To My Ma 20 Students: Some Notes on 2nd Long Exam

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Dear Ma 20 Students,

I arrived today (7/23/08) from my conference/meetings in France -- still having jetlag :(


For your 2nd Long Exam on Thursday, July 24, 2008, during our class period, you may focus your final review on the following:

1. Finding the derivative of a rational function using Chain Rule (15 pts.)

2. Problem involving marginal cost, marginal revenue, marginal profit and break even pts. (15 pts).

3. Problem involving continuity of a function (review theorems and properties) (15 pts.)

4. Graphing a rational function using the first and second derivative tests. Includes critical pts., y-intercept, x-intercept, horizontal and vertical asymptotes, concavity, and inflection points. (40 pts.)

5. Word problem involving maxima and minima (use of first and second derivative tests). (15 pts.)Pls. bring at least 8 sheets of clean bond paper and your scientific calculator (graphing calculators are not allowed).

I wish you all the best in your 2nd long exam on Thursday, July 24, 2008.

From,

Dr. Rafael Saldaña
in Quezon City
7/23/08

Ma 20 (Calculus): Congratulations to 'A' Students (Exam 1)

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Note: Last Tuesday, 1 July 2008, I gave the 1st Long Exam to my Ma 20 (Calculus) class. I started giving the results of the exam on 2 July (Wednesday) until 4 July (Friday).

Out of the 28 students in my class, three students got a grade of 'A' in the first exam. Their raw scores are as follows (perfect score is 100 pts.):

1. Miguel Antonio ROJAS : 96

2. Ann Jillian ADONA : 93

3. Jose Isaac MONJE: 92

CONGRATULATIONS!
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Lecture Notes in Ma 20 (Calculus): Marginal Cost as a Derivative, 6/30/08

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Lecture Notes in Ma 20 (Calculus for Economics)
30 June 2008


Topic: Marginal Cost as a Derivative

Suppose that a firm which makes electric light bulbs knows that in order to produce q light bulbs it will have to pay out C(q) dollars in wages, materials, overheads and so on. We say that C is the firm’s cost function.

In this case a change in production from q to q + 1 light bulbs is relatively small, and may be described as ‘marginal’. The corresponding increase in cost is C(q + 1) – C(q), and this can be thought of as the ‘marginal cost’ of making one more lightbulb, when the level of production is q. In the delta notation, we see that the marginal cost is the change delta C corresponding to a change delta q = 1. In general, the relationship between delta q and delta C is given by the approximation

delta C is approximately equal to C’(q) delta q

and so on, when delta q = 1, the resulting delta C is approximately C’(q) . For this reason it makes sense to define the marginal cost function to be the derivative of the cost function C.

(Note: In the traditional language of economics, the derivative of any function F is referred to as the marginal of F, and is often denoted by MF.)

What we have shown is that if the units are small, as for example in the production of light bulbs, C’(q) represents the cost of producing one more unit when the q units are being produced. The concept of marginal cost is equally useful when the units are larger, provided we remember this basic idea.

EXAMPLE:

Suppose that the costs of a firm making bicycles are $50,000 per week in overheads and $25 for every bicycle made. Then its cost function (in dollars) is

C(q) = 50,000 + 25q

So that C’(q) = 25

In this case the marginal cost is 25 dollars, independent of the level of production q.

Slightly more realistically, suppose that in order to produce a substantially larger weekly output of bicycles it would be necessary for the firm to incur extra costs, possibly because its increased consumption of a raw material would drive the price of that raw material upwards.

We can account for this by introducing an additional cost term, say 0.001q^2, which is trivial when q is small but which is more significant as q increases. In this case we have

C(q) = 5000 + 25q + 0.001q^2

So that

C’(q) = 25 + 0.002q

Thus the marginal cost is $25.2 if the output is 100 bicycles per week, but it would rise to $45 if the output were 10,000 bicycles per week.

Reference:

Martin Anthony and Norman Biggs (1996). Mathematics for Economics and Finance. Cambridge University Press, pp. 59 - 60.
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Lecture Notes in Ma 20 (Calculus): Sample Word Problem in Economics Involving the Derivative, 6/30/08

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Lecture Notes in Ma 20 (Calculus for Economics)
30 June 2008

Topic:

Sample Word Problem in Economics Involving the Derivative


Suppose that the demand set for cans of salmon is

D = {(q,p) where (p^3)q = 8000} ,

where q is the number of cans (in thousands per week) and p is the price per can in dollars.


a). If p is increased from $20 to $21, what will be the expected fall in sales, approximately?

b). If, on the other hand, production were to be increased from 1000 tins per week to 1100, what would be the expected fall in price?


SOLUTION:

The demand function q and its derivative q’ are

q(p) = 8000p^-3

q’(p) =(8000)(-3)p^-4 = -24000/p^4

a). Therefore when p = 20 and (delta)p = 1 we have

(delta)q is approximately equal to (-24000)/p^4)(delta)p = -24009/20^4 = -0.15

Remembering that the units are thousands of cans, it follows that 150 fewer cans will be sold per week.

b). In letter b, we have to consider p as a function of q, and so we need the inverse demand function and its derivative:

(p^3)q = 8000

p^3 = 8000/q

p = cuberoot of (8000/q)

p = 20q^(-1/3)

p’(q) = (-1/3)(20)q^(-4/3) = (-20/3)q^(-4/3)

So when q = 1 and (delta)q = 0.1 we have

(delta)p is approximately equal to [(-20/3)q^(-4/3)](0.1) = -2/3 or approx. 0.67

The conclusion is that the price falls by about 67 cents.

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Reference: Martin Anthony and Norman Biggs (1996). Mathematics for Eonomics and Finance. Cambridge University Press, pp. 57 - 58.
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Ma 20 (Calculus): Some Review Questions

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For my Ma 20 (Calculus for Economists) class, here are some review questions for your forthcoming 1st Long Test scheduled on Thursday, 26 June 2008.

