Friday, June 6, 2008

Ma 20 (Calculus), Lesson No. 1

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. [Note: Draft version, 7 June 2008]


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 )



  • 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

3 comments:

micoantonio said...


3. y' = 3x^(3 - 1), y' = 3x^2 (Note: c = 1)


Sir Raffy, why is c = 1?

Rafael P. Saldaña, Ph.D. said...

Hi, Mico Antonio:

In the given, y = x^3.

In the general form that we are using, y = cx^n.

Hence, c = 1, and n = 3.

Therefore, following the Power Rule [ y' = ncxn^(n-1) ], we have:

y' = 3(1)x^(3-1)

or y' = 3x^2

Note: If c = 1, we can omit writing (1).

I hope that this answers your question.

Raffy S.

Rafael P. Saldaña, Ph.D. said...

Correction:

There was a typographical error in my previous reply.

The statement should read:

Therefore, following the Power Rule [ y' = ncx^(n-1) ] ...

Raffy S.