## (Solved) Math 115 - Team Homework Assignment #6, Winter 2017 Due Date: 16 or 17 (Your instructor will tell you the exact date and time. Single Variable by...

Math 115 - Team Homework Assignment #6, Winter 2017
â€¢ Due Date: Mar. 16 or 17 (Your instructor will tell you the exact date and time.)
â€¢ This assignment covers material from Chapter 4 of the course textbook, which is the 6th edition of
Calculus: Single Variable by Hughes-Hallett, Gleason, McCallum, et al.. Make sure to understand
the material in this chapter before attempting these problems.
â€¢ It is important that you try these problems before your first meeting with your team.
Remember, the first meeting with your team is for the members of the team to discuss the
arguments used by each member of the team to find their solutions. It is also a time to address any
questions that each member of the team may have about the assignment.
â€¢ Remember to follow the guidelines from the â€œDoing Team Homeworkâ€ and â€œTeam HW Tutorialâ€
links in the sidebar of the course website.
â€¢ Do not forget to rotate roles and include a reporterâ€™s page each week.
â€¢ This is an assignment that evaluates mathematical skills and your ability to justify carefully your
solutions. 1. Let
(
6x5 âˆ’ 50x3 + 48
p(x) =
2
4eâˆ’x +7xâˆ’6 for x &lt; 1
for x â‰¥ 1. For each of the following questions, you should use calculus to find and justify your answer, and be
sure to show enough evidence that you have found all of the points/values.
(a) Find the critical points of p(x).
(b) Find and classify all local extrema of p(x).
(c) Find the inflection points of p(x).
(d) Find the x-values that give the global extrema (if they exist) on [âˆ’1, 6], and the value of the
function at each of these points.
(e) Find the x-values that give the global extrema of p(x) (if they exist) on [0, âˆž), and the value
of the function at each of these points. 1 2. Let m(w) be a function with domain (âˆ’4, 5), which is continuous at every point in its domain, and
with derivative given in the graph below.
y
4
3
y = m0 (w)
2
1
w
âˆ’4 âˆ’3 âˆ’2 âˆ’1
âˆ’1 1 2 3 4 5 âˆ’2
âˆ’3
âˆ’4
(a) Find all of the critical points of m(w).
(b) Find and classify all of the local extrema of m(w).
(c) Find all of the inflection points of m(w).
3. Duncanâ€™s person is making him a tent. She plans to make it a rectangular prism in shape, with
plastic tubing making a frame on the four vertical edges and the four edges on the top. At least one
of the sides (not the top) of the tent will be a square; let s be the length (in inches) of each side of
that square.
(a) She already has the fabric she needs, but she still needs to purchase the tubing, which costs
\$1.49 per foot. Duncan will be most comfortable in a tent with a volume of 720 cubic inches.
i. Find a formula for C(s), the cost (in dollars) of the tubing needed to build a tent when an
edge of the square side of the tent is s inches.
ii. In the context of this problem, what is the domain of C(s)?
iii. Find the dimensions that will give this volume while minimizing the cost (if such
dimensions exist), and the cost associated to those dimensions.
iv. Find the dimensions that will give this volume while maximizing the cost (if such
dimensions exist), and the cost associated to those dimensions.
(b) Later she discovers that she does have 8 feet of tubing, and decides that instead of making a
tent with a particular volume, she will make one using all of this tubing.
i. Find a formula for V (s), the volume (in cubic inches) of the tent when an edge of the
square side is s inches.
ii. In the context of this problem, what is the domain of V (s)?
iii. Find the dimensions that will use the tubing while minimizing the volume (if such
dimensions exist), and the volume associated to those dimensions.
iv. Find the dimensions that will use the tubing while maximize the volume (if such
dimensions exist), and the volume associated to those dimensions. 2 4. Mario and Luigi are each building a rain gutter. The rain gutters will have a length of 7 feet. Each
rain gutter will be constructed by bending upwards the sides of a single metal sheet as shown below.
The sheets are 7 feet long and 30 inches wide.
w
L A L
Î¸ Î¸ (a) Mario decides to build his rain gutter by bending up two sides of the sheet of length L at a
fixed angle Î¸ = Ï€4 radians.
i. What are the possible values of L in the context of this problem?
ii. Find a formula for the cross sectional area A of the gutter in terms of L.
iii. Find the value of L that gives the largest cross sectional area A. Use calculus to find and
(b) Luigi, on the other hand, decides that his rain gutter will be constructed by bending sides of
length L = 10 inches of the sheet, but he has not decided what angle Î¸ to use.
i. What are the possible values of Î¸ in the context of this problem?
ii. Find a formula for the cross sectional area A of the gutter in terms of Î¸.
iii. Find the value of Î¸ that gives the largest cross sectional area A. Use calculus to find and
(c) Use your answers in questions (a)(iii) and (b)(iii) to decide which of the rain gutters has the
larger cross sectional area. 3

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