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give the calculation step by step as like as full answer in 2 steps and explanation block in every steps. And final answer summary Definition of Parameters Solutions are reviewed by our quality team mainly based on 6 parameters: Relevancy, Completeness, Accuracy, Clarity, Structure, and Voice. Relevancy: This indicates how closely a solution aligns with the question asked. It evaluates whether the solution addresses the core aspects of the question and provides appropriate information. Completeness: This indicates how thoroughly the solution addresses various asks in the question. It evaluates whether the solution has all essential information like calculations, diagrams, procedural steps, & explanatory statements in the solution. Accuracy: It refers to how correct the provided solution is for the given question/s. This evaluates the correctness of the information/concept/method followed in the solution. Clarity: Clarity refers to how easily the solution can be understood by the student. This evaluates the language complexity, readability, and redundancy of a solution. Structure: Structure refers to how the solution is organized and presented in relation to the requirements of the question. This evaluates the necessity of steps and their effectiveness in helping the student grasp how to arrive at the solution. Voice: This refers to the language and tone of the solution. This primarily evaluates whether a solution sounds human or overly non-human (bot-like). don't use we, then,:.V'. Tavorpoimomis 3 Write down the Tylor polynomial for x0 x” abot x=0. 1) Using the polynomial in part a), estimate the negra J; xsin x? de with error of magnitude ss than 10-¢

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give the calculation step by step as like as full answer in 2 steps and explanation block in every steps. And final answer summary Definition of Parameters Solutions are reviewed by our quality team mainly based on 6 parameters: Relevancy, Completeness, Accuracy, Clarity, Structure, and Voice. Relevancy: This indicates how closely a solution aligns with the question asked. It evaluates whether the solution addresses the core aspects of the question and provides appropriate information. Completeness: This indicates how thoroughly the solution addresses various asks in the question. It evaluates whether the solution has all essential information like calculations, diagrams, procedural steps, & explanatory statements in the solution. Accuracy: It refers to how correct the provided solution is for the given question/s. This evaluates the correctness of the information/concept/method followed in the solution. Clarity: Clarity refers to how easily the solution can be understood by the student. This evaluates the language complexity, readability, and redundancy of a solution. Structure: Structure refers to how the solution is organized and presented in relation to the requirements of the question. This evaluates the necessity of steps and their effectiveness in helping the student grasp how to arrive at the solution. Voice: This refers to the language and tone of the solution. This primarily evaluates whether a solution sounds human or overly non-human (bot-like). don't use we, then,:.Uploaded ImageV'. Tavorpoimomis 3 Write down the Tylor polynomial for x0 x” abot x=0. 1) Using the polynomial in part a), estimate the negra J; xsin x? de with error of magnitude ss than 10-¢

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Asked by: Test User 100241
Created at: 2025-05-16 01:27:52

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### **Step-by-Step Solution** --- #### **Step 1: Find the Taylor Polynomial for \( x \sin(x^2) \) about \( x = 0 \)** **Explanation:** Begin by recalling the Taylor series for \( \sin(x) \) about \( x = 0 \): \[ \sin(x) = x - \frac{x^3}{3!} + \frac{x^5}{5!} - \cdots \] Substitute \( x^2 \) in place of \( x \): \[ \sin(x^2) = x^2 - \frac{(x^2)^3}{3!} + \frac{(x^2)^5}{5!} - \cdots = x^2 - \frac{x^6}{6} + \frac{x^{10}}{120} - \cdots \] Multiplying by \( x \): \[ x \sin(x^2) = x(x^2 - \frac{x^6}{6} + \frac{x^{10}}{120} - \cdots) = x^3 - \frac{x^7}{6} + \frac{x^{11}}{120} - \cdots \] The Taylor polynomial up to degree 11 is: \[ x \sin(x^2) \approx x^3 - \frac{x^7}{6} + \frac{x^{11}}{120} \] --- #### **Step 2: Estimate the Integral \( \int_0^1 x \sin(x^2) \, dx \) with Error Less Than \( 10^{-6} \)** **Explanation:** Integrate the polynomial from part (a): \[ \int_0^1 x \sin(x^2) \, dx \approx \int_0^1 \left( x^3 - \frac{x^7}{6} + \frac{x^{11}}{120} \right) dx \] Compute each term: - \( \int_0^1 x^3 dx = \left[ \frac{x^4}{4} \right]_0^1 = \frac{1}{4} \) - \( \int_0^1 x^7 dx = \left[ \frac{x^8}{8} \right]_0^1 = \frac{1}{8} \) - \( \int_0^1 x^{11} dx = \left[ \frac{x^{12}}{12} \right]_0^1 = \frac{1}{12} \) So, \[ \int_0^1 x \sin(x^2)dx \approx \frac{1}{4} - \frac{1}{6 \times 8} + \frac{1}{120 \times 12} \] \[ = \frac{1}{4} - \frac{1}{48} + \frac{1}{1440} \] Calculate each term numerically: - \( \frac{1}{4} = 0.25 \) - \( \frac{1}{48} \approx 0.020833 \) - \( \frac{1}{1440} \approx 0.000694 \) Combine: \[ 0.25 - 0.020833 + 0.000694 = 0.229861 \] Now, estimate the error. The next term in the Taylor expansion is \( -\frac{x^{15}}{5040} \). Its integral from 0 to 1: \[ \left| \int_0^1 -\frac{x^{15}}{5040} dx \right| = \frac{1}{5040} \cdot \frac{1}{16} = \frac{1}{80640} \approx 1.24 \times 10^{-5} \] This error is less than \( 10^{-4} \) but not yet \( 10^{-6} \). Include the next term: - The next term is \( +\frac{x^{19}}{362880} \), and its integral is \( \frac{1}{362880 \times 20} \approx 1.38 \times 10^{-7} \), making the error below \( 10^{-6} \). So, add this term: \[ \int_0^1 x^{19} dx = \frac{1}{20} \] \[ \frac{1}{362880 \times 20} = \frac{1}{7,257,600} \approx 1.38 \times 10^{-7} \] So, the improved estimate: \[ \int_0^1 x \sin(x^2) dx \approx \frac{1}{4} - \frac{1}{48} + \frac{1}{1440} - \frac{1}{80640} + \frac{1}{7,257,600} \] Numerically: - \( -\frac{1}{80640} \approx -0.0000124 \) - \( +\frac{1}{7,257,600} \approx 0.000000138 \) Total sum: \[ 0.25 - 0.020833 + 0.000694 - 0.0000124 + 0.000000138 \approx 0.2298497 \] --- ### **Final Answer Summary** **Taylor polynomial for \( x \sin(x^2) \) about \( x = 0 \):** \[ x^3 - \frac{x^7}{6} + \frac{x^{11}}{120} - \frac{x^{15}}{5040} + \frac{x^{19}}{362880} \] **Estimated value for \( \int_0^1 x \sin(x^2) dx \) with error less than \( 10^{-6} \):** \[ \boxed{0.22985} \] (Rounded to five decimal places; error is less than \( 10^{-6} \)).

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