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Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 2- 1 Chapter 2 Limits and Continuity Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall 2.1 Rates of Change and Limits Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Quick Review In Exercises 1 - 4, find f (2). 3 2 1. f (x)= 2 x - 5 x + 4 3. æ xö f (x)= sin ççp ÷ ÷ çè 2 ÷ ø 4. ìï 3x - 1, ïï f (x)= í 1 ïï 2 , ïî x - 1 2. 4 x2 - 5 f (x )= 3 x +4 x< 2 x³ 2 Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 2- 4 Quick Review In Exercises 5 - 8, write the inequality in the form a < x < b. 5. x<4 6. x < c2 7. x- 2 < 3 8. x- c < d 2 In Exercises 9 and 10, write the fraction in reduced form. 9. x 2 - 3x - 18 x+3 2x2 - x 10. 2 x2 + x - 1 Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 2- 5 Quick Review Solutions In Exercises 1 - 4, find f (2). 3 2 1. f (x)= 2 x - 5 x + 4 3. æ xö f (x)= sin ççp ÷ ÷ çè 2 ÷ ø 4. ìï 3x - 1, ïï f (x)= í 1 ïï 2 , ïî x - 1 0 2. 4 x2 - 5 f (x )= 3 x +4 11 12 0 x< 2 x³ 2 1 3 Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 2- 6 Quick Review Solutions In Exercises 5 - 8, write the inequality in the form a < x < b. 5. x<4 - 4< x < 4 6. x < c2 - c2 < x < c2 7. x- 2 < 3 - 1< x < 5 8. x- c < d 2 c- d 2 < x< c + d 2 In Exercises 9 and 10, write the fraction in reduced form. 9. x 2 - 3 x - 18 x+3 x- 6 Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall 2x2 - x 10. 2x2 + x - 1 x x+ 1 Slide 2- 7 What you’ll learn about Average and Instantaneous Speed Definition of Limit Properties of Limits One-Sided and Two-Sided Limits Sandwich Theorem …and why Limits can be used to describe continuity, the derivative and the integral: the ideas giving the foundation of calculus. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 2- 8 Average and Instantaneous Speed A body's average speed during an interval of time is found by dividing the distance covered by the elapsed time. Experiments show that a dense solid object dropped from rest to fall freely near the surface of the earth will fall y 16t 2 feet in the first t seconds. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 2- 9 Definition of Limit Let c and L be real numbers. The function f has limit L as x approaches c if, given any positive number , there is a positive number such that for all x, 0 x c f x L . We write lim f x L x c Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 2- 10 Definition of Limit continued The sentence lim f x L is read, "The limit of f of x as x x c approaches c equals L". the notation means that the values f x of the function f approach or equal L as the values of x approach (but do not equal) c. Figure 2.2 illustrates the fact that the existence of a limit as x c never depends on how the function may or may not be defined at c. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 2- 11 Definition of Limit continued The function f has limit 2 as x 1 even though f is not defined at 1. The function g has limit 2 as x 1 even though g 1 2. The function h is the only one whose limit as x 1 equals its value at x =1. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 2- 12 Properties of Limits If L, M , c, and k are real numbers and lim f x L x c 1. Sum Rule : and lim g x M , then x c lim f x g x L M x c The limit of the sum of two functions is the sum of their limits. 2. DifferenceRule : lim f x g x L M x c The limit of the difference of two functions is the difference of their limits. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 2- 13 Properties of Limits continued 3. Product Rule: lim f x g x L M x c The limit of the product of two functions is the product of their limits. 4. Constant Multiple Rule: lim k f x k L x c The limit of a constant times a function is the constant times the limit of the function. 5. Quotient Rule : lim x c f x g x L , M 0 M The limit of the quotient of two functions is the quotient of their limits, provided the limit of the denominator is not zero. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 2- 14 Properties of Limits continued 6. If r and s are integers, s 0, then Power Rule : r s r s lim f x L x c r s provided that L is a real number. The limit of a rational power of a function is that power of the limit of the function, provided the latter is a real number. Other properties of limits: lim k k x c lim x c x c Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 2- 15 Example Properties of Limits Use any of the properties of limits to find lim 3 x3 2 x 9 x c lim 3x3 2 x 9 lim3x3 lim 2 x lim9 x c x c x c 3c3 2c 9 x c sum and difference rules product and multiple rules Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 2- 16 Polynomial and Rational Functions 1. If f x an x n an1 x n1 ... a0 is any polynomial function and c is any real number, then lim f x f c an c n an1c n1 ... a0 x c 2. If f x and g x are polynomials and c is any real number, then lim x c f x g x f c g c , provided that g c 0. