Ma 19: Sample Long Exam 1 (2nd Semester, SY 2008-2009)

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SAMPLE LONG EXAM NO. 1. MA 19 (Applied Calculus for Busines)
(R. Saldaña)

Directions: Show complete solutions.

1. (15 pts.) Find the limit:
a.) lim f(x) if f(x) = | 3x - 2 |
x →-1⁺

b.) lim(((x²+7x+10)(x-1))/(x-2)),as x →2

c.) lim (((x+3)²+1)/(2(x-3)²-1)),as x →3

2. (15 pts.)

a. Find the derivative of f(x) using the limit definition of the derivative: f(x)= (5/(√(x²+5)))

b. Verify your answer in (2a) using a differentiation formula.

3. (15 pts.) Evaluate the derivative of f(x) at x = 1.

f(x) =((x³+5x-6)/((1+x³)²))

4. (15 pts.) Find the derivative of f(x).
f(x)=(([[3]√(x+3√x))]⁴)/(((2/(x³))+5)))

5. (15 pts.)Find the equation of the tangent line to the curve
y = 4x³ - 2x + 5, at x =1.

6. (25 pts.) The price-demand equation and the cost function for the production of a certain brand of computer are given, respectively, by:
x = 9,000 - 30p and C(x) = 150,000 + 30x

where x is the number of computers that can be sold at a price of p (in dollars) per computer, and C(x) is the total cost (in dollars) of producing x computers.

(a). Express the price p as a function of the demand x.

(b). Find the revenue function.

(c). Find the marginal revenue function.

(d). Find R'(3,000) and R'(6,000), and interpret the results.

(e). Find the marginal cost.

(f). Find the break even points.

(g). Graph the cost function and the revenue function on the same coordinate system for 0≼x≼9,000. Indicate the break-even points, and the regions of loss and profit.

Ma 19 (Applied Calculus), Lesson No. 1, November 10, 2008

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Image Captions: The 'Fathers' of Calculus: Sir Isaac Newton (left) and Gottfried Wilhelm Leibniz (right)(Image Source: http://en.wikipedia.org/wiki/Calculus ).

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Ma 19 (Applied Calculus)
Lesson No. 1
10 November 2008

Topic: Introduction to the Derivative

by Rafael P. Saldaña, Ph.D.

Outline:

1. Brief Introduction to Calculus
2. Operational Definition of the Derivative
3. The derivative of some functions
3.1 y = c (where c is a constant)
3.2 y = x
3.3 y = cx^n (where c is a constant, x is the independent variable, and n is an integer)
4. Examples
5. Exercises

1. Brief Introduction to Calculus (Source: http://en.wikipedia.org/wiki/Calculus )
Excerpts:

Calculus (Latin, calculus, a small stone used for counting) is a branch of mathematics that includes the study of limits, derivatives, integrals, and infinite series, and constitutes a major part of modern university education.

Historically, it was sometimes referred to as "the calculus of infinitesimals", but that usage is seldom seen today. Most basically, calculus is the study of change, in the same way that geometry is the study of space.

Calculus has widespread applications in science and engineering and is used to solve problems for which algebra alone is insufficient. Calculus builds on algebra, trigonometry, and analytic geometry and includes two major branches, differential calculus and integral calculus, that are related by the fundamental theorem of calculus.

In more advanced mathematics, calculus is usually called analysis and is defined as the study of functions.

History:

The development of Calculus is attributed to two great mathematicians: Sir Isaac Newton (from England) and Gottfried Wilhelm Leibniz (from Germany).

Leibniz and Newton pulled earlier ideas together into a coherent whole and they are usually credited with the independent and nearly simultaneous invention of calculus. Newton was the first to apply calculus to general physics and Leibniz developed much of the notation used in calculus today; he often spent days determining appropriate symbols for concepts. The basic insight that both Newton and Leibniz had was the fundamental theorem of calculus.

2. Operational Definition of the Derivative

Calculus may be divided into two parts: Differential Calculus and Integral Calculus.
Differential Calculus deals with derivatives of functions and differentation.

Integral Calculus deals with antiderivatives (or integrals) of functions and antidifferentiation (or integration)

Given a function y = f(x), we say that the "derivative" of y with respect to (wrt) x is given by
y' = dy/dx = f '(x)

Read: "y prime", "dee-why dee-eks", and "f prime of x"

The derivative a function is another function.

