THE COUNCIL OF COMMUNITY COLLEGES OF JAMAICA

COURSE NAME: Business Calculus

COURSE CODE: MATH23OI

CREDITS: 4

CONTACT HOURS: 60(60 hours theory)

PRE-REQUISITE(S): Pre-Calculus (MATH 1201)

CO-REQUISITE(S): None

SEIESTER:

COURSE DESCRIPTION:

This course seeks to establish the fundamental principles of calculus. k develops techniques for

students Lo apply in solving various business and economic problems. It promotes critical thinking

skills towards improving the decision making process.

COURSE OUTCOMES:

Upon successful completion of this course, students should:

I. understand the concepts of limits and continuity

2. apply concepts to determine limits and continuity

3. demonstrate competence in applying the rules of differentiation and integration

4. apply the concepts of (liffcrcntiation and integration w solving business and economic

problems

5. introduce multi variable functions and partial differentiation to solve problems and obtain

optimal values as required

6. understand double integration and apply in evaluating polynomial function in two

variables

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UNIT 1 - Limits (6 hours)

Learner Outcomes:

Upon successful completion of this unit, students should he able to:

1. define the limit of a polynomial, rational and piecewise functions

2. determine if the limit of polynomial, rational and piecewise functions exists at a given point

in its domain

3. evaluate the limits of polynomial. rational and piecewise functions at a given point in its

domain

4. calculate the limit of a rational function that results in an indeterminate form or undefined

5. evaluate the limits of polynomial and rational functions as the indcpcndcnt variable

approaches + co and/Or — co

Content:

1. Definition of limits of algebraic functions

a. Polynomial

b. Rational

e. Piecewise

2. Determination of the existence of limits of algebraic functions

a. Polynomial

b. Rational

c. Piecewise

3. Evaluation of limits of algebraic functions

a. Polynomial

b. Rational

e. Piecewise

4. Calculation of the limits of rational functions resulting in the indeterminate forms

()

a.

C)

00

b. —

00

5. Evaluation of the limits of Rational functions:

a.

b.

e. Undefined

6. Evaluation of limits using algebraic techniques (without using L’Hopital’s Rule.

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UNIT II - Continuity (6 hours)

Learner Outcomes:

Upon successful completion of this unit, students should be able lo:

1. state the conditions for continuity of a function at a given point

2. determine ifa rational or compound (piecewise) function is continuous or discontinuous at

given point(s)

3. identify point(s) of discontinuity in a graphical representation ola function

4. dctcrminc points of discontinuity

Content:

I. Statement of conditions for continuity

a. f(a) exits

h. Iimf(x)= limf(x) = Iimf(x)exists

C. f(a)=limf(X)

2. Determination of point(s) of discontinuity of rational or compound piecewise functions

3. Identification of point(s) of discontinuity on graphs

4. Computation of point(s) of discontinuity of rational and compound (piecewise) functions

UNIT III — Average Rate of Change (6 hours)

Learner Outcomes:

Upon successful completion of this unit, students should be able to:

. calculate the average rate of charge for continuous functions over given intervals

2. Solve business and economic problems involving average rate.

Content:

I. Calculation of average rate of change on the interval [x0,]= f(.)

XI Xii

2. Solution average rate of change in business and economic as functions (e.g. average total cost

per unit of production. average total revenue per unit of sales, etc)

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UNIT IV — Differentiation front First PrincipIes (6 hours)

Learner Outcomes:

Upon successful completion of this unit. students should be able lo:

1. recognize the differential coefficient as the slope, gradient and instantaneous rate of change

2. define the derivative oía [unction as f () f(x+AX)f(x)

3. differentiate linear and quadratic functions from first principle

Content:

I. Recognition of the differential coefficient as:

a. Slope

b. Gradicnt

c Instantaneous rate of change

. .. . . . .,. f(x+)—f(x)

1. Definition of differentiation: J (X) = lin

2. I)iffcrcntiation from first principles:

a. Linear functions

b. Quadratic functions

UNIT V — Differentiation by Rule (Single Variable) (9 hours)

Learner Outcomes:

Upon successful completion of this unit, students should be able to:

I. use rules to differentiate single variable algebraic functions, exponential functions and

logarithmic functions

2 differentiate single variable algebraic functions, exponential functions and logarithmic

functions by substitution

3. perform (iiffcrcnhialion to obtain higher derivatives (2’ and3td)

(“ontent:

I. Using rules:

a. a constant

b. a variable raised to a constant power (power rule)

e. a constant times a function

d. addition

e. product

f. quotient

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g. chain

h. a function raised to a constant power (power function rule)