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 2- 17 Example Limits Use Theorem 2 to find lim 4 x2 2 x 6 x5 lim 4 x 2 - 2 x 6 4 5 2 5 6 4 25 10 6 100 10 6 96 2 x 5 Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 2- 18 Evaluating Limits As with polynomials, limits of many familiar functions can be found by substitution at points where they are defined. This includes trigonometric functions, exponential and logarithmic functions, and composites of these functions. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 2- 19 Example Limits Find 1 sin x x 0 cos x lim Solve graphically: The graph of f x 1 sin x suggests that the limit exists and is 1. cos x Confirm Analytically: Find 1 sin x 1 sin 0 1 sin x lim x 0 lim x 0 cos x lim cos x cos 0 x 0 1 0 1 1 Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 2- 20 Example Limits 5 x 0 x Find lim 5 Solve graphically: The graph of f x suggests that x the limit does not exist. [-6,6] by [-10,10] Confirm Analytically : We can't use substitution in this example because when x is relaced by 0, the denominator becomes 0 and the function is undefined. This would suggest that we rely on the graph to see that the limit does not exist. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 2- 21 One-Sided and Two-Sided Limits Sometimes the values of a function f tend to different limits as x approaches a number c from opposite sides. When this happens, we call the limit of f as x approaches c from the right the right-hand limit of f at c and the limit as x approaches c from the left the left-hand limit. right-hand: lim f x The limit of f as x approaches c from the right. x c left-hand: lim f x The limit of f as x approaches c from the left. x c Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 2- 22 One-Sided and Two-Sided Limits continued We sometimes call lim f x the two-sided limit of f at c to distinguish it from x c the one-sided right-hand and left-hand limits of f at c. A function f x has a limit as x approaches c if and only if the right-hand and left-hand limits at c exist and are equal. In symbols, lim f x L lim f x L and x c x c Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall lim f x L. x c Slide 2- 23 Example One-Sided and Two-Sided Limits Find the following limits from the given graph. a. 4 o b. c. 1 2 3 d. e. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall lim f x x 0 0 lim f x Does Not Exist lim f x 4 x2 lim f x Does Not Exist x2 lim f x 0 x 3 x 2 Slide 2- 24 Sandwich Theorem If we cannot find a limit directly, we may be able to find it indirectly with the Sandwich Theorem. The theorem refers to a function f whose values are sandwiched between the values of two other functions, g and h. If g and h have the same limit as x c then f has that limit too. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 2- 25 Sandwich Theorem If g x f x h x for all x c in some interval about c, and lim g x = lim h x = L, x c x c then lim f x = L x c Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 2- 26 2.2 Limits Involving Infinity Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Quick Review In Exercises 1 - 4, find f - 1 and graph f , f - 1 and y = x in the same viewing window. 1. f (x)= 2 x - 3 2. f (x )= e x Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 2- 28 Quick Review 3. f (x)= tan- 1 x 4. f (x)= cot- 1 x Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 2- 29 Quick Review In Exercises 5 and 6, find the quotient q (x) and remainder r (x ) when f (x)is divided by g (x ). 5. f (x)= 2 x 3 - 3x 2 + x - 1, g (x )= 3x 3 + 4 x - 5 6. f (x)= 2 x 5 - x 3 + x - 1, g (x )= x 3 - x 2 + 1 Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 2- 30 Quick Review æ1 ÷ ö ç In Exercises 7 - 10, write a formula for (a ) f (- x ) and (b) f ç ÷ . çè x ÷ ø Simplify where possible. 7. f (x)= cos x 9. f (x)= ln x x 8. f (x )= e- x æ 10. f (x)= ççx + çè Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall 1ö ÷ sin x ÷ ÷ xø Slide 2- 31 Quick Review Solutions In Exercises 1 - 4, find f - 1 and graph f , f - 1 and y = x in the same viewing window. 1. f (x)= 2 x - 3 f - 1 2. x+ 3 (x)= 2 f (x)= e x f - 1 (x)= ln x [-12,12] by [-8,8] Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall [-6,6] by [-4,4] Slide 2- 32 Quick Review Solutions 3. f (x)= tan - 1 x 4. f (x)= cot - 1 x f - 1 (x)= tan x f - 1 (x )= cot x p p - < x< 2 2 0< x < p 3 , 3 by 2 , 2 Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall 0, by 1, Slide 2- 33 Quick Review Solutions In Exercises 5 and 6, find the quotient q (x ) and remainder r (x ) when f (x ) is divided by g (x ). 