Limits 4: Continuity

Transcription

1 Limits 4: Continuit 55 Limits 4: Continuit Model : Continuit I. II. III. IV. z V. VI. z a VII. VIII. IX. Construct Your Understanding Questions (to do in class). Which is the correct value of f (a) in Graph IV in Model : z or z? Circle one and eplain our reasoning. (Hint: Recall the meaning of an open versus a filled circle.). In Model, onl the functions in Graphs I-III are continuous over the interval shown. a. Each of the other functions (shown on Graphs IV-IX) is discontinuous at one or more points. Mark each point of discontinuit b adding a tick mark and a letter a along the ais as shown on Graph IV. If there are multiple points of discontinuit, mark them a, a, etc. b. Based on the information in Model, make up a definition for the term continuous function (on an interval).

2 56 Limits 4: Continuit 3. (Check our work) A crude test of continuit is to imagine an infinitel small ant walking along the graph of a function. If the ant can travel along the curve without interruption (e.g. without falling in a hole such as the one in Graph VII in Model ) the function is continuous over that interval. Is our definition consistent with this crude test? Eplain. 4. List each graph in Model where lim f ( ) lim f( ) for some point a on the interval shown. a a a. True or False: If lim f ( ) lim f( ) a a then f is discontinuous at a. If false, draw a function that is continuous at a but lim f ( ) lim f( ). a a b. (Check our work) It is a common error to assume that corners, such as those in Graph II in Model, are points of discontinuit. This leads some students to erroneousl conclude that part a of this question is false. A graph with corners (e.g. Graph II in Model ) is continuous, and part a of this question is true. c. Recall that is not a real number, so if lim f ( ) then this limit does not eist. a i. Which graph in Model has a point a where lim f ( ) lim f ( )? a a ii. Is this graph continuous at a? Eplain our reasoning. 5. True or False: If lim f ( ) lim f ( ) [a real number] then f is continuous at a. If a a false, give an eample from Model that demonstrates that this statement is false, and eplain our reasoning.

3 Limits 4: Continuit (Check our work) There are two graphs in Model that demonstrate the statement in the previous question is false. a. Identif the Roman numerals of these two graphs. b. For each of these two graphs, write down what ou know about the value of f ( a ). c. Each of these two graphs has a problem at = a that makes the function discontinuous at a. Describe how these two problems are different from one another. d. Graphs VII and VIII each have a point (or removable) discontinuit. In a wa, each of these functions is discontinuous at a because f ( a ) is not the value we epect based on the function near a. What would fi this? That is, the graph would be continuous at a if f ( a) Choose from: a f ( a ) lim f ( ) a f ( ) and eplain our reasoning. 7. (Check our work) What part of Summar Bo L4. confirms our answer to part d of the previous question? If it does not confirm our answer, reconsider our choice, above. Summar Bo L4.: Definitions of Continuit At a Point and On an Interval A function f is continuous at a if lim f ( ) f ( a). a A function f is continuous on an interval (c,d), if it is continuous at ever point in the interval. If this interval is closed [c,d], then it must be also be true that at c, the left endpoint, lim f ( ) f ( c), i.e. f ( ) is continuous from the right at c at d, the right endpoint, lim f ( ) f( d), i.e., f ( ) is continuous from the left at d c ; d. 8. With our group, come up with a real world eample of a process described b a continuous function, and one described b a function that contains one or more discontinuities. Sketch a graph of each of our functions, and be prepared to report our answer as part of a wholeclass discussion.

4 58 Limits 4: Continuit Model : Identifing a Discontinuit from an Equation I. f ( ) II. if f( ) if 3 7 if 3 III. 3 f ( ) 9 5 IV. if 6 f( ) 33 if 6 V. f( ) VI. 3 if f( ) if 3 if VII. f( ) 4 8 if 6 4 if 6 VIII. f ( ) 4 ( ) IX. f( ) Etend Your Understanding Questions (to do in or out of class) 9. At what value of does f ( ) have a discontinuit for a. Function V in Model? b. Function VIII in Model? Eplain our reasoning. 0. For Function VI in Model (a piece-wise defined function), what is a. f () b. c. lim f ( ) lim f ( ) d. f () e. f. lim f ( ) lim f ( ) Is Function VI continuous? Use our answers to parts a-f to justif our answer.. In the previous question, we checked Function VI for continuit at and. a. What two values of should be checked for continuit for Function II in Model? b. Is Function II in Model continuous? Show our work. c. What value(s) of should be checked for continuit for Function IV in Model? d. Is Function IV in Model continuous? Show our work.. Use what ou discovered above to mark Functions I, III, VII, and IX as continuous or not continuous (and note an values of where a function has a discontinuit).

5 Limits 4: Continuit (Check our work) The functions in Model are the same functions that appear as graphs in Model. Use this to check our answers to the previous question, and correct an answers. 4. Based on what ou know so far, sort the si discontinuous functions (IV-IX) in Models and into the three categories listed on the table below. (Put each Roman numeral in eactl one bo in the column labeled Roman numerals, below.) Tpe of Discontinuit Roman numerals of functions in Models and that have a.. Description of a graph of a function that has a Characteristics of an equation that should be checked for a Jump Infinite Point 5. In the column labeled Description of a graph (above), describe the ke defining characteristics of each tpe of discontinuit listed. 6. In the column labeled Characteristics of an equation (above), describe what ou look for in an equation that ma indicate the presence of each tpe of discontinuit listed. 7. (Check our work) Is our answer to the previous question consistent with Summar Bo L4.? Note an differences and discuss them with our group.

6 60 Limits 4: Continuit Summar Bo L4.: Some Properties of Continuous Functions Piecewise defined functions must be checked for continuit, and rational functions are discontinuous when the denominator is equal to zero; however, there are other functions that are discontinuous, for eample, f ( ) tan. If two functions f and g are continuous at = a, then all of the following are also continuous at a: f g, f g, cf ( c = constant), fg, f g ( g 0 ), and f g ( f continuous at g ( a ) ) 8. Circle the two statements below that are true. For the remaining three false statements, sketch a graph of a function demonstrating that it is false. a. If lim f ( ) f( a) a then f is continuous at a. b. If f is not continuous at a then f ( a) lim f( ). a c. If f is not continuous at a then f ( a ) is undefined. d. If f is not continuous at a then lim f ( ) does not eist. a e. If f is not continuous at a then either f ( a ) is undefined or lim f ( ) does not eist. a

7 Limits 4: Continuit 6 Confirm Your Understanding Questions (to do at home) 9. Consider the function f( ) a. Is this function continuous? If not, at what value(s) of is it discontinuous? Eplain our reasoning. b. The numerator of this function can be factored and the function can be epressed as ( 3) f( ). Note that this is not the same as the function f () 3. 3 Construct an eplanation for wh and how these two ver similar functions differ from one another. c. Sketch a graph of these two functions. 0. On what interval is the function f ( ) continuous? Eplain our reasoning.. For Graph IV in Model, the lim f ( ) f( a). a a. (Review) Eplain the significance of the superscript in the equation above. b. Use the term lim f ( ) a not continuous at a. in an eplanation for wh the function shown on Graph IV is

8 6 Limits 4: Continuit Notes