## Assignment Description and Instructions

Graded Individual Assignment (40%)
Due Date: (16th December 2020 2355 hrs ECT)
Assignment Description and Instructions
This is an Individual Assignment. It consists of structured-response problems. This Assignment
must be type-written in WORD and uploaded to the Dropbox as a WORD file. No hand-written
assignments will be accepted.
Answers ALL questions below, showing all working to support your answers
Relevant Course Objectives:

1. Apply the knowledge of functions to problems involving, supply, demand,
production, revenue and cost.
2. Identify the appropriate functions, equations and sequences which are to be used
in problem solving in the Social Sciences.
3. Use solutions to linear, quadratic, exponential and logarithmic equations to
determine market equilibrium price and quantity.
4. Solve problems involving rates of change and marginal change by the use of
derivatives
5. Write a linear system of equations in matrix form as a simple way to represent
multiple linear equations before solving them using a matrix approach.
6. Find and classify extreme points of a function for the purpose of identifying what
represents a minimum, a maximum or a point of inflexion.
7. Determine continuity or discontinuity of a function throughout its domain, since
some functions are not defined for all real numbers.
8. Solve a system of simultaneous equations with 3 variables using matrix inversion
and the Cramer’s Rule.
9. Compute and interpret the value of the derivative of a function
ππ«π¨ππ₯ππ¦ π
(a) Teddy J is a manufacturer of dish washing liquid . If his monthly demand function for 750ml
size is q = 4000 β 250p and his total cost function is C(q) = 500 + 0.2q.
(i) Derive an expression, R(q) for Teddy J
β²
s total revenue curve.
(ii) Derive an expression, Ξ (q) for Teddy J
β²
s profit function.
(iii) Determine whether Teddy Jβ²s profit is increasing or decreasing when
he produces 5 hundred, 750ml bottles of dish washing liquid.
(iv) How many 750ml bottles of dish washing liquid should Teddy J produce
per month if he wishes to maximize his profits.
(b) A firm has an average cost function
A(q) =
125
q
+
q
2
16 β 4.
where q is the firmβ²
s output.
(i) Determine the level of output for average costs are minimum.
(ii) Hence determine the range of values for which average costs are decreasing.
(iii) What part of the decreasing range is practically feasible?
(iv) Write an equation for the total cost function.
(v) Hence calculate the level of output for which total costs are minimum.
ππ«π¨ππ₯ππ¦ π
(a) The sales of a book publication are expected to grow according to the function
S = 300000(1 β e
β0.06t), where t is the time, given in days.
(i) Show using differentiation that the sales never attains an exact maximum value.
(ii) What is the limiting value approached by the sales function?
(b) A poll commissioned by a politician estimates that t days after he makes a statement
denegrating women,the percentage of his constituency (those who support him at the time he
made the statement) that still supports him is given by S(t) =
75(t
2 β 3t + 25)
t
2 + 3t + 25
The election is 10 days after he made the statement.
(i) If the derivative Sβ(t) may be thought of as an approval rate, derivate the a function
for his approval rate.
(ii) When was his support at its lowest level?
(iii) What was his minimum support level?
(iv) Was the approval rate positive or negative on the date of the election?
(c) Lara offers 100 autograph bats. If each is priced at p dollars, it is that the demand curve
for the bast will be p = 250 β
q
4
. If price elasticily is E(p) =
dq
q
Γ·
dp
p
.
When |E(p)| < 1, demand is inelastic and when |E(p)| > 1, demand is elastic.
(i) Find the price elasticity of demand for Laraβ²
s bats.
(ii) Is demand inelastic or elastic?
ππ«π¨ππ₯ππ¦ π
(a) A town has a population of 5000 persons, but is expected to grow by 2% every year.
(i) What wound be the population size in 7 years?
(ii) Find the sum of the first eight terms of the sequence
1
8
, β
1
4
,
1
2
, . . ..
(b) A landscape contractor is hired to cultivate ornamental plants in three new residential
developments. The contractor charges the developer for each tree cultivated, an hourly rate
to cultivate the ornamental plants, and a fixed delivery charge. In one development it took
211 labour hours to cultivate 244 ornamental plants for a cost of \$9394. In a second development
it took 128 labour hours to cultivate 283 ornamental plants for a cost of \$8270. In the final
development it took 165 labour hours to cultivate 386 ornamental plants for a cost of \$10938.
In total 504 labour hours were taken and 913 ornamental plants were cultivated.
Using Cramerβs Rule of the Inverse Method, determine the cost for each tree,the hourly
labour charge, and the delivery charge.
ππ«π¨ππ₯ππ¦ π
(a)
(i) Limxββ
5x
β3 β 4
2x
β2 + 9
(ii) Limxββ
(x β 3)
2
x
2
2
β 2x β 3
(b) During a nationwide program to immunize the population against a new strain of the flu,
public health officials determined that the cost of inoculating x% of the susceptible population
would be approximately C(x) =
1.85x
100 β x
million dollars.
(i) What would it cost to providing immunization to the first 20% of the susceptible
population?
(ii) What would it cost to providing immunization to the next 30% of the susceptible
population?
(iii) Suppose 17 million dollars are available for providing immunization. What percentage
of the susceptible population will not receive immunization?
(iv) If money was not a problem will they be able to providing immunization to the entire
susceptible population?
(c) Determine the values of x for which the function f(x) = {
2x
2 β 4 x < 2 x + 2 2 < x > 5
7 x β₯ 5
is discontinuous.
END OF ASSIGNMENT
Assignment Rubric
Criteria Excellent (9-10) Good (6-8) Satisfactory (3-
5)
Poor (0-2)
Understanding Demonstrates a
solid
understanding of
a major
approach to the
problem with
indications of
alternative
approaches, or
with sufficient
details to show
ease in
understanding.
Demonstrates a
solid
understanding
of a major
approach to the
problem. Major
concepts are
understood.
Demonstrates
an
understanding
of some major
concepts, but
misses others.
Misses
fundamental
concepts
underlying
problem.
Strategies,
Reasoning &
Procedures
Shows clear
evidence of plan
for solving
problem and all
strategies and
procedures are
clearly
Shows a plan
for solving the
problem is
clearly
understood and
main
procedures and
Can manage
common
strategies or
procedures for
solving problem
with some
minor
Does not
know common
strategies or
procedures for
solving
problem.
Reasoning is
understood.
Errors are
minimal, if
present.
Reasoning is
clear and correct
in details as well
as in main
aspects.
strategies are in
place.
Reasoning is
essentially
correct, except
for minor
aspects.
adjustments.
Reasoning
shows a
possible
approach to the
problem. Work
could lead to a
correct solution,
but is not there
yet.
muddled or
otherwise
incorrect.
Work cannot
lead to a
correct
solution.
Communication Explanation lays
out problem
solution clearly
and completely.
More than one
solution is
indicated, or
detail of
solution shows
deep
understanding.
Explanation is
clear and all
major steps are
present. Some
details may be
missed or some
language may
not be
completely
precise.
Explanation
shows some of
the steps
undertaken.
Needs help to
give full
explanation.
Explanation is
very sketchy
and/or shows
confusion or
cannot be
clarified.
Problem Solving Student arrives
at the correct
answer.
Student arrives
at a mostly
correct answer.
Student arrives
at a mostly
incorrect
answer.
Student
arrives at a
totally
incorrect
answer.