Copyright 2006 BrooksCole a division of Thomson Learning

ﺭﻭﺍﺀ ﺍﺑﺮﺍﻫﻴﻢ ﻋﻴﺴﻰ. ﻡ Copyright © 2006 Brooks/Cole, a division of Thomson Learning, Inc. ﺍﻟﻐﺎﻳﺎﺕ

Limit of a Function The limit of f (x), as x approaches a, equals L if we can make the value f (x) arbitrarily close to L by taking x to be sufficiently close to a. y L a x Copyright © 2006 Brooks/Cole, a division of Thomson Learning, Inc. written:

6 Note: f (-2) = 1 is not involved -2 x Copyright © 2006 Brooks/Cole, a division of Thomson Learning, Inc. Computing Limits Ex. y

Copyright © 2006 Brooks/Cole, a division of Thomson Learning, Inc. Properties of Limits

Ex. Copyright © 2006 Brooks/Cole, a division of Thomson Learning, Inc. Computing Limits Ex.

Ex. Notice Factor and cancel common factors Copyright © 2006 Brooks/Cole, a division of Thomson Learning, Inc. determinate Form: form

Limits at Infinity provided that Ex. is defined. Divide by Copyright © 2006 Brooks/Cole, a division of Thomson Learning, Inc. For all n > 0,

One-Sided Limit of a Function if we can make the value f (x) arbitrarily close to L by taking x to be sufficiently close to the right of a. L a Copyright © 2006 Brooks/Cole, a division of Thomson Learning, Inc. The right-hand limit of f (x), as x approaches a, equals L written:

One-Sided Limit of a Function if we can make the value f (x) arbitrarily close to M by taking x to be sufficiently close to the y left of a. M a x Copyright © 2006 Brooks/Cole, a division of Thomson Learning, Inc. The left-hand limit of f (x), as x approaches a, equals M written:

Find Copyright © 2006 Brooks/Cole, a division of Thomson Learning, Inc. One-Sided Limit of a Function Ex. Given Find

Continuity of a Function A function f is continuous at the point x = a if the following are true: f(a) a x Copyright © 2006 Brooks/Cole, a division of Thomson Learning, Inc. y

Properties of Continuous Functions The identity function f (x) = x is continuous everywhere. Copyright © 2006 Brooks/Cole, a division of Thomson Learning, Inc. The constant function f (x) is continuous everywhere. Ex. f (x) = 10 is continuous everywhere.

Properties of Continuous Functions A polynomial function y = P(x) is continuous at everywhere. A rational function at all x values in its domain. is continuous Copyright © 2006 Brooks/Cole, a division of Thomson Learning, Inc. If f and g are continuous at x = a, then

Intermediate Value Theorem f (b) f (c) = L f (a) x a c b Copyright © 2006 Brooks/Cole, a division of Thomson Learning, Inc. If f is a continuous function on a closed interval [a, b] and L is any number between f (a) and f (b), then there is at least one number c in [a, b] such that f(c) = L. y

Intermediate Value Theorem f (x) is continuous for all values of x and since f (1) < 0 and f (2) > 0, by the Intermediate Value Theorem, there exists a c on (1, 2) such that f (c) = 0. Copyright © 2006 Brooks/Cole, a division of Thomson Learning, Inc. Ex.

Existence of Zeros of a Continuous Function f(b) a b f(a) x Copyright © 2006 Brooks/Cole, a division of Thomson Learning, Inc. If f is a continuous function on a closed interval [a, b], and f(a) and f(b) have opposite signs, then there is at least one solution of the equation f(x) = 0 in the interval (a, b). y

Example (Existence of zeros of a continuous function) 1. Show that f(x) is a continuous function everywhere. The function is a polynomial function and is therefore continuous everywhere. 2. Show that f(x) = 0 has at least one solution on the interval (0, 2) Copyright © 2006 Brooks/Cole, a division of Thomson Learning, Inc.

Rates of Change Instantaneous rate of change of f at x Slope of the Tangent Line Copyright © 2006 Brooks/Cole, a division of Thomson Learning, Inc. Average rate of change of f over the interval [x, x+h] Slope of Secant Line

The Derivative It is read “f prime of x. ” Copyright © 2006 Brooks/Cole, a division of Thomson Learning, Inc. The derivative of a function f with respect to x is the function given by

The Derivative Four-step process for finding 2. Find 3. Find 4. Compute Copyright © 2006 Brooks/Cole, a division of Thomson Learning, Inc. 1. Compute

3. 4. Copyright © 2006 Brooks/Cole, a division of Thomson Learning, Inc. The Derivative Given 1. 2.

Example Find the slope of the tangent line to the graph of at any point (x, f(x)). Step 1. Step 2. Step 3. Step 4. Copyright © 2006 Brooks/Cole, a division of Thomson Learning, Inc.

If a function is differentiable at x = a, then it is continuous at x = a. y Not Continuous x Not Differentiable Still Continuous Copyright © 2006 Brooks/Cole, a division of Thomson Learning, Inc. Differentiability and Continuity

Example The function is not differentiable at x = 0 but it is continuous everywhere. y O x Copyright © 2006 Brooks/Cole, a division of Thomson Learning, Inc.