INTERMEDIATE VALUE THEOREM
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1 INTERMEDIATE VALUE THEOREM Section 1.4B Calculus AP/Dual, Revised 017 7/30/018 1:36 AM 1.4B: Intermediate Value Theorem 1
2 PROOF OF INTERMEDIATE VALUE THEOREM Can you prove that at one time, you were eactly feet tall? f b 5'5" k 5'3" 5' f a If f is continuous on a, b and k is between f(a) and f(b) then there eists a number c between a and b such that f(c) = k 10yrs a c 11yrs b 7/30/018 1:36 AM 1.4B: Intermediate Value Theorem
3 INTERMEDIATE VALUE THEOREM A. If f() is continuous on the closed interval a, b B. f a f b C. If k is between f a and f b then there eists a number c between a and b for f c = k 7/30/018 1:36 AM 1.4B: Intermediate Value Theorem 3
4 EXAMPLE 1 If f = + 1, prove the IVT holds through the indicated interval of 0, 5. If the IVT applies, find the value of c for f c = 11. What are the etremes? (other words f a and f b )? f 0 1 f f 0 1 f 1 Interval These are the two etremes. : 0,5 f 5 1 f /30/018 1:36 AM 1.4B: Intermediate Value Theorem 4 f 5 9
5 EXAMPLE 1 If f = + 1, prove the IVT holds through the indicated interval of 0, 5. If the IVT applies, find the value of c for f c = 11. f a k f b f 3 4 f c 11 4,3 or 4, 3 then IVT eists where c 3. 3 is in 0,5 7/30/018 1:36 AM 1.4B: Intermediate Value Theorem 5 A. f ( ) is continuous on the closed interval 0,5 where f 1 and f is f and f and for f c B. f 0 f 5 ; f 0 11 and f 5 11 C.If 11 between 0 5 then there eists a number "c" between Since f is continuous on 0,5 and 11 is between f 0 and f 5,
6 EXAMPLE If f = 6 + 8, prove the IVT holds through the indicated interval of 0, 3. If the IVT applies, find the value of c for f c = 3. Since f is continuous on 0,3 and 3 is between f 0 and f 5, then IVT eists where c 1. 7/30/018 1:36 AM 1.4B: Intermediate Value Theorem 6
7 If f = 1 YOUR TURN, use the Intermediate Value Theorem to prove for c on the interval 5, 7 if f c = Since f is continuous on,7 and f c is between f and f 4, 1 then IVT eists where f c 4 7/30/018 1:36 AM 1.4B: Intermediate Value Theorem 7
8 TO EARN FULL CREDIT: A. The function, f (or whatever they give) is identified and stated to be CONTINUOUS. B. Include the function is continuous in a, b where a and b are defined C. State the value of c, if asked to be defined. 7/30/018 1:36 AM 1.4B: Intermediate Value Theorem 8
9 PIECEWISE FUNCTIONS A. For a piecewise function to be continuous each function must be continuous on its specified interval and the it of the endpoints of each interval must be equal. 7/30/018 1:36 AM 1.4B: Intermediate Value Theorem 9
10 EXAMPLE 3 What value of k will make the given piecewise function f continuous +5 3 at = 3 of f = ቐ, 3 9 k, = /30/018 1:36 AM 1.4B: Intermediate Value Theorem 10? 1 3 3
11 EXAMPLE 3 What value of k will make the given piecewise function f continuous +5 3 at = 3 of f = ቐ 9 k, 3 3, 3 = /30/018 1:36 AM 1.4B: Intermediate Value Theorem k 7 6?
