M119 Project 3 (rev 10/09)

Date due: 25 points [5 pt. Deduction for each class day late]

Objective: Interpreting first and second derivatives as rates of change

Use calculus (and algebra) to solve for extrema (i.e. maximum or minimum). Show work!! You may check with the calculus.

I. The spread of a virus can be modeled by , where

N is the number of people infected in hundreds, and t is the time in weeks.

a) What is the maximum number of people projected to be infected?

b) When will the virus be spreading most rapidly?

II. Consider a college student who works from 7 p.m. to 11 p.m. assembling

mechanical components. The number N of components assembled after t

hours is given by the function . At

what time is the student assembling components at the greatest rate? Are

you maximizing N ?? or N’??

III. Make a rough sketch of each rectangular solid. Show your work to verify that

each rectangular solid has a surface area of 150 square inches. Find the

volume of each. Show work.

a) A box with dimensions of 3 X 3 X 11

b) A box with dimensions of 5 X 5 X 5

c) A box with dimensions of 6 X 6 X 3.25

IV. Make a rough sketch of each cylinder. Show your work to verify that each

right circular cylinder has a surface area of 24p. Find the volume of each

a) r = 1, h = 11 b) r = 2, h = 4 c) r = 3, h = 1

V. A manufacturer has the following cost function for producing x toasters in one

day for .

a) What is the average cost function, , per toaster if x toasters are

produced in one day?

b) Use calculus to find the intervals where the average cost per toaster is

decreasing, the intervals where the average cost per toaster is increasing,

and the local extrema.

VI. The average number of hours of TV usage in the US from 1987 to 1994 can

be modeled by the equation

where T = 7 corresponds to 1987.

(*Give results in years, that is 8.3 is 1988.3*)

a) Find the intervals on which N is increasing and decreasing

b) Find the absolute extrema on the interval [7,14].

c) Briefly interpret your results for parts a) and b) in the context of the

problem.