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Continuity 5 lessons + test day 2008-2009 9/16-9/25 2009-2010 9/15-9/22 5 lessons + test day2010 5 lessons + test 9/21-9/28 2014- 9/16 Contents lesson 1 Continuity ............................................................................................................ 2 Lesson 2 making it continuous and IVT ............................................................................ 8 lesson 3.............................................................................................................................. 12 Lesson 4 ............................................................................................................................ 14 Lesson 5 ............................................................................................................................ 15 Test CH 1 .......................................................................................................................... 16 lesson 1 Continuous Functions http://www.calculus-help.com/the-intermediate-value-theorem/ p. 84 #1-6, 11-18, p. 85 #19-24, sketch ๐(๐ฅ) = ( ๐ฅ 2 โ4, ๐ฅ (0,โ p. 86# 56-59 ) (โโ,0] continuous in its domain-square root new book uses infinite discontinuity as well as non removable Definition of Continuity: Functions are continuous at c if 1 f(c) is defined 2 lim ๐ (๐ฅ ) ๐๐ฅ๐๐ ๐ก๐ 3 lim ๐ (๐ฅ ) = ๐(๐) ๐ฅโ๐ ๐ฅโ๐ Informally you might say that a function is continuous on an open interval if its graph can be drawn with a pencil without lifting the pencil from the paper. Use a graphing calculator; graph each of the following function on the indicated interval. From the graphs, which functions would you say are continuous on the interval? Do you think you can trust the results you obtained graphically? Y= x2+1 ๐ฆ= Y= (-3,3) 1 ๐ฅโ2 ๐ฅ2 โ 4 ๐ฅ+2 (-3,3) (โ3,3) Point of Discontinuity โ If a function is discontinuous at a point, the discontinuity may be Removable or Nonremovable depending upon whether the limit of the function exists at the point of discontinuity. 5. Test for continuity at the indicated points- I must clearly see the three things that make a function continuous at a point. Definition of Continuity: Functions are continuous at c if 1 f(c) is defined 2 lim ๐(๐ฅ) ๐๐ฅ๐๐ ๐ก๐ 3 lim ๐(๐ฅ) = ๐(๐) ๐ฅโ๐ ๐ฅโ๐ These functions are not continuous at c on the interval (a,b). โ for which reason above There are two categories of discontinuity: Removable and Non-Removable. Removable because you can change the function to cover the hole such as: Here are a few more examples of removable discontinuity: Non Removable Here are some examples of non removable discontinuous functions: Find the x values at which the function is not continuous. Is it removable or non removable?๐(๐ฅ) = ๐ฅ1 this function has a non removable discontinuity at x=0 it has an asymptote ๐ฅ 2 โ1 g(x)= ๐ฅโ1 can be rewritten as g(x)= it is discontinuous at x=1 (๐ฅ+1)(๐ฅโ1) ๐ฅโ1 it is removable discontinuity if used reverse classroom after limits test, then this is the For each of the following, Find the vertical asymptote the removable discontinuity the non removable discontinuity the horizontal asymptote 1 f ( x) ๏ฝ 2x ๏ซ 2 x2 ๏ซ 1 f ( x) ๏ฝ 2 x ๏ญ1 x 2 ๏ซ 2x ๏ญ 8 F(x) = x2 ๏ญ 4 x2 ๏ญ x ๏ญ 2 f ( x) ๏ฝ x๏ญ2 Lesson 2 McClearys website- making a function continuous Making functions continuous p. 85 #47-50 and WS - IVT problems and asymptotes from old book p. 78 # 75-77, 83-85 Answers- 83) f(3)=11 84) f(2)=0 85) f(2)=4 http://tutorial.math.lamar.edu/Classes/CalcI/Continuity.aspx Example 1 - Given the graph of f(x), shown below, determine if f(x) is continuous at & if not give the reason why itโs not f(-2) f(0) f(3) Solution To answer the question for each point weโll need to get both the limit at that point and the function value at that point. If they are equal the function is continuous at that point and if they arenโt equal the function isnโt continuous at that point. What are the 3 conditions for continuity? Functions are continuous at c if 1 f(c) is defined 2 lim ๐(๐ฅ) ๐๐ฅ๐๐ ๐ก๐ 3 lim ๐(๐ฅ) = ๐(๐) ๐ฅโ๐ ๐ฅโ๐ Ask- limit exists when it is the same from both sides This is a piecewise function so we need to worry about what is happening where the 2 pieces meet In a piecewise function-to be continuous- the two functions must have the same y-value at the breaking point-or if the function is undefined at a point, we need to fill in the open circle What value of B would make this function continuous ๏ฌ๏ฏ ๏ผ 3 x ๏ซ B, x ๏ฃ 5 ๏ฏ ๏ญ ๏ฝ f(x) = 2 ๏ฏ๏ฎ x ๏ญ 1 , x ๏พ 5๏ฏ๏พ what IVT means If you are 5 ft on your 13th birthday and on your 14th you are 5โ6โ at some pt you had to be 5โ4โ http://www.calculus-help.com/the-intermediate-value-theorem/ Intermediate Value Theorem-( mcclearys website) Look at IVT on http://www.calculus-help.com/funstuff/phobe.html Class Work-Worksheet-continuity problems lesson 3 HW โ2.3 -WorksheetGo over hw- AP style test on ________________ Multiple choice- Free response ½ with calculator, ½ without calculator Asymptote power point http://online.math.uh.edu/HoustonACT/ 2.2 1. Use the Intermediate Value theorem to explain why the function f(x) = x2+2x-1 Must have at least one root on the interval [-1,1] 2. What is the value of k to make this a continuous function? ๏ฌ๏ ln x for 4. If f (x) ๏ฝ ๏ญ๏ 2 ๏ฎ๏ x ln x for (A) ln 2 0๏ผ x ๏ฃ 2 then lim f (x) is x๏ฎ 2 2๏ผ x ๏ฃ 4 (B) ln 8 (C) ln16 (D)4 E)nonexistent Lesson 4 Give back limits test โdo a conjugate problem and โinfinity square root problem HW-Worksheet โnext 2 pages of continuity packet CW- below Review questions-from mcclearys website, 1 2. 3 Find the value of a and b that makes this function continuous Functions are continuous at c if 1 f(c) is defined 2 lim ๐(๐ฅ) ๐๐ฅ๐๐ ๐ก๐ 3 lim ๐(๐ฅ) = ๐(๐) ๐ฅโ๐ ๐ฅโ๐ Lesson 5 Review problems p. 88-89 Go over HW- give answer books AP Style test ________________ cw- Review Problems How do you find the limits- analytically, graphically, and table Direct Substitution put the limit (c ) into the equation If #/# it is the limit If 0/0 it often has a limit, you must do some work- Factor, conjugatesโฆ If If #/0 0/# limit never exists DNE limit is 0 Vertical asymptotes occur where the denominator=0 ie- set the denominator = 0, solve for x Horizontal asymptotes occurHighest degree in denominator getting closer to 0, horizontal asymptote at y=0 Equal degree in numerator and denominator-horizontal asymptote is the leading coefficients Highest degree in numerator-no horizontal asymptote Whenever there is a vertical asymptote you have a limit of +โ or -โ Test CH 1