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:

- Apply the knowledge of functions to problems involving, supply, demand,

production, revenue and cost. - Identify the appropriate functions, equations and sequences which are to be used

in problem solving in the Social Sciences. - Use solutions to linear, quadratic, exponential and logarithmic equations to

determine market equilibrium price and quantity. - Solve problems involving rates of change and marginal change by the use of

derivatives - 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. - Find and classify extreme points of a function for the purpose of identifying what

represents a minimum, a maximum or a point of inflexion. - Determine continuity or discontinuity of a function throughout its domain, since

some functions are not defined for all real numbers. - Solve a system of simultaneous equations with 3 variables using matrix inversion

and the Cramer’s Rule. - Compute and interpret the value of the derivative of a function

ππ«π¨ππ₯ππ¦ 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

β²

s total revenue curve.

(ii) Derive an expression, Ξ (q) for Teddy J

β²

s profit function.

(iii) Determine whether Teddy Jβ²s 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 firmβ²

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.

ππ«π¨ππ₯ππ¦ π

(a) The sales of a book publication are expected to grow according to the function

S = 300000(1 β e

β0.06t), 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 Laraβ²

s bats.

(ii) Is demand inelastic or elastic?

ππ«π¨ππ₯ππ¦ 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 Cramerβs Rule of the Inverse Method, determine the cost for each tree,the hourly

labour charge, and the delivery charge.

ππ«π¨ππ₯ππ¦ π

(a)

(i) Limxββ

5x

β3 β 4

2x

β2 + 9

(ii) Limxββ

(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.