3. The Derivative of Some Functions

The derivative of y = c (where c is a constant)

If c is a constant, then the derivative of y = c wrt x is given by
y' = 0 (zero)

The derivative of y = x (where x is the independent variable)

If y = x, then the derivative of y wrt x is given by y' = 1

The derivative of y = cx^n (where c is a constant, x is the independent variable, and n is an integer)

If y = cx^n, then the derivative of y wrt x is given by y' = ncx^(n-1)

4. Examples:

Find the derivative y' of the following functions:

1. y = 5

2. y = 2x/2

3. y = x^3

4. y = 4x^5

5. y = bx^10, where b is a constant

Solutions:

1. y' = 0
2. y = 2x/2 = x, therefore y' = 1
3. y' = 3x^(3 - 1), y' = 3x^2 (Note: c = 1)
4. y' = 5(4)x^(5 - 1), y' = 20x^4
5. y' = 10bx^(10 -1), y' = 10bx^9
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Class List, Ma 19 L (Applied Calculus), 2nd Semester, SY 2008 - 2009

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MA 19 L - APPLIED CALCULUS FOR BUSINESS
School Year 2008 - 2009, Semester 2
SALDANA, RAFAEL P.


Class Number / Name

1. CHU, RHEANA STACY GO

2. CHUA CHING, JERINAE BRIZELLE PUA

3. DE LEON, CHAS MIL VICTORINO

4. DY TAN, KYLE RUSSELL ONG

5. FAUSTO, MARTIN JOSEPH FRES

6. FRANCO, KRISTOFFER PAUL LIM

7. HUR, SEUL KI

8. LEE, KATRINA VILLANUEVA

9. LIM, PATRICIA ANGELA AMPAG

10. MARTINEZ IV, GERMAN MARK AMA

11. MENDOZA, CATHERINE ISABEL MERCADO

12. OLIVAN, JR., DANIEL HAO CHUA

13. ONG, MARK VINCENT LEH

14. ONG, RYAN MARVIN WEE

15. ORTEGA, VICTORIA LUCIA CRISOSTOMO

16. PINEDA, SAMANTHA PATRICIA LACSON

17. QUIAMBAO, FRANCESCA MARIE RUIZ

18. RAVIDA, PAOLO MIGUEL MIGUEL

19. SAMONTE, MARIA SARAH FRANCESCA SIBAL

20. SIOCHI, TARA DOMINIQUE MANALANG

21. TAN, ALLIANA ANN CARPIO

22. TOLARBAS, GERARD NOEL RAGUERO

23. TUAZON, JAPHET AGRA

24. VICTA, FRANCIS BERTHOLET ROÑO

25. YU, HANS CHRISTIAN SO
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Course Syllabus, Ma 19 (Applied Calculus for Business)

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COURSE SYLLABUS for Ma 19

Catalogue No.: Ma 19 Course

Title: Applied Calculus for Business

Department: Mathematics

School: Science and Engineering, Ateneo de Manila University

Semester: 2nd

School Year: 2008 – 2009

Credit: 6 units

Prerequisite: Ma 11

A. Course Description

Ma 19 is a 6-unit course taken primarily by BS Management majors both for the standard and honors programs. It aims to introduce students to differential and integral calculus in preparation for various topics in higher management areas such as Business Statistics, Operations Management and Financial Management. Topics include: limits and continuity; derivatives and differentials, some applications of derivatives, antidifferentiation, the definite integral, partial derivatives and the total differential of functions of more than one variable.

B. Course Objectives

At the end of the course, the student should be able to:

Have an intuitive notion of the concepts of limit and continuity of a function; apply the limit theorems and obtain the asymptotes of the graph of a function.

Find the derivative of algebraic, exponential and logarithmic functions; interpret the derivative as a measure of slope and as a rate of change; use derivatives as an aid in sketching the graph of a function and to solve optimization problems.

Determine both the definite and the indefinite integrals of functions; interpret the integral as a measure of an area of a region; apply different techniques of integration; solve differential equations.

Find the partial derivatives and the higher order partial derivatives of functions of more than one variable; determine the maximum and minimum values of multivariate functions and use the method of Lagrange multipliers to solve constrained optimization problems.

Course Outline and Timeframe (based on the textbook, Calculus for Business, Economics, Life Sciences, and Social Sciences (9^th ed), by Barnett and others)

