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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.

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Presentation on theme: "Copyright © 2006 Brooks/Cole, a division of Thomson Learning, Inc. Functions, Limits, and the Derivative 2 Functions and Their Graphs The Algebra of Functions."— Presentation transcript:

1 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

2 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) Domain Range

3 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 writein place of y. This is read “f of x.” So NOTE: It is not f times x

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

5 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

6 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

7 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). f (1) = 2 (1, 2) x y

8 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. FunctionNot a Function It is crossed more than once.

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

10 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.

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

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

13 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.

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

15 Copyright © 2006 Brooks/Cole, a division of Thomson Learning, Inc. Application Example 1 (Functions ) A shirt producer has a fixed monthly cost of $5000. If each shirt costs $3 and sells for $12 find: a.The cost function b.The revenue function c.The profit from 900 shirts Cost: C(x) = 3x where x is the number of shirts produced. Revenue: R(x) = 12x where x is the number of shirts sold. Profit: P(x) = Revenue – Cost = 12x – (3x ) = 9x – 5000 P(900) = 9(900) – 5000 = 3100, or $3100.

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

17 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

18 Copyright © 2006 Brooks/Cole, a division of Thomson Learning, Inc. 3.What is the profit for the company if 2000 units are produced and sold each week? Since the profit function is we have

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

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

21 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. Average velocityt Answer: 12 ft/sec Notice how elapsed time approaches zero

22 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. a L x y

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

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

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

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

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

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

29 Copyright © 2006 Brooks/Cole, a division of Thomson Learning, Inc. 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. a M x y

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

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

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

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

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

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

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

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

38 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] Instantaneous rate of change of f at x Slope of the Tangent Line Slope of Secant Line

39 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.”

40 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

41 Copyright © 2006 Brooks/Cole, a division of Thomson Learning, Inc. The Derivative Given

42 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.

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

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


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