Section 4.5Limits “at Infinity”

Deal with the end behavior of a function.

Limits “at Infinity”

Remember these from MTH 112?

Horizontal Asymptotes

What happens in the sequences below?

What is equal to?

Critical Info

Find the limits.Additional Example 3

Section 4.6A Summary of Curve Sketching

Additional Example

-intercepts:

-intercept:

First derivative:

Second derivative:

End behavior:Critical numbers:Inflection pts.:

𝑓 (𝑥 )=𝑥2−𝑥3

Section 4.7Optimization Problems

An open box of maximum volume is to be made from a square piece of material, 24 inches on a side, by cutting equal squares from the corners and turning up the sides. What is the maximum volume of the box?

Example 1

Find two positive numbers such that the sum of the first number and twice the second number is 108 and the product is a maximum.

Example 2

Find the length and width of a rectangle that has a perimeter of 80 meters and a maximum area.

Example 3

Find the length and width of a rectangle that has an area of 32 square feet and a minimum perimeter.

Example 4

Which points on the graph of are closest to the point ?

Example 5

A farmer plans to fence in a rectangular pasture adjacent to a river. The pasture must contain 245,000 sq. meters in order to provide enough grass for the herd. What dimensions will require the least amount of fencing if no fencing is needed along the river?

Example 6

A Norman window is constructed by adjoining a semicircle to the top of an ordinary rectangular window. Find the dimensions of a Norman window of maximum area if the total perimeter is 16 feet.

Example 7

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