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a. Compute T₂(x) at x = 0.8 for y = e. T₂(x) b. Use a calculator to compute the error |e* - T₂(x)| at x = 0. (Round your answer to within six decimal places.) |e* - T₂(x)| =

a. Compute T₂(x) at x = 0.8 for y = e. T₂(x) b. Use a calculator to compute the error |e* - T₂(x)| at x = 0. (Round your answer to within six decimal places.) |e* - T₂(x)| =

Calculus: Early Transcendentals
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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--- ### Taylor Polynomial Approximation Practice #### Problem Statement a. **Compute \( T_2(x) \) at \( x = 0.8 \) for \( y = e^x \).** The second degree Taylor polynomial for the function \( e^x \) centered at \( x = 0 \) is given by: \[ T_2(x) = \] b. **Use a calculator to compute the error \( |e^x - T_2(x)| \) at \( x = 0 \).** *(Round your answer to within six decimal places.)* \[ |e^x - T_2(x)| = \] --- ### Explanation In this exercise, you will practice using Taylor polynomial approximations for the exponential function \( e^x \). **Step 1: Compute the Taylor Polynomial, \( T_2(x) \)** For a function \( y = e^x \), the second-degree Taylor polynomial centered at \( x = 0 \) is computed using the formula: \[ T_2(x) = 1 + x + \frac{x^2}{2} \] Compute \( T_2(x) \) at \( x = 0.8 \). **Step 2: Calculate the Error** Next, calculate the true value of \( e^x \) at \( x = 0 \) and compare it with the Taylor polynomial \( T_2(x) \) at the same point. Compute the absolute error and round it to within six decimal places. Use this value: \[ |e^x - T_2(x)| = \] ---
Transcribed Image Text:--- ### Taylor Polynomial Approximation Practice #### Problem Statement a. **Compute \( T_2(x) \) at \( x = 0.8 \) for \( y = e^x \).** The second degree Taylor polynomial for the function \( e^x \) centered at \( x = 0 \) is given by: \[ T_2(x) = \] b. **Use a calculator to compute the error \( |e^x - T_2(x)| \) at \( x = 0 \).** *(Round your answer to within six decimal places.)* \[ |e^x - T_2(x)| = \] --- ### Explanation In this exercise, you will practice using Taylor polynomial approximations for the exponential function \( e^x \). **Step 1: Compute the Taylor Polynomial, \( T_2(x) \)** For a function \( y = e^x \), the second-degree Taylor polynomial centered at \( x = 0 \) is computed using the formula: \[ T_2(x) = 1 + x + \frac{x^2}{2} \] Compute \( T_2(x) \) at \( x = 0.8 \). **Step 2: Calculate the Error** Next, calculate the true value of \( e^x \) at \( x = 0 \) and compare it with the Taylor polynomial \( T_2(x) \) at the same point. Compute the absolute error and round it to within six decimal places. Use this value: \[ |e^x - T_2(x)| = \] ---
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