CALCULUS 1 – NOTES
NAME:
OPTIMIZATION – DAY 1
DATE:
WHAT IS OPTIMIZATION?
To optimize something means to maximize or minimize some aspect of it.
Examples:
1.)
2.)
3.)
We answer these questions by finding the greatest or smallest value of some particular function using
techniques from 4.2 and 4.3 (increasing/decreasing and maximum/minimum)
STRATEGY FOR OPTIMIZATION
Step 1:
When possible, draw a picture and illustrate the scenario. Label parts that are important.
Step 2:
Write an equation for the quantity that is to be a maximum or a minimum.
**
**
Step 3:
TAKE DERIVATIVE: If 𝑦 = 𝑓 ′ (𝑥) is the quantity to be a max or a min, find the values of
𝑑𝑦
x for which 𝑑𝑥 = 𝑓 ′ 𝑥 = 0
Step 4:
Test Critical Points: Test each value of x for which 𝑓 ′ 𝑥 = 0 to determine whether it
provides a maximum, minimum, or neither. Usually…
a)
b)
c)
also…
𝑑𝑦
𝑑𝑥
𝑖𝑠
𝑝𝑜𝑠𝑖𝑡𝑖𝑣𝑒 𝑓𝑜𝑟 𝑥 < 𝑥𝑐
𝑧𝑒𝑟𝑜 𝑓𝑜𝑟 𝑥 = 𝑥𝑐
𝑛𝑒𝑔𝑎𝑡𝑖𝑣𝑒 𝑓𝑜𝑟 𝑥 > 𝑥𝑐
𝑑𝑦
𝑑𝑥
𝑖𝑠
𝑛𝑒𝑔𝑎𝑡𝑖𝑣𝑒 𝑓𝑜𝑟 𝑥 < 𝑥𝑐
𝑧𝑒𝑟𝑜 𝑓𝑜𝑟 𝑥 = 𝑥𝑐
𝑝𝑜𝑠𝑖𝑡𝑖𝑣𝑒 𝑓𝑜𝑟 𝑥 > 𝑥𝑐
Step 5:
If the derivative fails to exist at some point, examine this point as a possible max or min.
Step 6:
If y = f(x) is defined for only a limit range of values 𝑎 ≤ 𝑥 ≤ 𝑏, examine x = a and x = b
for extreme values of y.
CALCULUS 1 – NOTES
NAME:
OPTIMIZATION – DAY 1
DATE:
1.) Find two positive numbers whose sum is 20 and whose product is as large as possible.
2.) The product of two numbers is 210 and the sum of the first plus three times the second yields a
minimum sum.
3.) The area of a rectangle is 100m2. Find the dimensions if the perimeter is to be a minimum.
4.) A farmer has 2400ft of fencing to close off a rectangular field that borders a river. What
dimensions would yield the maximum area?
5.) Fence off 3 adjoining rectangular pens with equal areas. There is 500 feet of fencing. Find the
dimensions of the entire enclosure to maximize the area.
6.) A poster is to contain a printed area of 150in2, with clear margins of 3 inches on the top and
bottom and 2 inches on each side. What overall dimensions would minimize the paper used for the
poster?
OPTIMIZATION- DAY 1 - HOMEWORK
Numbers, Rectangles: Choose 6 Problems!
1. The product of two numbers is 16. Find the numbers if the sum of the one plus the square of the other is a minimum.
2. Find two numbers that add to 100 such that the product of the cube of one times the square of the other is a maximum.
3. The product of two positive numbers is 16. Find the numbers so that the sum is a minimum.
4. Find two numbers that add to 120 such that the product of one times the square of the other is a maximum.
5. The sum of two numbers is 72. Find the numbers if the product of one times the cube of the other is a maximum.
6. Find two positive numbers whose sum is 110 and whose product is a maximum.
7. Find two positive numbers whose product is 192 and whose sum is a minimum.
8. Find two positive numbers whose product is 192 and the sum of the first plus three times the second is a minimum.
9. Find the length and width of a rectangle of maximum area if the perimeter is 100 feet.
10. Find the length and the width of a rectangle of minimum perimeter if the area is 64 square feet.
11. What should be the length and width of a rectangular field that is to be enclosed with 20 miles of fencing if the area is
to be a maximum?
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Boxes, Pastures, and Posters (CHOOSE 6 PROBLEMS)
1. A box is made from a sheet of cardboard 8 in by 15 in by cutting squares from each corner and folding up the sides.
What are the dimensions of the box of greatest volume?
2. A sheet of cardboard 10 in on a side is to be made into a box by cutting squares from each corner and folding up the
sides. What are the dimensions of the box of greatest volume?
3. A box is to be made from a square piece of material 12 in on a side by cutting squares from each corner and folding up
the sides. What is the volume of the largest box that can be made?
4. An open box is made from a 16 in by 30 in piece of cardboard by cutting squares from each corner and folding up the
sides. What should the size of the squares be to obtain a box of maximum volume?
5. A square is cut of each corner of a piece of cardboard 20 in on a side and turned up to make a box. What should the
size of the squares be so that the volume of the box is a maximum?
6. A farmer has 300 ft of fencing with which he wants to use to enclose a rectangular pasture next to a barn. No fence is
needed along the barn. What are the dimensions of the pasture of maximum area?
7. A farmer has 600 m of fencing with which he wants to use to enclose a rectangular pasture adjacent to a wall. There is
no fence needed along the wall. What are the dimensions of the pasture if the area is a maximum?
8. A dairy farmer plans to fence in a rectangular pasture adjacent to a river. The pasture must contain 180,000 square
meters in order to provide enough grass for the herd. What dimensions would require the least amount of fencing if no
fencing is needed along the river?
9. A rancher has 200 feet of fencing with which to enclose two adjacent corrals. What dimensions should be used so that
the enclosed area will be a maximum?
10. The Philadelphia Zoo has 500 meters of top quality wood fencing to build 8 “rooms” for the gorillas, by first
constructing a fence around a rectangular region. This region would then be partitioned into 8 smaller areas by placing 7
fences parallel to one of the sides. Find the dimensions that will maximize the total area.
11. What is the smallest size for a sheet of paper that is to contain 20 square inches of printed material and have 2 inch
margins at the top and bottom and 1 inch margins on the sides?
12. A page has an area of 30 square inches. What are the dimensions of the printed area if it is to have margins of 1.5
inches on the top and bottom and 2 inches on the sides and the area is a maximum?
13. A page of a book contains 24 square inches of print and has 1.5 inch margins on the top and bottom and 1 inch
margins on the sides. Find the dimensions of the page so that its area is a minimum?
14. A sheet of paper for a poster is 30 square inches. The margins on the top and bottom are 2 inches and on the sides are
1 inch. What are the dimensions of the printed area if it is to be a maximum?