1. Using the graphical method, find the limit of the given function as x approaches zero. Show a complete solution. Given: f(x) = (absolute value of (3x - 2))/(x - 2).

2. Given: f(x) = 100x^5 - (square root of (x^2 + 2)) - pi. Find the limit of f(x) as x approaches 1. Cite the relevant limit theorems and properties.

3. Given: f(x) = [(x^2 + 7x +10)(x-1)/(2(x-2))]. Find the limit of f(x) as x approaches 2. Show a complete solution.

4. Given: f(x) = [(x^2 - 9)/(2(x + 3)]. Find the limit of f(x) as x approaches 1. Show a complete solution.

5. Given: f(x) = [5/(square root of (x^2 + 5))]. Find the derivative of f(x) using the limit definition of the derivative. Show a complete solution.

6. Given: f(x) = x^3 + 5x - 6, g(x) = 2(1 + x^2). Find the instantaneous rate of change of f(x) + g(x) at x = 1. Show a complete solution.

7. Given: f(x) = x^3 - 5, g(x) = x^2 + (2/x^2) - 100000, h(x) = f(x)*g(x). Find the derivative of h(x) using the product rule of differentiation. Show a complete solution. There is no need to simplify your final answer. [Note: the original f(x) = cube root of (x^3 - 5). I removed the cube root function to avoid the use of Chain Rule which has not yet been discussed in class... RPS, 6/23/08]

8. Refer to problem no. 7 for the value of f(x) and g(x). Let r(x) = f(x)/g(x). Find the derivative of h(x) using the quotient rule of differentiation. Show a complete solution. There is no need to simplify your final answer.

9. Find the equation of the tangent line to the curve y = 4x^3 - 2x +5, at x = 1. Show a complete solution.

10. (a) Give an example of a function with undefined limit. (b) Give an example of a function with a limit that is equal to negative infinity as x approaches zero.

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Notes on Ma 20 (Calculus), 6/18/08

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These are our activities in Ma 20 (Calculus) yesterday (6/17/08):

1. Boardwork (on an assigned problem given last 6/16/08) on finding the limits of a function using the graphical method.

2. Graded Seatwork No. 1 (8 pts). Given a function f(x) find the limit of f(x) using the graphical method.

Given: f(x) = ((absolute value of (x + 3)) - 1)/(x - 1)

a. Find the limit of f(x) as x approaches -1.
b. Find the limit of f(x) as x approaches 0.
c. Find the limit of f(x) as x approacher 1.

3. Discussed the Properties of Limits (Theorem 1) (See Barnett et al., p. 153)

Today (6/18/08) I will do the following in class:

1. Return the results of Graded Seatwork No. 1.
2. Give a short quiz on the topic finding the limit of a function using the graphical method.
3. Continue discussing Properties of Limits.

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Ma 20 (Calculus), Lesson No. 1

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Photo Captions: Sir Isaac Newton (left) and Gottfried Wilhelm Leibniz (right)
(Source: http://en.wikipedia.org/wiki/Calculus )

Ma 20 (Calculus for Economists)
Lesson No. 1
10 June 2008

Topic: Introduction to the Derivative

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 )

• 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

Syllabus for Ma 20, 1st Semester, SY 2008-2009, Ateneo de Manila University

SYLLABUS
Ma 20 (1st Sem 08-09)
Calculus for Economists

Professor: Dr. Rafael P. Saldaña
Class: 1200 – 1330 T-TH; 1130-1230 M-W-F
Consultation: 1500-1700 W-F or by appointment
Textbook: Calculus for Business, Economics, Life Sciences, and Social Sciences (9th ed), Barnett, et al.

Course Description: 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.

Specific objectives:

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

1. 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.

2. 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.

3. 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.

4. 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:

Topics / Sections / Tentative Exam Date

LT1 , The Derivative , 3.1-3.7 , June 26 (Th)
LT2 , Graphing and Optimization , 4.1-4.5 , July 17 (Th)
LT3 , Additional Derivative Topics , 5.1-5.5 , August 5 (T)
Midterms , August 8 (F)
LT4 , Integration , 6.1-6.5 , August 28 (Th)
LT5 , Additional Integration Topics , 7.1-7.3 , Sept 11(Th)
LT6, Multivariable Calculus, 8.1-8.4,8.6 Sept 30 (T)
Final Exams, Oct 6-11


Course Requirements:

6 Long Exams (100 pts each)
(the lowest will be cancelled) = 500 (max)
Midterm Exam (200 pts) = 200 (max)
Final Exam (200 pts) = 200 (max)
Quizzes, Homework, Recitation = 100 (max)

Grading System/Class Policies:

92 – 100 = A
86 – 91 = B+
77 – 85 = B
69 – 76 = C+
60 – 68 = C
50 – 59 = D
Below 50 = F

The final grade is computed as follows:

(total score/highest possible sum) x 100%

converted to a letter grade based on the above table.

· A student with a class standing of at least 90% prior to cancellation of the lowest exam may be exempted from taking the final exams.

· There will be no make-up quizzes or homework.

· Use short bond paper and pens in exams.

. Always bring a scientific calculator.

· Absence in a TTh class meeting is considered 1 ½ cuts; in an MWF meeting, 1 cut. Any student who accumulates more than 18 cuts will get a final grade of W. A late is considered a cut.

· No eating, sleeping, use of cellular phones during class. Cellular phones must be in silent mode. Avoid going out of class once classes have started unless there is an emergency. The teacher reserves the right to send anyone outside the classroom on the grounds of discourtesy to the teacher or to a fellow student, misbehavior in the classroom.

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