2. Differentiation by substitution

a. an cxponcniial lunction

h. a logarithmic function

3. Performing differentiation to obtain

a. Second derivatives

b. Third derivatives

UNIT VI — Application of Lifl’erentiation (6 hours)

Learner Outcomes:

Upon successful completion of this unit, students should be able to:

I. apply the second derivative test to identify stationary points (including points of inflexion)

2. locate and identify stationary points

3. sketch curves of quadratic and cubic functions

4. determine marginal functions for cost, revenue and profit functions

5. calculate marginal and optimal values for cost, revenue and protït as well optimal number of

units produced

Content:

1. Application of thc dcrivativc tests for y = f (x)

. . (df d2f

I. Maxirnumpoin(l—=OE —--<O

Il. Minimum point (-=o: ŠL>o)

dx dx

Ill. Pointof Inflexion =O: -4=o

dx

2. Location and Identification stationary points

a. Maximum point

b. Minimum point

c. Point of Inflexion

3. Sketching Curves

a. Quadratic Functions

b. Cubic Functions

4. Determination of:

a. Marginal Cost Functions

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b. Marginal Revenue Function

e. Marginal Profit Function

5. Calculation of:

a. Marginal Cost

b. Marginal Rcvcnuc

c. Marginal Protit

d. Optimal Profit

e. Optimal Revenue

f. Optimal Cost

g. Optimal Number of units produced

UNIT VII — Integration of Single Variable Function (9 hours)

Learner Outcomes:

Upon successful completion of this unit, students should be able to:

I. recognize the integral as the anti-derivative of a function

2. determine indefinite integral of polynomial function

3. evaluate definite integrals

4. compute area under the curve y = f (x) on the interval La, b]

5. derive total functions from marginal functions(e.g. cost, revenue and profit functions)

Content:

I. Recognition of integral as anti derivative

2. Determination of indefinite integrals using rules of integration

a. [k dx where k is a constant

b. f xdx were n—l

e. Integrate constant times a function

d. Integrate the sumldifference of functions

e. [ii”du where u is a polynomial function of x and n —I .

f. f u”du Ínu+c where u is a polynonìial function of x and n

g. f edu where u is a polynomial function of x.

h. f (x) ðvlog If(x)+c

f(x)

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3. Evaluation of definite integrals of functions in (I) above.( mention the fundamental theorem

of calculus)

f”f(x)th.= F(b)—F(a)

4. Computation of the area under the curve wheref(x) O on the interval la, bi

Jbj (x )ðv

5. Derivation of total functions from marginal functions (cg total cost from marginal cost etc)

UNIT VIII — Multivariable Calculus (8 hours)

Learner Outcomes:

Upon successful completion of this unit, students should be able to:

1. recognize mufti-variable functions

2. evaluate polynomial functions with two independent variables

3. compute partial-derivatives of polynomial functions with two independent variables

a.

b. f,

e. f

(I. L

e. fi,,

4. determine the critical values in a polynomial function with two independent variables

5. identify the nature of the stationary points of a multi- variable function

6. apply multi-variable calculus (as ab ve) to solve business and economic problems

Content:

I. Recognition of multi-variable functions

2. Evaluation of polynomial functions of f(x, v

3. Computation of partial derivatives of polynomial functions of f(x. y)

i. L

ii. fr

iii. f

iv. f1,

V.

4. Determination of stationary points of a multivariable function using

fjx,y)=O and f(x.y)=O

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5. Identification of the nature of stationary points using

D(x,y) =

6. Application optimal values of profit output and cost

UNIT IX —Integration of Multivariable Functions (4 hours)

[cariier Ouwwnes:

Upon successful completion of this unit, studens should be able to:

1. define a double integral over a region R(a xb,c y d)

2. interpret double integration or region as that of finding the representative area

3. compute double integrals

Content:

I. L)efinition

Double integral over a region R, may he given A = Jjf(x. y)dA = f h[ f (x. y)dy}Ix

2. Interpretation

The area of a region as a cross section or the area of a surface

3. Compute double integrals

METHODS OF I)ETT VERY:

t. Lectures

2. Case Studies

3. Demonstrations

4. Discussions

5. Group Work

6. Projects

MFTHOI)S OF ASSESSMENT AND EVALUATION:

I. Common Coursework 20%

2. Internal Tests 2O

3. Final Examination 60%

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RESOURCE MATERIAL:

Prescribed:

Hoffman. D., & Bradley. G. (2009). Applied calculus for business. economics and social and life

sciences (10’ ed.). OH: McGraw-Hill.

Barneu, R. & Zie2ler, M. (2007). calculus for business, economics, life sciences & social

science (I 1th ed.). NJ: Prentice Hall.

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