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!!!

Questions???