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
    𝐏𝐫𝐨𝐛π₯𝐞𝐦 𝟏
    (a) Teddy J is a manufacturer of dish washing liquid . If his monthly demand function for 750ml
    size is q = 4000 βˆ’ 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 βˆ’ 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.
    𝐏𝐫𝐨𝐛π₯𝐞𝐦 𝟐
    (a) The sales of a book publication are expected to grow according to the function
    S = 300000(1 βˆ’ e
    βˆ’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 βˆ’ 3t + 25)
    t
    2 + 3t + 25
    The election is 10 days after he made the statement.
    (i) If the derivative S’(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 βˆ’
    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?
    𝐏𝐫𝐨𝐛π₯𝐞𝐦 πŸ‘
    (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
    , βˆ’
    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 Cramer’s Rule of the Inverse Method, determine the cost for each tree,the hourly
    labour charge, and the delivery charge.
    𝐏𝐫𝐨𝐛π₯𝐞𝐦 πŸ’
    (a)
    (i) Limxβ†’βˆž
    5x
    βˆ’3 βˆ’ 4
    2x
    βˆ’2 + 9
    (ii) Limxβ†’βˆž
    (x βˆ’ 3)
    2
    x
    2
    2
    βˆ’ 2x βˆ’ 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 βˆ’ 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 βˆ’ 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.

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