Intermediate Value Theorem
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1 Stewart Section 2.5 Continuity p. 1/ Intermediate Value Theorem The intermediate value theorem states that, if a function f is continuous on a closed interval [a,b] (that is, an interval that includes both endpoints) and f(a) f(b), then for every N (a,b), there exists a c (a,b) with f(c) = N.
2 Stewart Section 2.5 Continuity p. 1/ Intermediate Value Theorem The intermediate value theorem states that, if a function f is continuous on a closed interval [a,b] (that is, an interval that includes both endpoints) and f(a) f(b), then for every N (a,b), there exists a c (a,b) with f(c) = N. Note that the theorem does not apply if f is not continuous at every point of [a,b].
3 Stewart Section 2.5 Continuity p. 1/ Intermediate Value Theorem The intermediate value theorem states that, if a function f is continuous on a closed interval [a,b] (that is, an interval that includes both endpoints) and f(a) f(b), then for every N (a,b), there exists a c (a,b) with f(c) = N. Note that the theorem does not apply if f is not continuous at every point of [a,b]. The theorem also does not apply if f(a) = f(b), that is, if the function value is the same at both endpoints.
4 Stewart Section 2.5 Continuity p. 2/ IVT The intermediate value theorem can be used to prove that an equation has a root, that is, a point where f(c) = 0, c [a,b]
5 Stewart Section 2.5 Continuity p. 2/ IVT The intermediate value theorem can be used to prove that an equation has a root, that is, a point where f(c) = 0, c [a,b] If f(a) < 0 and f(b) > 0, the intermediate value theorem guarantees that f assumes every value between f(a) and f(b) somewhere on the open interval (a,b).
6 Stewart Section 2.5 Continuity p. 2/ IVT The intermediate value theorem can be used to prove that an equation has a root, that is, a point where f(c) = 0, c [a,b] If f(a) < 0 and f(b) > 0, the intermediate value theorem guarantees that f assumes every value between f(a) and f(b) somewhere on the open interval (a,b). In this case, since f(a) < 0 < f(b), the theorem guarantees the existince of a c [a,b] with f(c) = 0.
7 Stewart Section 2.5 Continuity p. 3/ Example 1 Prove that the equation has a root in the interval (1, 3) 4 x 2 = 0
8 Stewart Section 2.5 Continuity p. 3/ Example 1 Prove that the equation has a root in the interval (1, 3) 4 x 2 = 0 Note that f(x) = 4 x 2 is continuous everywhere, so it is also continuous on the closed interval [1, 3]
9 Stewart Section 2.5 Continuity p. 3/ Example 1 Prove that the equation has a root in the interval (1, 3) 4 x 2 = 0 Note that f(x) = 4 x 2 is continuous everywhere, so it is also continuous on the closed interval [1, 3] Now note that f(1) = = 3 > 0 and f(3) = = 5 < 0, so by the intermediate value theorem f has a root in (1, 3).
10 Stewart Section 2.5 Continuity p. 4/ Example 2 Prove that the equation has a root in the interval (1, 8) 1 x + 2x = 0
11 Stewart Section 2.5 Continuity p. 4/ Example 2 Prove that the equation has a root in the interval (1, 8) 1 x + 2x = 0 Note that f(x) = 1 x + 2x is continuous on [0, ), so it is also continuous on the closed interval [1, 8]
12 Stewart Section 2.5 Continuity p. 4/ Example 2 Prove that the equation has a root in the interval (1, 8) 1 x + 2x = 0 Note that f(x) = 1 x + 2x is continuous on [0, ), so it is also continuous on the closed interval [1, 8] Now note that f(1) = sqrt2 > 0 and f(8) = < 0, so f has a root in (1, 8).
13 Stewart Section 2.5 Continuity p. 5/ Example 2 Let f(x) = 3x + 2 5x 1
14 Stewart Section 2.5 Continuity p. 5/ Example 2 Let As x, f(x) 3 5 f(x) = 3x + 2 5x 1 from above.
15 Stewart Section 2.5 Continuity p. 5/ Example 2 Let As x, f(x) 3 5 f(x) = 3x + 2 5x 1 from above. As x, f(x) 3 5 from below.
16 Stewart Section 2.5 Continuity p. 6/ Example 3 Let f(x) = 2 + e x
17 Stewart Section 2.5 Continuity p. 6/ Example 3 Let f(x) = 2 + e x As x, f(x) 2 from above.
18 Stewart Section 2.5 Continuity p. 7/ Question 1 Find the horizontal asymptote(s) of f(x) = x 2 1. y = 0 4. y = 2 2. y = 1 5. y = 1 3. y = 2 6. f has no horizontal asymptote
19 Stewart Section 2.5 Continuity p. 7/ Question 1 Find the horizontal asymptote(s) of f(x) = x 2 1. y = 0 4. y = 2 2. y = 1 5. y = 1 3. y = 2 6. f has no horizontal asymptote 3. y = 2.
20 Stewart Section 2.5 Continuity p. 8/ Question 2 Find the horizontal asymptote(s) of f(x) = x y = 0 4. y = 3 2. y = 1 5. y = 3 3. y = 2 6. f has no horizontal asymptote
21 Stewart Section 2.5 Continuity p. 8/ Question 2 Find the horizontal asymptote(s) of f(x) = x y = 0 4. y = 3 2. y = 1 5. y = 3 3. y = 2 6. f has no horizontal asymptote 4. y = 3.
22 Stewart Section 2.5 Continuity p. 9/ Question 3 Find the horizontal asymptote(s) of f(x) = 2x5 + 7x 3 15x + 10 x 5 8x 4 + 2x y = 0 4. y = 3 2. y = 1 5. y = 3 3. y = 2 6. f has no horizontal asymptote
23 Stewart Section 2.5 Continuity p. 9/ Question 3 Find the horizontal asymptote(s) of f(x) = 2x5 + 7x 3 15x + 10 x 5 8x 4 + 2x y = 0 4. y = 3 2. y = 1 5. y = 3 3. y = 2 6. f has no horizontal asymptote 3. y = 2.
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