5. f (x )= 2 x 3 - 3 x 2 + x - 1, 2 q (x )= , 3 6. g (x )= 3x 3 + 4 x - 5 5 7 r (x )= - 3x - x + 3 3 f (x )= 2 x 5 - x 3 + x - 1, q (x )= 2 x 2 + 2 x + 1, 2 g (x)= x3 - x 2 + 1 r (x )= - x 2 - x - 2 Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 2- 34 Quick Review Solutions æ1 ÷ ö ç In Exercises 7 - 10, write a formula for (a ) f (- x ) and (b) f ç ÷ . çè x ÷ ø Simplify where possible. 7. f (x )= cos x f (- x )= cos x, ln x 9. f (x )= x æ 10. f (x )= ççx + çè 8. æ1 ÷ ö 1 ç fç ÷ = cos çè x ÷ ø x f (- x )= - ö 1÷ sin x ÷ ÷ ø x f (x )= e- x æ1 ö÷ - x1 f (- x )= e , f çç ÷ =e èç x ø÷ x ln (- x ) x æ f (- x )= ççx + çè Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall , æ1 ö 1 f çç ÷ = x l n ÷ çè x ÷ ø x ö 1÷ sin x, ÷ ÷ ø x æ1 ÷ ö æ ç fç ÷ = çx + çè x ø÷ èçç 1 ö÷ 1 ÷sin ÷ ø x x Slide 2- 35 What you’ll learn about Finite Limits as x→±∞ Sandwich Theorem Revisited Infinite Limits as x→a End Behavior Models Seeing Limits as x→±∞ …and why Limits can be used to describe the behavior of functions for numbers large in absolute value. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 2- 36 Finite limits as x→±∞ The symbol for infinity (∞) does not represent a real number. We use ∞ to describe the behavior of a function when the values in its domain or range outgrow all finite bounds. For example, when we say “the limit of f as x approaches infinity” we mean the limit of f as x moves increasingly far to the right on the number line. When we say “the limit of f as x approaches negative infinity (- ∞)” we mean the limit of f as x moves increasingly far to the left on the number line. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 2- 37 Horizontal Asymptote The line y b is a horizontal asymptote of the graph of a function y f x if either lim f x b x or Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall lim f x b x Slide 2- 38 Example Horizontal Asymptote Use a graph and tables to find a lim f x and x c Identify all horizontal asymptotes. f x f x . b xlim x 1 x f x 1 a lim x f x 1 b xlim c Identify all horizontal asymptotes. [-6,6] by [-5,5] y 1 Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 2- 39 Example Sandwich Theorem Revisited The sandwich theorem also holds for limits as x . cos x graphically and using a table of values. x x Find lim The graph and table suggest that the function oscillates about the x-axis. cos x 0 x x Thus y 0 is the horizontal asymptote and lim Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 2- 40 Properties of Limits as x→±∞ If L, M and k are real numbers and lim f x L x 1. Sum Rule : and lim g x M , then x lim f x g x L M x The limit of the sum of two functions is the sum of their limits. 2. Difference Rule : lim f x g x L M x The limit of the difference of two functions is the difference of their limits Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 2- 41 Properties of Limits as x→±∞ 3. Product Rule: lim f x g x L M x The limit of the product of two functions is the product of their limits. 4. Constant Multiple Rule: lim k f x k L x The limit of a constant times a function is the constant times the limit of the function. 5. Quotient Rule : lim x f x g x L , M 0 M The limit of the quotient of two functions is the quotient of their limits, provided the limit of the denominator is not zero. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 2- 42 Properties of Limits as x→±∞ 6. If r and s are integers, s 0, then Power Rule : r s r s lim f x L x r s provided that L is a real number. The limit of a rational power of a function is that power of the limit of the function, provided the latter is a real number. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 2- 43 Infinite Limits as x→a If the values of a function f ( x) outgrow all positive bounds as x approaches a finite number a, we say that lim f x . If the values of f become large xa and negative, exceeding all negative bounds as x approaches a finite number a, we say that lim f x . xa Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 2- 44 Vertical Asymptote The line x a is a vertical asymptote of the graph of a function y f x if either lim f x or lim f x x a x a Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 2- 45 Example Vertical Asymptote Find the vertical asymptotes of the graph of f ( x) and describe the behavior of f ( x) to the right and left of each vertical asymptote. 