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1 Problem 2 (a), (b) from the textbook [8 points] Prove that the following equations have at least one solution in the given interval. (a) √√x cos(x) = 0, interval [0, 1]. - (b) ex²+3x-2 = 0, interval [0, 1]. 2 Problem 4 (b), (c) from the textbook [8 points] Find intervals containing solutions to the following equations. (a) 4x² - ex = 0, (b) x3 2x24x + 2 = 0. 3 Problem 6 (d) from the textbook [8 points] Find maxo≤x≤1 |f(x)|, where f(x) = x√√3x². Hint: One of the possible method is to consi g(x) = f(x) instead of f. 4 Taylor series [8 points] Let f(x) = cos(x³)+(x+1)². Find the second Taylor polynomial P2(x) and sixth Taylor polynom P6(x) at x=0 and use it to approximate f(0.1). 5 Problem 14 (a), (b) from the textbook [8 points] Let f(x) = 2x cos(2x) - (x-2)². Find the third Taylor polynomial P3(x) at x0 = 0 and use it approximate f(0.4). Then use the error formula in Taylor's Theorem to find an upper bound the error |f(0.4) − P3(0.4)| < 0.1.

1 Problem 2 (a), (b) from the textbook [8 points] Prove that the following equations have at least one solution in the given interval. (a) √√x cos(x) = 0, interval [0, 1]. - (b) ex²+3x-2 = 0, interval [0, 1]. 2 Problem 4 (b), (c) from the textbook [8 points] Find intervals containing solutions to the following equations. (a) 4x² - ex = 0, (b) x3 2x24x + 2 = 0. 3 Problem 6 (d) from the textbook [8 points] Find maxo≤x≤1 |f(x)|, where f(x) = x√√3x². Hint: One of the possible method is to consi g(x) = f(x) instead of f. 4 Taylor series [8 points] Let f(x) = cos(x³)+(x+1)². Find the second Taylor polynomial P2(x) and sixth Taylor polynom P6(x) at x=0 and use it to approximate f(0.1). 5 Problem 14 (a), (b) from the textbook [8 points] Let f(x) = 2x cos(2x) - (x-2)². Find the third Taylor polynomial P3(x) at x0 = 0 and use it approximate f(0.4). Then use the error formula in Taylor's Theorem to find an upper bound the error |f(0.4) − P3(0.4)| < 0.1.

Algebra & Trigonometry with Analytic Geometry
Algebra & Trigonometry with Analytic Geometry
13th Edition
ISBN:9781133382119
Author:Swokowski
Publisher:Swokowski
Chapter7: Analytic Trigonometry
Section7.2: Trigonometric Equations
Problem 30E
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1 Problem 2 (a), (b) from the textbook [8 points] Prove that the following equations have at least one solution in the given interval. (a) √√x cos(x) = 0, interval [0, 1]. - (b) ex²+3x-2 = 0, interval [0, 1]. 2 Problem 4 (b), (c) from the textbook [8 points] Find intervals containing solutions to the following equations. (a) 4x² - ex = 0, (b) x3 2x24x + 2 = 0. 3 Problem 6 (d) from the textbook [8 points] Find maxo≤x≤1 |f(x)|, where f(x) = x√√3x². Hint: One of the possible method is to consi g(x) = f(x) instead of f. 4 Taylor series [8 points] Let f(x) = cos(x³)+(x+1)². Find the second Taylor polynomial P2(x) and sixth Taylor polynom P6(x) at x=0 and use it to approximate f(0.1). 5 Problem 14 (a), (b) from the textbook [8 points] Let f(x) = 2x cos(2x) - (x-2)². Find the third Taylor polynomial P3(x) at x0 = 0 and use it approximate f(0.4). Then use the error formula in Taylor's Theorem to find an upper bound the error |f(0.4) − P3(0.4)| < 0.1.
Transcribed Image Text:1 Problem 2 (a), (b) from the textbook [8 points] Prove that the following equations have at least one solution in the given interval. (a) √√x cos(x) = 0, interval [0, 1]. - (b) ex²+3x-2 = 0, interval [0, 1]. 2 Problem 4 (b), (c) from the textbook [8 points] Find intervals containing solutions to the following equations. (a) 4x² - ex = 0, (b) x3 2x24x + 2 = 0. 3 Problem 6 (d) from the textbook [8 points] Find maxo≤x≤1 |f(x)|, where f(x) = x√√3x². Hint: One of the possible method is to consi g(x) = f(x) instead of f. 4 Taylor series [8 points] Let f(x) = cos(x³)+(x+1)². Find the second Taylor polynomial P2(x) and sixth Taylor polynom P6(x) at x=0 and use it to approximate f(0.1). 5 Problem 14 (a), (b) from the textbook [8 points] Let f(x) = 2x cos(2x) - (x-2)². Find the third Taylor polynomial P3(x) at x0 = 0 and use it approximate f(0.4). Then use the error formula in Taylor's Theorem to find an upper bound the error |f(0.4) − P3(0.4)| < 0.1.
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