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

Functions, Limits, and the Derivative 2 Functions and Their Graphs The Algebra of Functions Functions and Mathematical Models Limits One-Sided Limits and Continuity The Derivative Copyright © 2006 Brooks/Cole, a division of Thomson Learning, Inc.

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

Function A rule that assigns to each element in a set A (the domain), one and only one element in a set B (the range) Range Domain -1 1 -6 1 3 -4 Copyright © 2006 Brooks/Cole, a division of Thomson Learning, Inc.

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

Function Notation is a function, with values of x as the domain and values of y as the range. We write in place of y. This is read “f of x.” So NOTE: It is not f times x Copyright © 2006 Brooks/Cole, a division of Thomson Learning, Inc.

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

Function Notation Example Solution Plug in –2 Copyright © 2006 Brooks/Cole, a division of Thomson Learning, Inc.

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

Domain of a Function The domain of a function is the set of values for x for which f (x) is a real number. Ex. Find the domain of Since division by zero is undefined we must The domain can be expressed as the intervals Copyright © 2006 Brooks/Cole, a division of Thomson Learning, Inc.

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

Domain of a Function Ex. Find the domain of Since the square root of a negative number is undefined we must have The domain can be expressed as the interval Copyright © 2006 Brooks/Cole, a division of Thomson Learning, Inc.

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

Graph of a Function The graph of a function is the set of all points (x, y) such that x is in the domain of f and y = f (x). Given the graph of y = f (x), find f (1). y f (1) = 2 (1, 2) x Copyright © 2006 Brooks/Cole, a division of Thomson Learning, Inc.

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

Graph of a Function Vertical Line Test: The graph of a function can be crossed at most once by any vertical line. Function Not a Function It is crossed more than once. Copyright © 2006 Brooks/Cole, a division of Thomson Learning, Inc.

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

Example Sketch the graph of the function: Copyright © 2006 Brooks/Cole, a division of Thomson Learning, Inc.

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

Algebra of Functions Domain: Domain of f intersected with the domain of g. Domain: Domain of f intersected with the domain of g with the exclusion of all values of x, such that g(x) = 0. Copyright © 2006 Brooks/Cole, a division of Thomson Learning, Inc.

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

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**Composition of Functions**

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

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

Types of Functions Polynomial Functions n is a nonnegative integer, each is a constant. Ex. Rational Functions polynomials Ex. Copyright © 2006 Brooks/Cole, a division of Thomson Learning, Inc.

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

Types of Functions Power Functions ( r is any real number) Ex. Ex. Copyright © 2006 Brooks/Cole, a division of Thomson Learning, Inc.

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**Application Example 1 (Functions )**

A shirt producer has a fixed monthly cost of $ If each shirt costs $3 and sells for $12 find: a. The cost function Cost: C(x) = 3x where x is the number of shirts produced. b. The revenue function Revenue: R(x) = 12x where x is the number of shirts sold. c. The profit from 900 shirts Profit: P(x) = Revenue – Cost = 12x – (3x ) = 9x – 5000 P(900) = 9(900) – 5000 = 3100, or $3100. Copyright © 2006 Brooks/Cole, a division of Thomson Learning, Inc.

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**Application Example 2 (Functions)**

A division of Chapman Corporation manufactures a pager. The weekly fixed cost for the division is $20,000, and the variable cost for producing x pagers/week is The company realizes a revenue of Copyright © 2006 Brooks/Cole, a division of Thomson Learning, Inc.

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

1. Find the total cost function. The total cost function is the variable cost plus the fixed cost: 2. Find the total profit function. The profit is the revenue minus the total cost Copyright © 2006 Brooks/Cole, a division of Thomson Learning, Inc.

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

What is the profit for the company if 2000 units are produced and sold each week? Since the profit function is we have Copyright © 2006 Brooks/Cole, a division of Thomson Learning, Inc.

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**Introduction to Calculus**

There are two main areas of focus: 1. Finding the tangent line to a curve at a given point. y x tangent line 2. Finding the area of a planar region bounded by a given curve. y Area x Copyright © 2006 Brooks/Cole, a division of Thomson Learning, Inc.

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

Velocity Over any time interval Average If I travel 200 miles in 5 hours, my average velocity is 40 miles/hour. As elapsed time approaches zero Instantaneous When I see the police officer, my instantaneous velocity is 60 miles/hour. Copyright © 2006 Brooks/Cole, a division of Thomson Learning, Inc.

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

Velocity Ex. Given the position function where t is in seconds and s(t) is measured in feet, find: a. The average velocity for t = 1 to t = 3. b. The instantaneous velocity at t = 1. t Average velocity Notice how elapsed time approaches zero Answer: 12 ft/sec Copyright © 2006 Brooks/Cole, a division of Thomson Learning, Inc.

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

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

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

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

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

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

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

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

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

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

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

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

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**One-Sided Limit of a Function**

The right-hand limit of f (x), as x approaches a, equals L written: 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.

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**One-Sided Limit of a Function**

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

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**One-Sided Limit of a Function**

Ex. Given Find Find Copyright © 2006 Brooks/Cole, a division of Thomson Learning, Inc.

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**Continuity of a Function**

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

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**Properties of Continuous Functions**

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

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**Properties of Continuous Functions**

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

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**Intermediate Value Theorem**

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 f (b) f (c) = L f (a) x a c b Copyright © 2006 Brooks/Cole, a division of Thomson Learning, Inc.

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**Intermediate Value Theorem**

Ex. 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.

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**Existence of Zeros of a Continuous Function**

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 f(b) x a b f(a) Copyright © 2006 Brooks/Cole, a division of Thomson Learning, Inc.

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**Example (Existence of zeros of a continuous function)**

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.

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

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

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

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

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

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

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

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

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

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.

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**Differentiability and Continuity**

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

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

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

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