Section 4.7
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Transcript of Section 4.7
Section 4.7Optimization Problems
A rectangular package to be sent by a postal service can have a maximum combined length and girth (perimeter of a cross section) of 108 inches. Find the dimensions of the package of maximum volume that can be sent.
Example 8
Section 4.8Differentials
What is the equation of a line tangent to at the given point ?
This is called the tangent line approximation (or linear approximation) of at .
Tangent Line Approximations
Find the equation of the tangent line to the graph of at the given point. Use this linear approximation to complete the table.
Example 1
1.9 1.99
2 2.01
2.1
Using our tangent line approximation,
,
when is small we have that
is typically expressed as and is called the differential of .
is denoted and is called the differential of .
Differentials
Use the info to evaluate and compare and .
Example 2
Use the info to evaluate and compare and .
Example 3
Find the differential .
Example 4
Find the differential .
Example 5
Find the differential .
Example 6
Use differentials and the graph of to approximate and .
Example 7
The measurement of the radius of the end of a log is found to be 16 inches, with a possible error of ¼ inch. Use differentials to approximate the possible propagated error in computing the area of the end of the edge.
Example 8
Brief Review of 4.1-4.6
Locate the absolute extrema of the function on the closed interval.
4.1 (p. 209 #25)
Locate the absolute extrema of the function on the closed interval.
4.1 (p. 209 #27)
Determine if Rolle’s can be applied. If so, find the in such that .
4.2 (p. 216 #15)
Determine if Rolle’s can be applied. If so, find the in such that .
4.2 (p. 216 #19)
Determine if the MVT can be applied. If so, find the related -value.
4.2 (p. 217 #43)
Determine if the MVT can be applied. If so, find the related -value.
4.2
Identify the intervals where the function is increasing or decreasing and locate all relative extrema.
4.3 (p. 226 #41)
Identify the intervals where the function is increasing or decreasing and locate all relative extrema.
4.3 (p. 226 #25)
Find the points of inflection and discuss the concavity of the graph of the function.
4.4 (p. 235 #19)
Find the points of inflection and discuss the concavity of the graph of the function.
4.4 (p. 235 #27)
Find all relative extrema. Use the Second Derivative Test where applicable.
4.4 (p. 235 #47)
Find all relative extrema. Use the Second Derivative Test where applicable.
4.4 (p. 235 #55)
Find the limit.
4.5
Find the limit.
4.5
Find the limit.
4.5
4.6 (p. 256 #29) -intercepts:
-intercept:
First derivative:
Second derivative:
End behavior:Critical numbers:Inflection pts.:
𝑓 (𝑥 )=3 𝑥4+4 𝑥3
4.6 (p. 256 #6) -intercepts:-intercept:Asymptotes:
First derivative:
Second derivative:
End behavior:Critical numbers:Inflection pts.:
𝑓 (𝑥 )= 𝑥𝑥2+1
Study hard and good luck!!!
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