INTERMEDIATE VALUE THEOREM Section 1.4B Calculus AP/Dual, Revised 017 viet.dang@humbleisd.net 7/30/018 1:9 AM 1.4B: Intermediate Value Theorem 1
DEFINITION OF CONTINUITY A function is continuous at the point x = c if and only if: 1. f(c) is continuous ) lim f x exists x c 3) lim f x = f(c) x c 7/30/018 1:30 AM 1.4B: Intermediate Value Theorem
EXAMPLES OF DISCONTINUOUS 7/30/018 1:30 AM 1.4B: Intermediate Value Theorem 3
PROOF OF INTERMEDIATE VALUE THEOREM Can you prove that at one time, you were exactly 5.578 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 exists a number c between a and b such that f(c) = k 10yrs a c 11yrs b 7/30/018 1:30 AM 1.4B: Intermediate Value Theorem 4
INTERMEDIATE VALUE THEOREM A. If f(x) 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 exists a number c between a and b for f c = k 7/30/018 1:30 AM 1.4B: Intermediate Value Theorem 5
EXAMPLE 1 Use the IVT to prove that the function f x = x is 7 on the interval between, 5. 5 7 4 a c 5 b f x a b f b r A. If is continuous on the closed interval, B. f a C. If k is between f a and f b then there exists a number c between a and b fo f c k c f c Since f x is continuous on,5 and 7 is between f and f 5, then IVT exists where a number between and 5 for 7 7/30/018 1:30 AM 1.4B: Intermediate Value Theorem 6
EXAMPLE If f x = ln x, prove by the IVT that there is a root on the interval of 1, 3. 1 A. f ( x ) is continuous on the closed interval,3 where lim and lim are real numbers f x f x x0.5 x3 1 1 B. f f 3 ;ln 0;ln 3 0 1 1 C.If k is between f and f (3) then there exists a number " c" between and 3 for f ( c) k 1 Since f x is continuous on,3, then IVT exists 1 where a number "c" between and 3 for f c k 7/30/018 1:30 AM 1.4B: Intermediate Value Theorem 7
EXAMPLE 3 If f x = x + x 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 extremes? (other words f a and f b )? f 0 x x 1 f 0 0 0 1 f 0 1 f x x x 1 Interval These are the two extremes. : 0,5 f 5 x x 1 f 5 5 5 1 7/30/018 1:30 AM 1.4B: Intermediate Value Theorem 8 f 5 9
EXAMPLE 3 If f x = x + x 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 1 11 9 11 x x1 0 x x1 f x x 3 x 4 f c 11 4,3 or x 4, x 3 then IVT exists where c 3. 3 is in 0,5 7/30/018 1:30 AM 1.4B: Intermediate Value Theorem 9 A. f ( x ) is continuous on the closed interval 0,5 where lim f x 1 and lim f x 9 x0 x5 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 exists a number "c" between 0 5 11 Since f x is continuous on 0,5 and 11 is between f 0 and f 5,
EXAMPLE 4 If f x = x +x, prove the IVT holds through the indicated interval of 5 x 1, 4 if f c = 6. If the IVT applies, find the value of c for f c = 6. 5 5 Since f x is continuous on,4 and f c is between f and f 4, then IVT exists where c 3 7/30/018 1:30 AM 1.4B: Intermediate Value Theorem 10
EXAMPLE 4 (EXTENSION) Would the IVT hold for f x = x +x, through the indicated interval of 3, 7? Explain why. x 1 Since f x has a non-removable discontinuity at x 1 on 3,7 then IVT does not exist where. 7/30/018 1:30 AM 1.4B: Intermediate Value Theorem 11
EXAMPLE 5 If f x = x 6x + 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 x is continuous on 0,3 and 3 is between f 0 and f 5, then IVT exists where c 1. 7/30/018 1:30 AM 1.4B: Intermediate Value Theorem 1
If f x = 1 x YOUR TURN, use the Intermediate Value Theorem to prove for c on the interval 5, 7 if f c = 1 4. 5 5 Since f x is continuous on,7 and f c is between f and f 4, 1 then IVT exists where f c 4 7/30/018 1:30 AM 1.4B: Intermediate Value Theorem 13
TO EARN FULL CREDIT: A. The function, f x (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 1:30 AM 1.4B: Intermediate Value Theorem 14
PIECEWISE FUNCTIONS A. For a piecewise function to be continuous each function must be continuous on its specified interval and the limit of the endpoints of each interval must be equal. 7/30/018 1:30 AM 1.4B: Intermediate Value Theorem 15
EXAMPLE 6 What value of k will make the given piecewise function f x continuous x +5x 3 at x = 3 of f x = ቐ, x 3 x 9 k, x = 3 x 5x3 lim x3 x 9 x1 x3 lim x3 x3 x3 7/30/018 1:30 AM 1.4B: Intermediate Value Theorem 16? x 1 lim x3 x 3
EXAMPLE 6 What value of k will make the given piecewise function f x continuous x +5x 3 at x = 3 of f x = ቐ x 9 k, lim x3 lim x3, x 3 x = 3 x 1 x 3 7/30/018 1:30 AM 1.4B: Intermediate Value Theorem 17 3 1 3 3 k 7 6?
EXAMPLE 6 What value of k will make the given piecewise function f x continuous x +5x 3 at x = 3 of f x = ቐ x 9 k,, x 3 x = 3? k 7 6 7/30/018 1:30 AM 1.4B: Intermediate Value Theorem 18
IN CONCLUSION A function exists when: 1. Point Exists. Limit Exists 3. Limit = Point 7/30/018 1:30 AM 1.4B: Intermediate Value Theorem 19
AP MULTIPLE CHOICE PRACTICE QUESTION 1 (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(x) 3 for all x 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 1:30 AM 1.4B: Intermediate Value Theorem 0
AP MULTIPLE CHOICE PRACTICE QUESTION 1 (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 exist. 7/30/018 1:30 AM 1.4B: Intermediate Value Theorem 1
ASSIGNMENT Worksheet 7/30/018 1:30 AM 1.4B: Intermediate Value Theorem