8 f x 4 x2 The values of the function approach to the left of x 2. The values of the function approach + to the right of x 2. The values of the function approach + to the left of x 2. The values of the function approach to the right of x 2. 8 8 lim and lim 2 2 x 2 4 x x 2 4 x 8 8 lim and lim 2 2 x2 4 x x 2 4 x So, the vertical asymptotes are x 2 and x 2 [-6,6] by [-6,6] Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 2- 46 End Behavior Models The function g is a a right end behavior model for f if and only if lim x b a left end behavior model for f if and only if lim x Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall f x g x f x g x 1. 1. Slide 2- 47 Example End Behavior Models Find an end behavior model for 3x 2 2 x 5 f x 4 x2 7 Notice that 3 x 2 is an end behavior model for the numerator of f , and 4 x 2 is one for the denominator. This makes 3x 2 3 = an end behavior model for f . 2 4x 4 Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 2- 48 End Behavior Models If one function provides both a left and right end behavior model, it is simply called an end behavior model. In general, g x an x n is an end behavior model for the polynomial function f x an x n an 1 x n 1 ... a0 , an 0 Overall, all polynomials behave like monomials. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 2- 49 End Behavior Models 3 is also a horizontal 4 asymptote of the graph of f . We can use the end behavior model of a rational function to identify any horizontal asymptote. A rational function always has a simple power function as In this example, the end behavior model for f , y an end behavior model. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 2- 50 Example “Seeing” Limits as x→±∞ We can investigate the graph of y f x as x by investigating the 1 graph of y f as x 0. x 1 Use the graph of y f to find lim f x and lim f x x x x 1 for f x x cos . x cos x 1 The graph of y f = is shown. x x 1 lim f x lim f x x 0 x 1 lim f x lim f x x 0 x Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 2- 51 Quick Quiz Sections 2.1 and 2.2 You may use a graphing calculator to solve the following problems. 1. A B C D E x2 x 6 Find lim if it exists x 3 x3 1 1 2 5 does not exist Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 2- 52 Quick Quiz Sections 2.1 and 2.2 You may use a graphing calculator to solve the following problems. 1. A B C D E x2 x 6 Find lim if it exists x 3 x3 1 1 2 5 does not exist Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 2- 53 Quick Quiz Sections 2.1 and 2.2 2. A B C D E 3 x 1, Find lim f x = 5 x2 x 1 , 5 3 13 3 7 does not exist x2 x2 if it exists Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 2- 54 Quick Quiz Sections 2.1 and 2.2 2. A B C D E 3 x 1, Find lim f x = 5 x2 x 1 , 5 3 13 3 7 does not exist x2 x2 if it exists Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 2- 55 Quick Quiz Sections 2.1 and 2.2 3. Which of the following lines is a horizontal asymptote for 3x3 x 2 x 7 f x 2 x3 4 x 5 3 A y x 2 B y 0 C y D E 2 3 7 5 3 y 2 y Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 2- 56 Quick Quiz Sections 2.1 and 2.2 3. Which of the following lines is a horizontal asymptote for 3x3 x 2 x 7 f x 2 x3 4 x 5 3 A y x 2 B y 0 2 3 C y D 7 5 3 y 2 E y Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 2- 57 2.3 Continuity Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Quick Review 1. 3x 2 - 2 x + 1 Find lim x® - 1 x3 + 4 2. Let f (x )= int x. Find each limit. f (x ) (a ) xlim ®-1 - f (x ) (b) xlim ®-1 + f (x ) (c) xlim ®-1 3. ìï x 2 - 4 x + 5, Let f (x )= ïí ïïî 4 - x, Find each limit. (d ) f (- 1) x< 2 x³ 2 f (x ) (a ) xlim ®2 f (x ) (b) xlim ®2 f (x ) (c) lim x® 2 (d ) f (2) - + Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 2- 59 Quick Review In Exercises 4 - 6, find the remaining functions in the list of functions: f , g, f o g, g o f . 2x - 1 1 4. f (x)= , g (x)= + 1 x+ 5 x 5. f (x)= x 2 , (g o f )(x)= sin x 2 , domain of g = [0, ¥ ) Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 2- 60 Quick Review 7. 1 g (x)= x - 1, (g o f )(x)= , x > 0 x Use factoring to solve 2 x 2 + 9 x - 5 = 0 8. Use graphing to solve 6. x3 + 2 x - 1= 0 Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 2- 61 Quick Review ìï 5 - x, x£ 3 ï In Exercises 9 and 10, let f (x)= í ïïî - x 2 + 6 x - 8, x > 3 9. Solve the equation f (x)= 4 10. Find a value of c for which the equation f (x)= c has no solution. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 2- 62 Quick Review Solutions 1. 3x 2 - 2 x + 1 Find lim x® - 1 x3 + 4 2. Let f (x )= int x. Find each limit. f (x ) (a ) xlim ®-1 - 2 f (x ) - 1 (b) xlim ®-1 f (x ) (c) xlim ®-1 no limit (d ) f (- 1) - 1 - 3. 2 + ìï x 2 - 4 x + 5, Let f (x )= ïí ïîï 4 - x, Find each limit. f (x ) (a ) xlim ®2 - f (x ) (c) lim x® 2 x< 2 x³ 2 1 no limit f (x ) (b) xlim ®2 + (d ) f (2) Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall 2 2 Slide 2- 63 Quick Review Solutions In Exercises 4 - 6, find the remaining functions in the list of functions: f , g, f o g, g o f . 4. f (x )= 2x- 1 1 , g (x )= + 1 x+ 5 x ( f o g )(x) = 5. x+ 2 , 6x + 1 x¹ 0 (g o f )(x ) = 3x + 4 , 2x- 1 x¹ 5 f (x )= x 2 , (g o f )(x )= sin x 2 , domain of g = [0, ¥ ) g (x )= sin x, x ³ 0 ( f o g )(x)= sin 2 x, x ³ 0 Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 2- 64 Quick Review Solutions 6. g (x)= x - 1, f (x)= 1 + 1, 2 x 1 (g o f )(x)= , x x> 0 x> 0 ( f o g )(x)= 7. Use factoring to solve 2 x2 + 9 x - 5 = 0 8. Use graphing to solve x3 + 2 x - 1= 0 Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall x , x- 1 x= x> 1 1 , - 5 2 x » 0.453 Slide 2- 65 Quick Review Solutions ìï 5 - x, x£ 3 ï In Exercises 9 and 10, let f (x)= í 2 ïïî - x + 6 x - 8, x > 3 9. Solve the equation f (x)= 4 x= 1 10. Find a value of c for which the equation f (x)= c has no solution. Any c in [1, 2) Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 2- 66 What you’ll learn about Continuity at a Point Continuous Functions Algebraic Combinations Composites Intermediate Value Theorem for Continuous Functions …and why Continuous functions are used to describe how a body moves through space and how the speed of a chemical reaction changes with time. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 2- 67 Continuity at a Point Any function y f x whose graph can be sketched in one continuous motion without lifting the pencil is an example of a continuous function. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 2- 68 Example Continuity at a Point Find the points at which the given function is continuous and the points at which it is discontinuous. o Points at which f is continuous At x 0 At x 6 lim f x f 0 x 0 lim f x f 6 x 6 At 0 < c < 6 but not 2 c 3 lim f x f c x c Points at which f is discontinuous At x 2 lim f x does not exist x2 At c 0, 2 c 3, c 6 these points are not in the domain of f Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 2- 69 Continuity at a Point Interior Point: A function y f x is continuous at an interior point c of its domain if lim f x f c x c Endpoint: A function y f x is continuous at a left endpoint a or is continuous at a right endpoint b of its domain if lim f x f a x a or lim f x f b x b Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall respectively. Slide 2- 70 Continuity at a Point If a function f is not continuous at a point c , we say that f is discontinuous at c and c is a point of discontinuity of f. Note that c need not be in the domain of f. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 2- 71 Continuity at a Point The typical discontinuity types are: a) Removable (2.21b and 2.21c) b) Jump (2.21d) c) Infinite (2.21e) d) Oscillating (2.21f) Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 2- 72 Continuity at a Point Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 2- 73 Example Continuity at a Point Find and identify the points of discontinuity of y 3 x 1 2 There is an infinite discontinuity at x 1. [-5,5] by [-5,10] Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 2- 74 Continuous Functions A function is continuous on an interval if and only if it is continuous at every point of the interval. A continuous function is one that is continuous at every point of its domain. A continuous function need not be continuous on every interval. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 2- 75 Continuous Functions The given function is a continuous function because it is continuous at every point of its domain. It does have a point of discontinuity at x 2 because it is not defined there. y 2 x 2 2 [-5,5] by [-5,10] Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 2- 76 Properties of Continuous Functions If the functions f and g are continuous at x c, then the following combinations are continuous at x c. 1. Sums : f g 2. Differences: f g 3. 4. Products: Constant multiples: f g k f , for any number k 5. Quotients: f , provided g c 0 g Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 2- 77 Composite of Continuous Functions If f is continuous at c and g is continuous at f c , then the composite g f is continuous at c. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 2- 78 Intermediate Value Theorem for Continuous Functions A function y f x that is continuous on a closed interval [a, b] takes on every value between f a and f b . In other words, if y0 is between f a and f b , then y0 f c for some c in [a, b]. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 2- 79 Intermediate Value Theorem for Continuous Functions The Intermediate Value Theorem for Continuous Functions is the reason why the graph of a function continuous on an interval cannot have any breaks. The graph will be connected, a single, unbroken curve. It will not have jumps or separate branches. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 2- 80 2.4 Rates of Change and Tangent Lines Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Quick Review In Exercises 1 and 2, find the increments D x and D y from point A to point B. 1. A(- 5, 2), B (3,5) 2. A(1,3), B (a, b) In Exercises 3 and 4, find the slope of the line determined by the points. 3. (- 2,3), (5, - 1) 4. (- 3, - 1), Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall (3,3) Slide 2- 82 Quick Review In Exercises 5 - 9, write an equation for the specified line. 3 2 5. through (- 2,3) with slope = 6. through (1,6) and (4, - 1) 7. through (1, 4) and parallel to y = - 3 x+ 2 4 Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 2- 83 Quick Review 3 x+ 2 4 8. through (1, 4) and perpendicular to y = - 9. through (- 1,3) and parallel to 2 x + 3 y = 5 10. For what value of b will the slope of the line through (2,3) and (4, b) be 5 ? 3 Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 2- 84 Quick Review Solutions In Exercises 1 and 2, find the increments D x and D y from point A to point B. 1. A(- 5, 2), B (3,5) 2. D x = 8, D y = 3 D x = a - 1, A(1,3), B (a, b) D y = b- 3 In Exercises 3 and 4, find the slope of the line determined by the points. 3. (- 2,3), (5, - 1) - 4 7 4. (- 3, - 1), (3,3) 2 3 Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 2- 85 Quick Review Solutions In Exercises 5 - 9, write an equation for the specified line. 3 2 5. through (- 2,3) with slope = 6. through (1,6) and (4, - 1) 7. through (1, 4) and parallel to y = - y= y= - 3 x+ 2 4 Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall 3 x+ 6 2 7 25 x+ 3 3 y= - 3 19 x+ 4 4 Slide 2- 86 Quick Review Solutions 8. through (1, 4) and perpendicular to y = - 3 x+ 2 4 4 8 y= x+ 3 3 9. through (- 1,3) and parallel to 2 x + 3 y = 5 2 7 x+ 3 3 10. For what value of b will the slope of the line through (2,3) y= - and (4, b) be 5 ? 3 b= 19 3 Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 2- 87 What you’ll learn about Average Rates of Change Tangent to a Curve Slope of a Curve Normal to a Curve Speed Revisited …and why The tangent line determines the direction of a body’s motion at every point along its path. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 2- 88 Average Rates of Change The average rate of change of a quantity over a period of time is the amount of change divided by the time it takes. In general, the average rate of change of a function over an interval is the amount of change divided by the length of the interval. Also, the average rate of change can be thought of as the slope of a secant line to a curve. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 2- 89 Example Average Rates of Change Find the average rate of change of f x 2 x 2 3 x 7 over the interval -2,4 f 2 f 4 2 2 3 2 7 2 4 3 4 7 2 4 21 27 6 1 6 6 2 2 2 4 Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 2- 90 Tangent to a Curve In calculus, we often want to define the rate at which the value of a function y = f(x) is changing with respect to x at any particular value x = a to be the slope of the tangent to the curve y = f(x) at x = a. The problem with this is that we only have one point and our usual definition of slope requires two points. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 2- 91 Tangent to a Curve The process becomes: 1. Start with what can be calculated, namely, the slope of a secant through P and a point Q nearby on the curve. 2. Find the limiting value of the secant slope (if it exists) as Q approaches P along the curve. 3. Define the slope of the curve at P to be this number and define the tangent to the curve at P to be the line through P with this slope. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 2- 92 Example Tangent to a Curve Given y x 2 2 at x 1 find: the slope of the curve and an equation of the tangent line. Then draw a graph of the curve and tangent line in the same viewing window. a Write an expression for the slope of the secant line and find the limiting value of the slope as Q approaches P along the curve. When x 1, y x 2 2 3 so =P 1,3 1 h 2 2 1 2 y 1 h y 1 lim lim h 0 h 0 h h h h 2 3 2h h 2 3 h 2 2h lim lim lim lim h 2 2 h 0 h 0 h 0 h 0 h h h 2 Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 2- 93 Example Tangent to a Curve b The tangent line has slope 2 and passes through 1,3 . The equation of the tangent line is y 3 2 x 1 y 2 x 1 3 curve y 2x 2 3 y 2x 1 y x2 2 tangent y 2x 1 Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 2- 94 Slope of a Curve To find the tangent to a curve y = f(x) at a point P(a,f(a)) calculate the slope of the secant line through P and a point Q(a+h, f(a+h)). Next, investigate the limit of the slope as h→0. If the limit exists, it is the slope of the curve at P and we define the tangent at P to be the line through P with this slope. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 2- 95 Slope of a Curve Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 2- 96 Slope of a Curve at a Point The slope of the curve y f x at the point P a, f a is the number m lim h 0 f a h f a h provided the limit exists. The tangent line to the curve at P is the line through P with this slope. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 2- 97 Slope of a Curve All of the following mean the same: 1. the slope of y f ( x) at x a 2. the slope of the tangent to y f ( x ) at x a 3. the (instantaneous) rate of change of f ( x) with respect to x at x a 4. lim h 0 f a h f a h The expression f a h f a h is the difference quotient of f at a. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 2- 98 Normal to a Curve The normal line to a curve at a point is the line perpendicular to the tangent at the point. The slope of the normal line is the negative reciprocal of the slope of the tangent line. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 2- 99 Example Normal to a Curve Given y x 2 2 at x 1 write the equation of the normal line. Draw a graph of the curve, the tangent line and the normal line in the same viewing window. From an earlier example, the slope of the tangent line was found to be 2 so the slope of the normal is y 3 1 x 1 2 1 y x 1 3 2 1 1 6 y x 2 2 2 1 7 y x 2 2 1 . 2 tangent y 2x 1 Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall curve y x2 2 normal line 1 7 y x 2 2 Slide 2- 100 Speed Revisited The function y 16t 2 is an object's position function. An object's average speed along a coordinate axis for a given period of time is the average rate of change of its position y f (t ). It's instantaneous speed at any time t is the instantaneous rate of change of position with respect to time at time t , or lim h 0 Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall f t h f t h . Slide 2- 101 Quick Quiz Sections 2.3 and 2.4 You may use a graphing calculator to solve the following problems. 1. Which of the following values is the average rate of change of f x x 1 over the interval 0,3 ? A B 1 C 3 1 3 1 3 D E 3 Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 2- 102 Quick Quiz Sections 2.3 and 2.4 You may use a graphing calculator to solve the following problems. 1. Which of the following values is the average rate of change of f x x 1 over the interval 0,3 ? A B 3 1 1 C 3 1 D 3 E 3 Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 2- 103 Quick Quiz Sections 2.3 and 2.4 2. Which of the following statements is false for the function 3 0 x4 4 x, x4 f x 2, x 7, 4 x6 6 x8 ? 1, f x exists A lim x4 B f 4 exists f x exists C lim x 6 f x exists D xlim 8 E f is continuous at x 4 Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 2- 104 Quick Quiz Sections 2.3 and 2.4 2. Which of the following statements is false for the function 3 0 x4 4 x, x4 f x 2, x 7, 4 x6 6 x8 ? 1, f x exists A lim x4 B f 4 exists f x exists C lim x 6 f x exists D xlim 8 E f is continuous at x 4 Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 2- 105 Quick Quiz Sections 2.3 and 2.4 3. Which of the following is an equation for the tangent line to f x 9 x 2 at x 2? 1 9 A y x 4 2 B y 4 x 13 C D E y 4x 3 y 4x 3 y 4 x 13 Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 2- 106 Quick Quiz Sections 2.3 and 2.4 3. Which of the following is an equation for the tangent line to f x 9 x 2 at x 2? 1 9 A y x 4 2 B y 4 x 13 C D E y 4x 3 y 4x 3 y 4 x 13 Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 2- 107 Chapter Test In Exercises 1 and 2, find the limits. 1 1 2+ x 2 1. lim 2. x® 0 x lim x® ¥ x + sin x x + cos x In Exercises 3 and 4, determine whether the limit exists on the basis of the graph of y = f (x ). The domain of f is the set of real numbers. 3. lim f (x ) x ® c- 4. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall lim f (x ) x® b Slide 2- 108 Chapter Test In Exercise 5, use the graph of the function with domain - 1£ x £ 3. 