Limits and Derivatives

Concept of a Function

y is a function of x, and the relation y = x 2 describes a function. We notice that with such a relation, every value of x corresponds to one (and only one) value of y. y = x 2

Since the value of y depends on a given value of x, we call y the dependent variable and x the independent variable and of the function y = x 2.

Notation for a Function : f(x)

The Idea of Limits

Consider the function The Idea of Limits x f(x)f(x)

Consider the function The Idea of Limits x f(x)f(x) un- defined

Consider the function The Idea of Limits x g(x)g(x) x y O 2

If a function f(x) is a continuous at x 0, then. approaches to, but not equal to

Consider the function The Idea of Limits x g(x)g(x)

Consider the function The Idea of Limits x h(x)h(x) un- defined 1234

does not exist.

A function f(x) has limit l at x 0 if f(x) can be made as close to l as we please by taking x sufficiently close to (but not equal to) x 0. We write

Theorems On Limits

Exercise 12.1 P.7

Limits at Infinity

Consider

Generalized, if then

Theorems of Limits at Infinity

Exercise 12.2 P.13

Theorem where θ is measured in radians. All angles in calculus are measured in radians.

Exercise 12.3 P.16

The Slope of the Tangent to a Curve

The slope of the tangent to a curve y = f(x) with respect to x is defined as provided that the limit exists.

Exercise 12.4 P.18

Increments The increment △ x of a variable is the change in x from a fixed value x = x 0 to another value x = x 1.

For any function y = f(x), if the variable x is given an increment △ x from x = x 0, then the value of y would change to f(x 0 + △ x) accordingly. Hence thee is a corresponding increment of y( △ y) such that △ y = f(x 0 + △ x) – f(x 0 ).

Derivatives (A) Definition of Derivative. The derivative of a function y = f(x) with respect to x is defined as provided that the limit exists.

The derivative of a function y = f(x) with respect to x is usually denoted by

The process of finding the derivative of a function is called differentiation. A function y = f(x) is said to be differentiable with respect to x at x = x 0 if the derivative of the function with respect to x exists at x = x 0.

The value of the derivative of y = f(x) with respect to x at x = x 0 is denoted by or.

To obtain the derivative of a function by its definition is called differentiation of the function from first principles.

Exercise 12.5 P.21