4.4 Homework Assignment
1. The product of two positive numbers is 100. Find the two numbers so that the sum of the numbers is as small as
possible.
2. The area of a rectangle is 81 𝑐𝑚2 . Find the length and width so that the rectangle has a minimum perimeter.
3. The perimeter of a rectangle is 80 feet. Find the length and width so that the rectangle has a maximum area.
4. Two adjacent rectangular corrals are to be made using 240 feet of fencing.
The fence must extend around the outer perimeter and across the middle
as shown in the diagram. Find the dimensions so that the total enclosed
area is as large as possible.
x
y
y
x
5. A shelter as a bus stop is to be made with three Plexiglas sides & a
Plexiglas top. If the volume of the shelter is 486 cubic feet, find the
dimensions that require the least amount of Plexiglas.
6. A box is made by cutting small squares from each corner of a piece of
square materials 12 inches on each side and then folding up the
flaps. Find the side of the square cutouts that will product the
greatest volume box.
1
2
7. A rectangle is positioned with one vertex on the line 𝑦 = − 𝑥 + 3
as shown. Find the point so that the rectangle has a maximum
area.
8. Use the graph of 𝑓 ′ shown to graph 𝑓 " and a graph of f with the
starting point (0,2).
9. Use the graph of 𝑦 = 𝑓(𝑥) at the right for these problems.
a. Find lim f  x  .
x0
b. Find lim f  x  .
x 0
c. Find lim f  x  .
x 4
d. Find lim f  x  .
x 
x
x
e. Find lim f  x  .
x3
f. List the discontinuities of f .
g. Which of these discontinuities are removable?
h. Find the absolute maximum of f (x) on  2,3.
i. Find the absolute minimum of f (x) on  2,3.
j. Find f  1 .
k. Find f  1 .
l. List all x-values where f   x  does not exist.
m. List all x-values at which f  x  has a local minimum.
n. List all x-values at which f  x  has a local maximum.
10. A rectangle is positioned with two points on the semicircle 𝑦 = √36 − 𝑥 2
as shown. Find the point (𝑥, 𝑦) so that the area of the rectangle is a maximum.
11. Find the relative extrema & points of inflection and graph f,
if 𝑓(𝑥) = −𝑥 4 + 4𝑥 3 − 16𝑥 + 2 and 𝑓 ′ (𝑥) = −4(𝑥 + 1)(𝑥 − 2)2 .
12. 𝑓(𝑥) = −𝑥 3 + 4
a. Write an equation of the tangent line shown.
b. Use this tangent line equation to approximate 𝑓(1.1).
c. What is the actual value of 𝑓(1.1)?
13. 𝑓(𝑥) = 𝑥 2 , (1,1)
a. Draw the tangent line at the point (1,1)
b. Write an equation of this tangent line.
c. Label a point on your tangent line with an x-coordinate of .9 as point A.
d. Use your equation of the tangent line to approximate 𝑓(.9) by finding
the y-coordinate of your point A.
e. Label a point B on the parabola with an x-coordinate of .9. What is the
actual value of 𝑓(.9)?
f. Use the same tangent line to approximate 𝑓(.6). How accurate is your
approximation?
14. Without using a calculator, find vertical asymptotes, relative extrema, & end behavior, & sketch of graph of 𝑓(𝑥) =
𝑥 2 +2
𝑥 2 −9
2 x3  5
.
x  10 x 3  6 x  3
15. Find lim