12 EXAMPLE 3 What value of k will make the given piecewise function f continuous +5 3 at = 3 of f = ቐ 9 k,, 3 = 3? k 7 6 7/30/018 1:36 AM 1.4B: Intermediate Value Theorem 1
13 INFINITE LIMITS A. Infinite Limits are its in which f() increases or decreases without bound as approaches c or c f() = ±, does not eist. The statement, approaches to infinity really describes the it to reach a constant number. B. Let f and g be continuous on an open interval containing c. If f(c) 0, g(c) = 0, then h = f g has a vertical asymptote at = c. 7/30/018 1:36 AM 1.5 & 3.5: Infinite Limits and Limits at Infinity 13
14 STEPS A. Simplify the rational function B. Establish the it from the left side and right side C. If the its are the same, it is the established it. D. If the its are not the same, the it fails to eist. 7/30/018 1:36 AM 1.5 & 3.5: Infinite Limits and Limits at Infinity 14
15 REVIEW Determine 3 and /30/018 1:36 AM 1.5 & 3.5: Infinite Limits and Limits at Infinity 15
16 EXAMPLE 4 Solve the vertical asymptotes (if any) of f = sin and 0 = sin 0 sin sin sin n 0 n DNE, left side right side 1 sin 7/30/018 1:36 AM 1.5 & 3.5: Infinite Limits and Limits at Infinity 16
17 EXAMPLE 4 Solve the vertical asymptotes (if any) of f = sin and 0 = sin DNE, left side right side 1 sin 7/30/018 1:36 AM 1.5 & 3.5: Infinite Limits and Limits at Infinity 17
18 YOUR TURN Solve the vertical asymptotes (if any) of f = 1 and 0 = 1 VA: /30/018 1:36 AM 1.5 & 3.5: Infinite Limits and Limits at Infinity 18
19 INFINITE LIMITS A. Infinite Limits are its in which f() increases or decreases without bound as approaches c or c f() = ±, does not eist. The statement, approaches to infinity really describes the it to reach a constant number. B. Let f and g be continuous on an open interval containing c. If f(c) 0, g(c) = 0, then h = f g has a vertical asymptote at = c. 7/30/018 1:36 AM 1.5 & 3.5: Infinite Limits and Limits at Infinity 19
20 PROPERTIES OF INFINITE LIMITS A. As we let f = and g = L : c c B. Types: 1. Sum or difference: f ± g = ± L = c. Product: f g = if g or L > 0 c c 3. Quotient: c f g = if L < 0 c g f = 0 C. Similar properties hold for one sided its and for functions for which the it of f as approaches c is +. 7/30/018 1:36 AM 1.5 & 3.5: Infinite Limits and Limits at Infinity 0
21 A. If LIMITS AT INFINITY f = L = f, then the line y = L is a horizontal asymptote for the graph of f(). There is not a negative left and right bound side. B. With rational functions and establishing vertical asymptotes, determine the highest eponents on top, bottom, and the same. C. If r is a positive rational number, c is any number, and r is defined c c when < 0 of r = 0 = r 7/30/018 1:36 AM 1.5 & 3.5: Infinite Limits and Limits at Infinity 1
22 EXAMPLE 5 If = and =, solve /( 1). 7/30/018 1:37 AM 1.5 & 3.5: Infinite Limits and Limits at Infinity g c f Smaller 0 Bigger 1 g goes first over f because it of g = constant
23 YOUR TURN If = and 1 cot π =, solve 1 +1 cot π. 0 7/30/018 1:37 AM 1.5 & 3.5: Infinite Limits and Limits at Infinity 3
24 EXAMPLE 6 Solve f() ??? /30/018 1:37 AM 1.5 & 3.5: Infinite Limits and Limits at Infinity 4
25 EXAMPLE 6 Solve /30/018 1:37 AM 1.5 & 3.5: Infinite Limits and Limits at Infinity 5
26 YOUR TURN Solve f()??? /30/018 1:37 AM 1.5 & 3.5: Infinite Limits and Limits at Infinity 6
27 INFINITE LIMITS A. When a it does not eist, it means it approaches towards negative infinity or positive infinity. B. If a it does not eist, write DNE with a reason C. However, if the it does not eist and you are asked why, you can respond ± if it approaches there 7/30/018 1:37 AM 1.5 & 3.5: Infinite Limits and Limits at Infinity 7
28 STEPS A. Establish the highest eponent B. Divide each term by the biggest eponent of the denominator 1. Same Eponents: Divide the leading coefficients. Smaller Eponent/Bigger Eponent: (S/B) = 0 3. Bigger Eponent/Smaller Eponent: (B/S) = ± C. Simplify 7/30/018 1:37 AM 1.5 & 3.5: Infinite Limits and Limits at Infinity 8
29 EXAMPLE 7 Solve 3 +1 and f() ; /30/018 1:37 AM 1.5 & 3.5: Infinite Limits and Limits at Infinity 9
