bibliography
Karassev, Alex. "Increasing and Decreasing Functions." Curve Sketching Tutorial. Nipissing University, n.d. Web. 15 May 2014.
Dawkins, Paul. "Pauls Online Notes : Calculus I  Minimum and Maximum Values." Pauls Online Notes : Calculus I  Minimum and Maximum Values. Paul Dawkins, 2003. Web. 15 May 2014.
"Lecture Notes: Relative Extrema." Hidegkuti, Powell, 2010. Web. 15 May 2014.
Husch, Lawrence S. "Visual Calculus  Drill  Concavity." Visual Calculus  Drill  Concavity. University of Tennessee, 1995. Web. 16 May 2014.
"Calculus AB: Applications of the Derivative." SparkNotes. SparkNotes, 2014. Web. 16 May 2014.
3
2
2
12
inflection point
Inflection Point
Plug in 1 into x:
3
2
f (x) = x 3x + 2 on [ 3, 2]
f '(x) = 3x 3
3(x 1) = 0
3(x1) (x+1) = 0
x= 1, x= 1
# (x1) (x+ 2) f(x)
2
0
2


+

+

+
+
+
1
1
3
max @ x= 1
min @ x= 1
3
2
2