2. A continuous function f is defined on the interval [-2, 2]. The values of f at some of the points of the interval are given by the following table: -2 -1 1 2 f (x) -1 -1 (a) Using only this information, what can be concluded about the roots of f, that is, the solutions of f(x) = 0, in the interval [-2, 2]? The answer should be something like: f has at least 8 roots in [-2,2], Hint: Use the Intermediate Value Theorem on each of the intervals [-2, –1], [–1,0], [0, 1], and [1, 2]. or f has at most 6 roots in (-2,2]. (b) If f(x) = x4 – 4x2 + 2, verify that the relevant values of f are given by the table above. (a) Sketch the graph of y = f(x) in the viewing window [-2.5, 2.5]×[-3, 3]. (b) How many roots does f have in the interval [-2,2]? Find the roots algebraically. Suggestion: Let t = x2 and solve with the quadratic formula. Then find x. (c) If f(x) = x4 – 4x2 + 2+ 5(2x – 1)æ(x² – 1)(x² – 4). Verify that the relevant values of f are given by the table above. (a) Sketch the graph of y = f(x) in the viewing window [-2.5, 2.5]x[-80, 80]. (b) Explain why f has at least one root in each of the intervals (-2, 1), (–1,0), (0,1), and (1,2). (c) Sketch the graph of y = f(x) in the viewing window [0,1]×[-1, 3]. (d) How many roots does f have in the interval [0, 1]? Approximate the roots of f in [0, 1] to three decimal places using a calculator. (d) Having done (b) and (c), was your original conclusion in part (a) correct?
2. A continuous function f is defined on the interval [-2, 2]. The values of f at some of the points of the interval are given by the following table: -2 -1 1 2 f (x) -1 -1 (a) Using only this information, what can be concluded about the roots of f, that is, the solutions of f(x) = 0, in the interval [-2, 2]? The answer should be something like: f has at least 8 roots in [-2,2], Hint: Use the Intermediate Value Theorem on each of the intervals [-2, –1], [–1,0], [0, 1], and [1, 2]. or f has at most 6 roots in (-2,2]. (b) If f(x) = x4 – 4x2 + 2, verify that the relevant values of f are given by the table above. (a) Sketch the graph of y = f(x) in the viewing window [-2.5, 2.5]×[-3, 3]. (b) How many roots does f have in the interval [-2,2]? Find the roots algebraically. Suggestion: Let t = x2 and solve with the quadratic formula. Then find x. (c) If f(x) = x4 – 4x2 + 2+ 5(2x – 1)æ(x² – 1)(x² – 4). Verify that the relevant values of f are given by the table above. (a) Sketch the graph of y = f(x) in the viewing window [-2.5, 2.5]x[-80, 80]. (b) Explain why f has at least one root in each of the intervals (-2, 1), (–1,0), (0,1), and (1,2). (c) Sketch the graph of y = f(x) in the viewing window [0,1]×[-1, 3]. (d) How many roots does f have in the interval [0, 1]? Approximate the roots of f in [0, 1] to three decimal places using a calculator. (d) Having done (b) and (c), was your original conclusion in part (a) correct?
Calculus: Early Transcendentals
8th Edition
ISBN:9781285741550
Author:James Stewart
Publisher:James Stewart
Chapter1: Functions And Models
Section: Chapter Questions
Problem 1RCC: (a) What is a function? What are its domain and range? (b) What is the graph of a function? (c) How...
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![2. A continuous function f is defined on the interval [-2, 2]. The values of f at some of the points
of the interval are given by the following table:
-2
-1
1
2
|f(x)
2
-1
2
-1
2
(a) Using only this information, what can be concluded about the roots of f, that is, the
solutions of f(x) = 0, in the interval [-2, 2]? The answer should be something like: f has
at least 8 roots in [-2,2], or ƒ has at most 6 roots in [-2, 2].
Hint: Use the Intermediate Value Theorem on each of the intervals [-2, –1], [-1,0], [0,1],
and [1, 2].
(b) If f(x) = xª – 4.x² + 2, verify that the relevant values of f are given by the table above.
(a) Sketch the graph of y = f(x) in the viewing window [-2.5, 2.5]×[-3,3].
(b) How many roots does f have in the interval [-2,2]? Find the roots algebraically.
Suggestion: Let t = x? and solve with the quadratic formula. Then find x.
(c) If f(x) = x+ – 4x? + 2+ 5(2x – 1)x(x² – 1)(x² – 4). Verify that the relevant values of f
are given by the table above.
(a) Sketch the graph of y = f(x) in the viewing window [-2.5, 2.5] ×[-80, 80)].
(b) Explain why f has at least one root in each of the intervals (-2, 1), (–1,0), (0,1), and
(1,2).
(c) Sketch the graph of y = f(x) in the viewing window [0,1]×[-1, 3].
(d) How many roots does f have in the interval [0, 1]? Approximate the roots of f in [0, 1]
to three decimal places using a calculator.
(d) Having done (b) and (c), was your original conclusion in part (a) correct?](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fc31d976c-acd4-451e-8eb9-fde305025b60%2F7b3f1058-25ae-4581-8493-4aba6c6b5feb%2F69v805q_processed.png&w=3840&q=75)
Transcribed Image Text:2. A continuous function f is defined on the interval [-2, 2]. The values of f at some of the points
of the interval are given by the following table:
-2
-1
1
2
|f(x)
2
-1
2
-1
2
(a) Using only this information, what can be concluded about the roots of f, that is, the
solutions of f(x) = 0, in the interval [-2, 2]? The answer should be something like: f has
at least 8 roots in [-2,2], or ƒ has at most 6 roots in [-2, 2].
Hint: Use the Intermediate Value Theorem on each of the intervals [-2, –1], [-1,0], [0,1],
and [1, 2].
(b) If f(x) = xª – 4.x² + 2, verify that the relevant values of f are given by the table above.
(a) Sketch the graph of y = f(x) in the viewing window [-2.5, 2.5]×[-3,3].
(b) How many roots does f have in the interval [-2,2]? Find the roots algebraically.
Suggestion: Let t = x? and solve with the quadratic formula. Then find x.
(c) If f(x) = x+ – 4x? + 2+ 5(2x – 1)x(x² – 1)(x² – 4). Verify that the relevant values of f
are given by the table above.
(a) Sketch the graph of y = f(x) in the viewing window [-2.5, 2.5] ×[-80, 80)].
(b) Explain why f has at least one root in each of the intervals (-2, 1), (–1,0), (0,1), and
(1,2).
(c) Sketch the graph of y = f(x) in the viewing window [0,1]×[-1, 3].
(d) How many roots does f have in the interval [0, 1]? Approximate the roots of f in [0, 1]
to three decimal places using a calculator.
(d) Having done (b) and (c), was your original conclusion in part (a) correct?
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