Mathematics for Business Decisions, Part II
Homework 3
for
Math 115b, Section:
Instructor: James Barrett
Date:
by
Team:
We, the undersigned, maintain that each of us participated fully and equally in the completion of
this assignment and that the work contained herein is original. Furthermore, we acknowledge
that sanctions will be imposed jointly if any part of this work is found to violate the Student
Code of Conduct, the Code of Academic Integrity, or the policies and procedures established for
this course.
Name Typed
Signature
% of Grade
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1. For the nonlinear function, g(x) = 2x3-10x+5: (i) use the method of
difference quotients to find a formula for g’(x). (ii) Use your answer to (i) to
estimate the instantaneous rate of change of g(x) at x = 5. Show all of your
work.
Solution:
2. Let f ( x) 
5x
. (i) Use a difference quotient, with an increment of h=0.01 to
x 1
estimate f (4). (ii) Find the equation of the line that is tangent to the graph of
f ( x) 
5x
at x=4.
x 1
Solution:
3. The demand and cost functions for one model of an audio speaker are given by
D(q) = -.00006q2 + 250 and C(q) = 60,000 + 110q. (i) Use derivative rules to find
formulas for MC(q), MR(q) and MP(q). (ii) Plot the marginal cost function,
MC(q) and the marginal revenue function, MR(q) on the same set of axes. (iii)
What value of q will make MR(q) = MC(q)? (iv) Graph MP(q) and use it to
estimate the value of q that maximizes profit. (v) What is the maximum profit?
(vi) At what price should the speakers be sold in order to realize the maximum
profit?
Solution:
4. The demand and cost functions for a good are given by D(q)  0.1  q  150 and
C(q)  12,000  1300  q respectively. Use Differentiating.xls to plot MP (q ) .
Solution:
5. Refer to the demand and cost functions for audio speakers given in problem 3.
(i) Use Differentiating.xlsm to plot the marginal profit, P’ over the interval from 0
to 2100. (ii) Experiment with the Computation boxes to find a value for q (to the
nearest integer) at which P’(q) = 0.
Solution:
6. Let f ( x)  1  x / 2 . (i) Find the slope of the graph of at the point (4,f(4)), and (ii)
find an equation for the tangent line at that point. (iii) Use Excel to show both the
graph of f and the graph of the tangent line in a single plot. Plot both functions
over the interval [0,6] . (iv) Use the formulas for differentiation to explain what
happened in Part (iii).
Solution:
7. (i) Use your team’s data to plot MR, MC, and MP for its product. (ii) Prepare
computational cells, as in the sheet Marginal of Marketing Focus.xls and use
these to answer your team’s Questions 1-4 for its product. Use the same units as in
the work on the class project in Marketing Focus.xls.?
Question 1 Find the value of q that will maximize profit by
setting MR(q) = MC(q)
Question 2 Use the value of q from question 1 to find the price
that will maximize profit
Question 3 Use your revenue and cost functions to find the
maximum profit.
Solution: