Chapter 3 Limits and the Derivative Section 3 Continuity (Part 1)

Learning Objectives for Section 3.3 Continuity The student will understand the concept of continuity The student will be able to apply the continuity properties The student will be able to solve word problems. The student will be able to solve inequalities Barnett/Ziegler/Byleen Business Calculus 12e

Continuity In this lesson, we’ll take a closer look at graphs that are discontinuous due to: Holes Gaps Asymptotes Barnett/Ziegler/Byleen Business Calculus 12e

Definition of Continuity A function f is continuous at a point x = c if it meets these three criteria: 1. 2. f (c)  undefined 3. lim 𝑥→𝑐 𝑓(𝑥)≠𝐷𝑁𝐸 Barnett/Ziegler/Byleen Business Calculus 12e

Example 1 Is f(x) continuous at x = 2 ? ? lim 𝑥→2 𝑓 𝑥 =𝑓(2) 𝐷𝑁𝐸 𝑢𝑛𝑑𝑒𝑓𝑖𝑛𝑒𝑑 f(x) is not continuous at x = 2 Is f(x) continuous at x = 3 ? ? lim 𝑥→3 𝑓 𝑥 =𝑓(3) 1 𝑢𝑛𝑑𝑒𝑓𝑖𝑛𝑒𝑑 f(x) is not continuous at x = 3 Continuous over the interval: −∞, 2 ∪(2, 3)∪(3,∞) Barnett/Ziegler/Byleen Business Calculus 12e

Example 2 Is f(x) continuous at x = -2 ? ? lim 𝑥→−2 𝑓 𝑥 =𝑓(−2) 3 −1 f(x) is not continuous at x = -2 Continuous over the interval: −∞, −2 ∪(−2,∞) Barnett/Ziegler/Byleen Business Calculus 12e

Example 3 𝑓 𝑥 = 2𝑥 2 +𝑥−1 Is f(x) continuous at x = 3? ? lim 𝑥→3 𝑓 𝑥 =𝑓(3) 20 20 f(x) is continuous at x = 3 Barnett/Ziegler/Byleen Business Calculus 12e

Example 4 𝑓 𝑥 = 𝑥 2 −4 𝑥+2 Is f(x) continuous at x = -2? ? 𝑓 𝑥 = 𝑥 2 −4 𝑥+2 Is f(x) continuous at x = -2? lim 𝑥→−2 𝑥 2 −4 𝑥+2 =𝑓(−2) ? 1. lim 𝑥→−2 𝑥 2 −4 𝑥+2 = lim 𝑥→−2 (𝑥+2)(𝑥−2) 𝑥+2 = lim 𝑥→−2 (𝑥−2) = −4 2. 𝑓 −2 = 𝑢𝑛𝑑𝑒𝑓𝑖𝑛𝑒𝑑 f(x) is NOT continuous at x = -2 Barnett/Ziegler/Byleen Business Calculus 12e

Example 5 𝑓 𝑥 = 𝑥−5 𝑥−5 Is f(x) continuous at x = 5? Explain. 𝑓 𝑥 = 𝑥−5 𝑥−5 Is f(x) continuous at x = 5? Explain. lim 𝑥→5 𝑥−5 𝑥−5 =𝑓(5) ? lim 𝑥→ 5 − 𝑥−5 𝑥−5 = lim 𝑥→ 5 − −(𝑥−5) 𝑥−5 = −1 1. lim 𝑥→5 𝑥−5 𝑥−5 lim 𝑥→ 5 + 𝑥−5 𝑥−5 = lim 𝑥→ 5 + (𝑥−5) 𝑥−5 = 1 lim 𝑥→5 𝑓 𝑥 =𝐷𝑁𝐸 2. 𝑓 5 = 𝑢𝑛𝑑𝑒𝑓𝑖𝑛𝑒𝑑 f(x) is NOT continuous at x = 5 Barnett/Ziegler/Byleen Business Calculus 12e

Example 6 𝑓 𝑥 = 𝑥−5 𝑥−5 Is f(x) continuous at x = 2? 𝑓 𝑥 = 𝑥−5 𝑥−5 Is f(x) continuous at x = 2? lim 𝑥→2 𝑥−5 𝑥−5 =𝑓(2) ? 1. lim 𝑥→2 𝑥−5 𝑥−5 = 2−5 2−5 = −1 2. 𝑓 2 = −1 f(x) is continuous at x = 2 Barnett/Ziegler/Byleen Business Calculus 12e

Continuity Where is a graph continuous? Where there are no asymptotes or holes. Where the function is defined. Barnett/Ziegler/Byleen Business Calculus 12e

Example 7 Where is 𝑓 𝑥 = 2𝑥+6 continuous? 2𝑥+6≥0 𝑥≥−3 𝑓(𝑥) is continuous over the interval: [−3 ,∞) Barnett/Ziegler/Byleen Business Calculus 12e

Example 8 Where is 𝑓 𝑥 = (𝑥+1)(𝑥−4) (𝑥+1)(𝑥−2) continuous? x≠−1, 2 𝑓(𝑥) is continuous over the interval: (−∞,−1)∪(−1,2)∪(2 ,∞) Barnett/Ziegler/Byleen Business Calculus 12e