30 EXAMPLE 7 Solve 3 +1 Same Co /30/018 1:37 AM 1.5 & 3.5: Infinite Limits and Limits at Infinity 30
31 EXAMPLE 8 Solve Same Same /3 1/ /3 1/3 1/3 6 1/3 1/3 3 7/30/018 1:37 AM 1.5 & 3.5: Infinite Limits and Limits at Infinity 31
32 YOUR TURN Solve /30/018 1:37 AM 1.5 & 3.5: Infinite Limits and Limits at Infinity 3
33 EXAMPLE 9 1 Solve +1 Same Co /30/018 1:37 AM 1.5 & 3.5: Infinite Limits and Limits at Infinity 33
34 EXAMPLE 10 Solve S B = /30/018 1:37 AM 1.5 & 3.5: Infinite Limits and Limits at Infinity 34
35 EXAMPLE 11 Solve B S = ± y DNE, as it approaches 4 1 7/30/018 1:37 AM 1.5 & 3.5: Infinite Limits and Limits at Infinity 35
36 YOUR TURN Solve DNE, as it approaches 7 4 7/30/018 1:37 AM 1.5 & 3.5: Infinite Limits and Limits at Infinity 36
37 IN CERTAIN ORDER Identify the following of functions in order of bigness and how they grow e, ln, c,, sin or cos SMALL c,sin, 33 cc or orcos,ln,,,,,,, etc,..., e BIG 7/30/018 1:37 AM 1.5 & 3.5: Infinite Limits and Limits at Infinity 37
38 EXAMPLE 1 Solve e ln e ln B DNE S DNE, as it approaches 7/30/018 1:37 AM 1.5 & 3.5: Infinite Limits and Limits at Infinity 38
39 EXAMPLE 13 Solve /30/018 1:37 AM 1.5 & 3.5: Infinite Limits and Limits at Infinity 39 3
40 EXAMPLE 14 Solve /30/018 1:37 AM 1.5 & 3.5: Infinite Limits and Limits at Infinity
41 YOUR TURN Solve /30/018 1:37 AM 1.5 & 3.5: Infinite Limits and Limits at Infinity 41
42 EXAMPLE 15 Identify the horizontal asymptote of g = 4 1 HA : y /30/018 1:37 AM 1.5 & 3.5: Infinite Limits and Limits at Infinity 4
43 EXAMPLE 15 Identify the horizontal asymptote of g = HA : y 1 7/30/018 1:37 AM 1.5 & 3.5: Infinite Limits and Limits at Infinity 43
44 EXAMPLE 16 Identify the horizontal asymptote of g = e +1 1+e e e e 1 e e e e e 0 0 e why? Look at the graph. 7/30/018 1:37 AM 1.5 & 3.5: Infinite Limits and Limits at Infinity 44
45 EXAMPLE 16 Identify the horizontal asymptote of g = e +1 1+e HA : y 0, y 1 7/30/018 1:37 AM 1.5 & 3.5: Infinite Limits and Limits at Infinity 45
46 YOUR TURN Identify the horizontal asymptote of h = HA : y 0 7/30/018 1:37 AM 1.5 & 3.5: Infinite Limits and Limits at Infinity 46
47 EXAMPLE 17 Identify the horizontal asymptotes of f = 4+6e 5 9e 4 6e 5 9 e 0 0 6e 9 e y 3 4 6e 5 9 e y , y 3 5 7/30/018 1:37 AM 1.5 & 3.5: Infinite Limits and Limits at Infinity y 4 5
48 Solve t t 4/3 +t 1/3 4t /3 +1 (A) 1 16 (B) 1 4 (C) 0 (D) 1 16 AP MULTIPLE CHOICE PRACTICE QUESTION 1 (NON-CALCULATOR) 7/30/018 1:37 AM 1.5 & 3.5: Infinite Limits and Limits at Infinity 48
49 Solve t t 4/3 +t 1/3 4t /3 +1 AP MULTIPLE CHOICE PRACTICE QUESTION 1 (NON-CALCULATOR) Vocabulary Connections and Process Answer and Justifications Limit t approaches, Make Table t t t t t /3 1/3 4/3 1/3 D t 4/3 /3 t t /3 /3 4t 14t 1 t 1 4/3 t 16 4/3 t 16 7/30/018 1:37 AM 1.5 & 3.5: Infinite Limits and Limits at Infinity 49
50 AP MULTIPLE CHOICE PRACTICE QUESTION (NON-CALCULATOR) Let f be a continuous function on the closed interval 3, 6. If f 3 = 1 and f 6 = 3, then the Intermediate Value Theorem guarantees that: (A) f c = 4 for at least one c between 3 and 6 9 (B) 1 f() 3 for all between 3 and 6 (C) f(c) = 1 for at least one c between 3 and 6 (D) f(c) = 0 for at least one c between 1 and 3 7/30/018 1:37 AM 1.4B: Intermediate Value Theorem 50
51 AP MULTIPLE CHOICE PRACTICE QUESTION (NON-CALCULATOR) Let f be a continuous function on the closed interval 3, 6. If f 3 = 1 and f 6 = 3, then the Intermediate Value Theorem guarantees that: Vocabulary Connections and Process Answer and Justifications Continuous IVT Interval: 3, 6 f b f a Point: 3, 1, 6,3 c A) f ' MVT b a B) 1 y 3 Not necessarily true. How do we know? C) Is there a point in the I 3,6 when the y-value is 1? True. Since the range of the points are 1 and 3. D) Is there a point in the I 1,3 when the y-value of the point = 0? Possible. We don't know for sure. C There's a f c in the I 3,6 when the y-value is 1. Since the range of the points are 1 and 3, 1 must eist. 7/30/018 1:37 AM 1.4B: Intermediate Value Theorem 51
52 ASSIGNMENT Page odd, odd, odd Page all, odd 7/30/018 1:36 AM 1.5 & 3.5: Infinite Limits and Limits at Infinity 5
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