Continuity, End Behavior, and Limits. Unit 1 Lesson 3
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1 Unit Lesson 3
2 Students will be able to: Interpret ke features of graphs and tables in terms of the quantities, and sketch graphs showing ke features given a verbal description of the relationship. Ke Vocabular: Discontinuit, A limit, End Behavior
3 The graph of a continuous function has no breaks, holes, or gaps. You can trace the graph of a continuous function without lifting our pencil. One condition for a function f to be continuous at = c is that the function must approach a unique function value as -values approach c from the left and right sides.
4 The concept of approaching a value without necessaril ever reaching it is called a limit. If the value of f approaches a unique value L as approaches c from each side, then the limit of f as approaches c is L. lim c f = L
5 Functions that are not continuous are discontinuous. Graphs that are discontinuous can ehibit: Infinite discontinuit A function has an infinite discontinuit at = c, if the function value increases or decreases indefinitel as approaches c from the left and right.
6 Functions that are not continuous are discontinuous. Graphs that are discontinuous can ehibit: Jump discontinuit A function has a jump discontinuit at = c if the limits of the function as approaches c from the left and right eist but have two distinct values.
7 Functions that are not continuous are discontinuous. Graphs that are discontinuous can ehibit: Removable discontinuit, also called point discontinuit Function has a removable discontinuit if the function is continuous everwhere ecept for a hole at = c..
8 Continuit Test A function f is continuous at = c, if it satisfies the following conditions:. f is defined at c. f(c)eists.. f approaches the same function value to the left and right of c. lim f eists c 3. The function value that f approaches from each side of c is f c. lim f = f (c) c
9 Sample Problem : Determine whether each function is continuous at the given -values. Justif using the continuit test. If discontinuous, identif the tpe of discontinuit. a. f = at =
10 Sample Problem : Determine whether each function is continuous at the given -values. Justif using the continuit test. If discontinuous, identif the tpe of discontinuit. a. f = at = f = = 3 f eists 0-30
11 Sample Problem : Determine whether each function is continuous at the given -values. Justif using the continuit test. If discontinuous, identif the tpe of discontinuit. a. f = at = 30 < f
12 Sample Problem : Determine whether each function is continuous at the given -values. Justif using the continuit test. If discontinuous, identif the tpe of discontinuit. a. f = at = 30 A f
13 Sample Problem : Determine whether each function is continuous at the given -values. Justif using the continuit test. If discontinuous, identif the tpe of discontinuit. a. f = at = 30 0 f = 3 and 3 from both side of = lim = f () f = is continuous at =
14 Sample Problem : Determine whether each function is continuous at the given -values. Justif using the continuit test. If discontinuous, identif the tpe of discontinuit. b. f = at =
15 Sample Problem : Determine whether each function is continuous at the given -values. Justif using the continuit test. If discontinuous, identif the tpe of discontinuit. b. f = at = f 0 = 3 0 = 0 0 The function is undefined at =
16 Sample Problem : Determine whether each function is continuous at the given -values. Justif using the continuit test. If discontinuous, identif the tpe of discontinuit. b. f = at = < f
17 Sample Problem : Determine whether each function is continuous at the given -values. Justif using the continuit test. If discontinuous, identif the tpe of discontinuit. b. f = at = A f -3-5
18 Sample Problem : Determine whether each function is continuous at the given -values. Justif using the continuit test. If discontinuous, identif the tpe of discontinuit. b. f = 5 3 at = f = has jump discontinuit at = 0 since values are and on opposite sides of = 0
19 Sample Problem : Determine whether each function is continuous at the given -values. Justif using the continuit test. If discontinuous, identif the tpe of discontinuit. c. f = at =
20 Sample Problem : Determine whether each function is continuous at the given -values. Justif using the continuit test. If discontinuous, identif the tpe of discontinuit. c. f = at = + 5 f = + = f is undefined at = f = + is discontinuous at =
21 Sample Problem : Determine whether each function is continuous at the given -values. Justif using the continuit test. If discontinuous, identif the tpe of discontinuit. c. f = at = + 5 < f
22 Sample Problem : Determine whether each function is continuous at the given -values. Justif using the continuit test. If discontinuous, identif the tpe of discontinuit. c. f = at = + 5 A f
23 Sample Problem : Determine whether each function is continuous at the given -values. Justif using the continuit test. If discontinuous, identif the tpe of discontinuit. c. f = at = f = + has point discontinuit at = since value is on opposite sides of =
