MATH 234, Calculus for BusinessLecture 6, Textbook Sections 3.2, 3.3Continuity and Rates of ChangeUniversity of Illinois, Urbana-ChampaignFebruary 6th, 2017Continuity and Rates of Change (University of Illinois, Urbana-Champaign)MATH 234, Calculus for Business February 6th, 2017 1 / 23AnnouncementsQuiz 3 is tomorrow in section. The coverage is Lectures 4 and 5;GWs 4 and 5; HW2 problems 7-10, HW3 problems 1- 6.HW 3 is due Thursday, 2/9, 11:59pm; late HWs are not accepted.Information for Exam 1 will be posted later this week (stay tuned).Continuity and Rates of Change (University of Illinois, Urbana-Champaign)MATH 234, Calculus for Business February 6th, 2017 2 / 23ReviewExampleCompute limx→−∞2x3+ x − 43x2− xSolutionlimx→−∞2x3+ x − 43x2− x= limx→−∞2x3+ x − 43x2− x∗x−2x−2= limx→−∞2x + x−1− 4x−23 −x−1=limx→−∞2x + x−1− 4x−2limx→−∞3 −x−1= −∞since numerator approaches −∞, while denominator approaches 3. Thus,limx→−∞2x3+ x − 43x2− xdoes not existContinuity and Rates of Change (University of Illinois, Urbana-Champaign)MATH 234, Calculus for Business February 6th, 2017 3 / 23ReviewExampleCompute limh→0(5 + h)2− 52hSolutionlimh→0(5 + h)2− 52h= limh→0(25 + 10h + h2) −25h= limh→010h + h2h= limh→010 + h = 10Continuity and Rates of Change (University of Illinois, Urbana-Champaign)MATH 234, Calculus for Business February 6th, 2017 4 / 23Section 3.2, Continuity Definition and ExamplesDefinitionA function f (x) is continuous at x === c if it satisfies all three conditions:(1) f (c) is defined (which is to say, x = c is in the domain of f (x))(2) limx→cf (x) exists(3) limx→cf (x) = f (c)If any of these conditions fail, then f (x) is discontinuous at x === c.ExampleIs f (x) = x2continuous at x = 1?Solution (Check all three conditions)(Condition 1) f (1) is defined; indeed, f (1) = 12= 1.(Condition 2) Since f (x) = x2is a polynomial, limx→1f (x) exists.(Condition 3) Limit laws says limx→1f (x) = f (1).All 3 conditions hold, so f (x) is continuous at x = 1.Continuity and Rates of Change (University of Illinois, Urbana-Champaign)MATH 234, Calculus for Business February 6th, 2017 5 / 23Section 3.2, Continuity Definition and ExamplesDefinition (... from previous slide)Continuity of f (x) at x = c requires all of the conditions to be true:(1) f (c) is defined, (2) limx→cf (x) exists, and (3) limx→cf (x) = f (c)ExampleLet f (x) =√x. Is f (x) continuous at x = −3?Solutionf (−3) =√−3 is undefined: Condition 1 fails.So f (x) is discontinuous at x = −3.Continuity and Rates of Change (University of Illinois, Urbana-Champaign)MATH 234, Calculus for Business February 6th, 2017 6 / 23Section 3.2, Continuity Definition and ExamplesDefinition (... from previous slide)Continuity of f (x) at x = c requires all of the conditions to be true:(1) f (c) is defined, (2) limx→cf (x) exists, and (3) limx→cf (x) = f (c)ExampleConsider the piece-wise functionf (x) =x2− 2 if x < 2x −2 if x ≥ 2Is f (x) continuous at x = 2?Solutionf (2) is defined (indeed, f (2) = 0). Since the left and right limitslimx→2−f (x) = 2 and limx→2+f (x) = 0are different, therefore limx→2f (x) does not exist (Condition 2 fails).So, f (x) is discontinuous at x = 2.Continuity and Rates of Change (University of Illinois, Urbana-Champaign)MATH 234, Calculus for Business February 6th, 2017 7 / 23Section 3.2, Continuity Definition and ExamplesDefinition (... from previous slide)Continuity of f (x) at x = c requires all of the conditions to be true:(1) f (c) is defined, (2) limx→cf (x) exists, and (3) limx→cf (x) = f (c)ExampleConsider the piece-wise functionf (x) =x2if x 6= 13 if x = 1Is f (x) continuous at x = 1?Solutionf (1) is defined (indeed, f (1) = 3). Also, limx→1f (x) = 1. Butlimx→1f (x) 6= f (1) Condition 3 fails.So, f (x) is discontinuous at x = 1; this is a removable discontinuity,(i.e., it is possible to change the value of f (1) to get continuity at x = 1)Continuity and Rates of Change (University of Illinois, Urbana-Champaign)MATH 234, Calculus for Business February 6th, 2017 8 / 23Section 3.2, Continuity Definition and ExamplesTheoremFunction type General Form Example Continuous for...Polynomial anxn+ an−1xn−1+ 3x4+ 5 all xfunction ··· + a1x + a0Rational r(x) =p(x)q(x)wherex2− 3x −2all x where q(x) 6= 0function p(x), q(x) are poly. (ex: all x except x = 2)Root√ax + b√3x + 7 all x where ax + b ≥ 0function (ex: all x ≥ −73)Exponential ax15xall xfunction where a > 0Logarithmic loga(x) log15(x) all x > 0function where a > 0, a 6= 1Continuity and Rates of Change (University of Illinois, Urbana-Champaign)MATH 234, Calculus for Business February 6th, 2017 9 / 23Section 3.2, Continuity Definition and ExamplesGraph of 3x4+ 5 isGraph of 15xisGraph ofx2− 3x − 2isGraph of log15(x) isGraph of√3x + 7 isContinuity and Rates of Change (University of Illinois, Urbana-Champaign)MATH 234, Calculus for Business February 6th, 2017 10 / 23Section 3.2, Continuity Definition and ExamplesDefinitionThe open interval (a, b) is the set of all values x such that a < x < b.The closed interval [a, b] is the set of all values x such that a ≤ x ≤ b.(Difference is closed intervals contain boundary/endpoints of the interval)DefinitionA function f (x) is continuous on (a,b) if it is continuous at each pointa < x < b in the interval.DefinitionA function f (x) is continuous on [a,b] if all 3 conditions hold:(1) f (x) is continuous on (a, b).(2) limx→a+f (x) = f (a)(3) limx→b−f (x) = f (b)If any of the 3 conditions fail, the f (x) is discontinuous on [a,b].Continuity and Rates of Change (University of Illinois, Urbana-Champaign)MATH 234, Calculus for Business February 6th, 2017 11 / 23Section 3.2, Continuity Definition and ExamplesExample (A polynomial function)Discuss the continuity of f (x) = 3x4+ 5 on intervals (2, 7) and [2, 7].Solution3x4+ 5 is continuous for all x. So, f (x) is continuous on (2, 7) and [2, 7].Example (An exponential function)Discuss the continuity of f (x) = 15xon intervals (2, 7) and [2, 7].Solution15xis continuous for all x. So, f (x) is continuous on (2, 7) and [2, 7].Continuity and Rates of Change (University of Illinois, Urbana-Champaign)MATH 234, Calculus for Business February 6th, 2017 12 / 23Section 3.2, Continuity Definition and ExamplesExample (A rational function)Discuss the continuity of f (x) =x2− 3x − 2on intervals (2, 7) and [2,