Topics (# of hours does NOT include exam day)	Chapter/Section
The Derivative (14½ hours) Rate of Change and Slope Limits The Derivative Derivatives of Constants, Power Forms, and Sums Derivatives of Products and Quotients Chain Rule: Power Form Marginal Analysis in Business and Economics Long Exam #1 Tentative Schedule: Nov. 27 (Th)	3.1 3.2 3.3 3.4 3.5 3.6 3.7
Graphing and Optimization (14½ hours) Continuity and Graphs First Derivative and Graphs Second Derivative and Graphs Curve Sketching Techniques: Unified and Extended Optimization; Absolute Maxima and Minima Long Exam #2 Tentative Schedule: Dec. 18 (Th)	4.1 4.2 4.3 4.4 4.5
Additional Derivative Topics (14 hours) The Constant e and Continuous Compound Interest Derivatives of Logarithmic and Exponential Functions Chain Rule: General Form Implicit Differentiation Related Rates Long Exam #3 Tentative Schedule: Jan. 20 (T)	5.1 5.2 5.3 5.4 5.5
Midterm Exam Tentative Schedule: Jan. 23, 2009 (F) 6:00 – 8:00 pm
4. Integration (14½ hours) Antiderivatives and Indefinite Integrals Integration by Substitution Differential Equations – Growth and Decay (no slope fields) A Geometric-Numeric Introduction to the Definite Integral Definite Integral as a Limit of a Sum; Fundamental Theorem of Calculus (no error bounds) Long Exam #4 Tentative Schedule: Feb. 12, 2009 (Th)	6.1 6.2 6.3 6.4 6.5 *Goldstein can be used as supplement
Additional Integration Topics (10½ hours) Area between Curves Applications in Business and Economics (no probability density functions) Integration by Parts Long Exam #5 Tentative Schedule: Feb. 26, 2009 (Th)	7.1 7.2 7.3
Multivariable Calculus (14 hours) Functions of Several Variables Partial Derivatives Maxima and Minima Maxima and Minima Using Lagrange Multipliers Double Integrals Over Rectangular Regions Long Exam #6 Tentative Schedule: Mar 17, 2009 (T)	8.1 8.2 8.3 8.4 8.6
Final Exam Schedule: c/o Registrar’s Office week of Mar. 23-27, 2009

D. Required Textbook:

Calculus for Business, Economics, Life Sciences, and Social Sciences (9^th ed.), by R.A. Barnett, M.R. Ziegler and K.E. Byleen.

E. References:

Bittinger, M.L., Calculus (5^th ed.), Addison-Wesley, 1992

Edwards, C.H. and E. Penney, Calculus with Analytic Geometry (6^th ed.), Prentice-

Hall, 2002.

Goldstein, L.J., D.C. Lay and D.I. Schneider, Calculus and its Applications (7^th ed.),

Prentice-Hall, 1996.

Harshbarger, R.J. and J.J. Reynolds, Calculus with Applications, D.C. Heath and

Company, 1990.

Thomas, G.B. and R.L. Finney, Calculus and Analytic Geometry (9^th ed.), Addison-

Wesley, 1996.

F. Course Requirements and Grading System:

The final mark of the student will be based principally on the following:

• 6 long exams (100 pts. each),
  

• the departmental midterm exam (200 pts.),
  
• the departmental final exam (200 pts.)
  
• and (optional) quizzes/assignments (100 pts.).

Percentage of the total score	Equivalent letter grade
92% - 100%	A
86% - 91%	B+
77% - 85%	B
69% - 76%	C+
60% - 68%	C
50% - 59%	D
49% and below	F

Students whose class standing before the final exam is 90% or above (prior to cancellation of one (1) lowest long exam score) may be exempted from taking the final exam subject to the discretion of the teacher.

Some Dates to be noted:

Date	Day	Event
Nov. 10, 2008	Monday	2nd Semester Classes begin
Nov. 14, 2008	Friday	2nd Semester Faculty Day
Dec. 1, 2008	Monday	Bonifacio Day (Monday nearest Nov. 30) (Regular non-working holiday)
Dec. 8, 2008	Monday	Solemnity of the Immaculate Conception (Non-working School Holiday)
Dec. 20, 2008 – Jan. 4, 2009	Saturday to Sunday	C H R I S TM A S B R E A K
Jan. 5, 2009	Monday	Classes resume
Feb. 2, 2009	Monday	President’s Day (Non-working School Holiday)
March 23-28, 2009		Final Exam Week

H. General Classroom Policies

1. A student who misses an exam MAY be given a make-up exam depending on the student’s reason(s), and upon the Department Chairperson’s approval. Make-up exams are more difficult than the regular exam.

2. As a departmental policy, there will be no take home long exams and no bonuses.
   
   

I. Consultation Hours:

MW: 1500 - 1630

F: 1500 - 1600

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Laboratory Exercise No. 4 (Ma195a15, Numerical Analysis): Mathematica and Bisection Method

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For my class in Ma 195a15 (Seminar on Numerical Analysis) today (1 July 2008) I discussed the Secant Method and later I taught my students how to write a Mathematica loop construct using the While[] function. The example I gave is the Mathematica implementation of the Bisection Method. See figure above. (Note: Click on the image to see a larger view).