24 Sample Problem : Determine whether each function is continuous at the given -values. Justif using the continuit test. If discontinuous, identif the tpe of discontinuit. d. f = at =
25 Sample Problem : Determine whether each function is continuous at the given -values. Justif using the continuit test. If discontinuous, identif the tpe of discontinuit. d. f = at = f 0 = 3 0 = f is undefined at = 0 f = 3 is discontinuous at = 0
26 Sample Problem : Determine whether each function is continuous at the given -values. Justif using the continuit test. If discontinuous, identif the tpe of discontinuit. d. f = 3 5 at = 0 0 < f , ,
27 Sample Problem : Determine whether each function is continuous at the given -values. Justif using the continuit test. If discontinuous, identif the tpe of discontinuit. d. f = 3 5 at = 0 0 A f , ,
28 Sample Problem : Determine whether each function is continuous at the given -values. Justif using the continuit test. If discontinuous, identif the tpe of discontinuit. d. f = at = f = 3 has infinit discontinuit at = 0 since value is + when 0
29 Intermediate Value Theorem If f is a continuous function and a < b and there is a value n such that n is between f a and f b, then there is a number c, such that a < c < b and f c = n
30 The Location Principle If f is a continuous function and f a and f b have opposite signs, then there eists at least one value c, such that a < c < b and f c = 0. That is, there is a zero between a and b.
31 Sample Problem : Determine between which consecutive integers the real zeros of function are located on the given interval. a. f = 3 [0, 6]
32 Sample Problem : Determine between which consecutive integers the real zeros of function are located on the given interval. a. f = 3 [0, 6] f 0 is positive and f is negative, f change sign in 0 f is negative and f 6 is positive, f change sign in 6 f has zeros in intervals: 0 and 6
33 End Behavior The end behavior of a function describes what the - values do as becomes greater and greater. When becomes greater and greater, we sa that approaches infinit, and we write +. When becomes more and more negative, we sa that approaches negative infinit, and we write.
34 The same notation can also be used with or f and with real numbers instead of infinit. Left - End Behavior (as becomes more and more negative): lim <Y f Right - End Behavior (as becomes more and more positive): lim AY f The f values ma approach negative infinit, positive infinit, or a specific value.
35 Sample Problem 3: Use the graph of each function to describe its end behavior. Support the conjecture numericall. a. f =
36 Sample Problem 3: Use the graph of each function to describe its end behavior. Support the conjecture numericall. a. f = From the graph, it appears that: f as f as The table supports this conjecture
37 Sample Problem 3: Use the graph of each function to describe its end behavior. Support the conjecture numericall. b. f =
38 Sample Problem 3: Use the graph of each function to describe its end behavior. Support the conjecture numericall. 5 b. f = From the graph, it appears that: f as 0 8 f as The table supports this conjecture
39 Increasing, Decreasing, and Constant Functions A function f is increasing on an interval I if and onl if for ever a and b contained in I, a < f b, whenever a < b. A function f is decreasing on an interval I if and onl if for ever a and b contained in I, f a > f b whenever a < b. A function f remains constant on an interval I if and onl if for ever a and b contained in I, f a = f b whenever a < b. Points in the domain of a function where the function changes from increasing to decreasing or from decreasing to increasing are called critical points.
40 Sample Problem : Determine the interval(s) on which the function is increasing and the interval(s) on which the function is decreasing. a. f =
41 Sample Problem : Determine the interval(s) on which the function is increasing and the interval(s) on which the function is decreasing. a. f = 3 + From the graph, it appears that: 5 A function 3 + is decreasing 3 for <. 5 A function 3 + is increasing for >
42 Sample Problem : Determine the interval(s) on which the function is increasing and the interval(s) on which the function is decreasing. a. f = The table supports this conjecture
43 Sample Problem : Determine the interval(s) on which the function is increasing and the interval(s) on which the function is decreasing. b. f =
44 Sample Problem : Determine the interval(s) on which the function is increasing and the interval(s) on which the function is decreasing. b. f = From the graph, it appears that: A function is increasing: <. 66 and > A function is decreasing:. 66 < <
45 Sample Problem : Determine the interval(s) on which the function is increasing and the interval(s) on which the function is decreasing. b. f = The table supports this conjecture
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