Example 9 Where is 𝑓 𝑥 continuous? 𝑓(𝑥) is continuous over the interval: (−∞,−2)∪(−2,1)∪(1,2)∪(2 ,∞) Barnett/Ziegler/Byleen Business Calculus 12e

#3-3A Pg 161 (7-14, 23, 27, 29, 31, 49-59 odd) Homework Barnett/Ziegler/Byleen Business Calculus 12e

Chapter 3 Limits and the Derivative Section 3 Continuity (Part 2)

Learning Objectives for Section 3.3 Continuity The student will understand the concept of continuity The student will be able to apply the continuity properties The student will be able to solve word problems. The student will be able to solve inequalities Barnett/Ziegler/Byleen Business Calculus 12e

Application: Media A music website called MyTunes charges $0.99 per song if you download less than 100 songs per month and $0.89 per song if you download 100 or more songs per month. Write a piecewise function f(x) for the cost of downloading x songs per month. Graph the function. Is f(x) continuous at x = 100? Explain. Barnett/Ziegler/Byleen Business Calculus 12e

Solution 𝑓 𝑥 = 0.99𝑥 0≤𝑥<100 0.89𝑥 𝑥≥100 ? lim 𝑥→100 𝑓 𝑥 =𝑓(100) 𝑓 𝑥 = 0.99𝑥 0≤𝑥<100 0.89𝑥 𝑥≥100 ? $99 $89 $79 100 110 lim 𝑥→100 𝑓 𝑥 =𝑓(100) 𝐷𝑁𝐸 89 f(x) is NOT continuous at x=100 Barnett/Ziegler/Byleen Business Calculus 12e

Application: Natural Gas Rates The table shows the monthly rates for natural gas charged by MyGas Company. The charge is based on the number of therms used per month. (1 therm = 100,000 Btu) Write a piecewise function f(x) of the monthly charge for x therms. Graph f(x). Is f(x) continuous at x = 40? Give a mathematical reason. Amount Cost Base Charge $8.00 First 40 therms $0.60 per therm Over 40 therms $0.35 per therm Barnett/Ziegler/Byleen Business Calculus 12e

Solution 𝑓 𝑥 = 8+0.60𝑥 0≤𝑥≤40 8+.60 40 +.35(𝑥−40) 𝑥>40 Amount Cost Base Charge $8.00 First 40 therms $0.60 per therm Over 40 therms $0.35 per therm 𝑓 𝑥 = 8+0.60𝑥 0≤𝑥≤40 8+.60 40 +.35(𝑥−40) 𝑥>40 = 8+0.60𝑥 0≤𝑥≤40 .35𝑥+18 𝑥>40 Barnett/Ziegler/Byleen Business Calculus 12e

Solution 𝑓(𝑥)= 8+0.60𝑥 0≤𝑥≤40 .35𝑥+18 𝑥>40 ? lim 𝑥→40 𝑓 𝑥 =𝑓(40) 32 𝑓(𝑥)= 8+0.60𝑥 0≤𝑥≤40 .35𝑥+18 𝑥>40 $90 $60 $30 40 80 ? lim 𝑥→40 𝑓 𝑥 =𝑓(40) 32 32 f(x) IS continuous at x=40 Barnett/Ziegler/Byleen Business Calculus 12e

Solving Inequalities Up until now, we have solved inequalities using a graphical approach. 𝑓 𝑥 <0  Where is the graph below the x-axis? 𝑓 𝑥 >0  Where is the graph above the x-axis? Now we will learn an algebraic approach that is based on continuity properties. Barnett/Ziegler/Byleen Business Calculus 12e

Constructing Sign Charts Find all numbers which are: a. Holes or vertical asymptotes. Plot these as open circles on the number line. b. x-intercepts Plot these according to the inequality symbol. Select a test number in each interval and determine if f (x) is positive (+) or negative (–) in the interval. Determine your answer using the signs and the inequality symbol and write it using interval notation. Barnett/Ziegler/Byleen Business Calculus 12e

Polynomial Inequalities Ex 3: Solve and write your answer in interval notation. 𝑥 2 −4𝑥−12>0 Factor f(x) and solve for zeros. (x−6)(x+2)>0 Graph the zeros on a number line. Use open circles for < or >, use closed circles for  or . + - + 6 −2 Test numbers on all sides of the zeros by plugging them into the inequality. 𝐴𝑛𝑠𝑤𝑒𝑟:(−∞,−2)∪(6,∞) Since f(x) > 0 we want the positive intervals. Barnett/Ziegler/Byleen Business Calculus 12e

Rational Inequalities Ex 4: Solve and write your answer in interval notation. 𝑥 2 −4 𝑥+4 ≤0 Factor the top and bottom. Graph the holes and vertical asymptotes as open circles. (𝑥+2)(𝑥−2) 𝑥+4 ≤0 Graph the x-intercepts. Use open circles for < or >, use closed circles for  or . − + − + Test numbers on all sides of the points by plugging them into the reduced inequality. 2 −2 −4 𝐴𝑛𝑠𝑤𝑒𝑟: −∞,−4 ∪[−2,2] Since f(x)  0, we want the negative intervals. Barnett/Ziegler/Byleen Business Calculus 12e

Homework #3-3B Pg. 162 (36-43, 45, 81, 85, 86) Barnett/Ziegler/Byleen Business Calculus 12e