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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