5. Determine g (x ) (a ) xlim ®3 - (b) g (3). (c) whether g (x) is continuous at x = 3. (d ) the points of discontinuity of g (x). (e) Writing to Learn whether any points of discontinuity are removable. If so, describe the extended function. If not, explain why not. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 2- 109 Chapter Test In Exercise 6, (a ) find the vertical asymptotes of the graph of y = f (x ), and (b) describe the behavior of f (x ) to the right and left of any vertical asymptote. 6. x- 1 f (x)= 2 x (x + 2) Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 2- 110 Chapter Test 7. Given ìï 1, x £ - 1 ïï ïï - x, - 1 < x < 0 ï f (x )= ïí 1, x = 0 ïï ïï - x, 0 < x < 1 ïï ïî 1, x ³ 1 (a ) Find the right-hand and left-hand limits of f at x = - 1, 0 and 1. (b) Does f have a limit as x approaches - 1? 0? 1? If so, what is it? If not, why not? (c) Is f continuous at x= - 1? 0? 1? Explain. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 2- 111 Chapter Test In Exercise 8, find (a ) a power function end behavior model and (b) any horizontal asymptotes. 2x + 1 x2 - 2x + 1 8. f (x)= 9. Find the average rate of change of f (x)= 1+ sin x over the interval [0, p ]. 2 Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 2- 112 Chapter Test 10. Let f (x)= x 2 - 3x and P = (1, f (1)). Find (a ) the slope of the curve y = f (x ) at P, (b) an equation of the tangent at P and (c) an equation of the normal at P. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 2- 113 Chapter Test Solutions In Exercises 1 and 2, find the limits. 1. 1 1 2+ x 2 lim x® 0 x - 1 4 2. lim x® ¥ x + sin x x + cos x 1 In Exercises 3 and 4, determine whether the limit exists on the basis of the graph of y = f (x ). The domain of f is the set of real numbers. 3. lim f (x ) x ® c- Exists Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall 4. lim f (x ) x® b Exists Slide 2- 114 Chapter Test Solutions In Exercise 5, use the graph of the function with domain - 1£ x £ 3. 5. Determine g (x ) (a ) xlim ®3 - 1 (b) g (3). (c) whether g (x) is continuous at x = 3. (d ) the points of discontinuity of g (x). 1.5 No at x = 3 and points not in the domain (e) Writing to Learn whether any points of discontinuity are removable. If so, describe the extended function. If not, explain why not. Removable at x = 3 by assigning the value 1 to g (3). Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 2- 115 Chapter Test Solutions In Exercise 6, (a ) find the vertical asymptotes of the graph of y = f (x ), and (b) describe the behavior of f (x ) to the right and left of any vertical asymptote. x- 1 6. f (x )= 2 vertical asymptotes: x = 0, x = - 2 x (x + 2 ) At x = 0 Left-hand limit = limx® 0 x- 1 2 =- ¥ x ( x + 2) Right-hand limit = lim+ x® 0 x- 1 2 =- ¥ x ( x + 2) At x = - 2 Left-hand limit = limx® 2 x- 1 2 x ( x + 2) =¥ Right-hand limit = lim+ Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall x® 2 x- 1 2 =- ¥ x ( x + 2) Slide 2- 116 Chapter Test Solutions 7. Given ìï 1, x £ - 1 ïï ïï - x, - 1 < x < 0 ï f (x )= ïí 1, x = 0 ïï ïï - x, 0 < x < 1 ïï ïî 1, x ³ 1 (a ) Find the right-hand and left-hand limits of f at x = - 1, 0 and 1. (b) Does f have a limit as x approaches - 1? 0? 1? If so, what is it? If not, why not? (c) Is f continuous at x= - 1? 0? 1? Explain. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 2- 117 Chapter Test Solutions 7. a At x 1: Left-hand limit = lim f x lim 1 1 x 1 x 1 Right-hand limit = lim f x lim x 1 x 1 x 1 At x 0 : Left-hand limit = lim f x lim x 0 x 0 x 0 Right-hand limit = lim f x lim x 0 x 0 x 0 At x 1 Left-hand limit = lim f x lim x 1 x 1 x 1 Right-hand limit = lim f x lim 1 1 x 1 x 1 Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 2- 118 Chapter Test Solutions 7. b At x 1: Yes, the limit is 1. At x 0 : Yes, the limit is 0. At x 1: No, the limit doesn't exist because the two one-sided limits are different. c At x 1: Continuous because f 1 the limit. At x 0 : Discontinuous because f 0 the limit. At x 1: Discontinuous because the limit does not exist. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 2- 119 Chapter Test Solutions In Exercise 8, find (a ) a power function end behavior model and (b) any horizontal asymptotes. 2x + 1 x2 - 2x + 1 (a ) 2 x (b) y = 0 8. f (x)= 9. Find the average rate of change of f (x )= 1+ sin x é pù over the interval ê0, ú. êë 2 ú û 2 p Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 2- 120 Chapter Test Solutions 10. Let f (x)= x 2 - 3x and P = (1, f (1)). Find (a ) the slope of the curve y = f (x ) at P, (b) an equation of the tangent at P and (c) an equation of the normal at P. (a ) m = - 1 (b) y = - x - 1 Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall (c) y = x - 3 Slide 2- 121