Let f(x)=e−xf(x)=e−x. We're going to calculate the Taylor series for this function near a=0a=0 .Calculate f(0),f′(0),f′′(0),f(0),f′(0),f″(0), and f(3)(0)f(3)(0) . What patterns do you notice here?Write down the degree 3 Taylor polynomial approximation for f(x)f(x) near a=0a=0 .Make a graph that shows the function and the degree 3 Taylor polynomial. For what interval of x-values is this a relatively good approximation? For which x-values is this not a very good approximation?Based on the patterns you observed in (a), find a general formula for the k-th derivative of this function when evaluated at 0.f(k)(0)=?f(k)(0)=? Write the Taylor series fo f(x)f(x) near a=0a=0, using sigma notation.What is the interval of convergence for this series?What would be different about your result if we instead calculated the Taylor series for f(x)f(x) near a=4a=4 ? 1. Let f (x) = e-7. We're going to calculate the Taylor series for this function near a = 0.a. Calculate f (0), ƒ' (0), f" (0), and f(3) (0) . What patterns do you notice here?b. Write down the degree 3 Taylor polynomial approximation for f (x) near a = 0.c. Make a graph that shows the function and the degree 3 Taylor polynomial. For what interval of x-values is this a relatively good approximation? Forwhich x-values is this not a very good approximation?d. Based on the patterns you observed in (a), find a general formula for the k-th derivative of this function when evaluated at 0.f(k) (0) =?e. Write the Taylor series fo f (x) near a = 0, using sigma notation.f. What is the interval of convergence for this series?g. What would be different about your result if we instead calculated the Taylor series for f (x) near a = 4?

Since you have posted question with multiple subparts we provide only solution first 3 for…

Answered: 1. Let f (x) = e¯ª. We're going to…

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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Let f(x)=e−xf(x)=e−x. We're going to calculate the Taylor series for this function near a=0a=0 .

Calculate f(0),f′(0),f′′(0),f(0),f′(0),f″(0), and f(3)(0)f(3)(0) . What patterns do you notice here?
Write down the degree 3 Taylor polynomial approximation for f(x)f(x) near a=0a=0 .
Make a graph that shows the function and the degree 3 Taylor polynomial. For what interval of x-values is this a relatively good approximation? For which x-values is this not a very good approximation?
Based on the patterns you observed in (a), find a general formula for the k-th derivative of this function when evaluated at 0.
f(k)(0)=?f(k)(0)=?
Write the Taylor series fo f(x)f(x) near a=0a=0, using sigma notation.
What is the interval of convergence for this series?
What would be different about your result if we instead calculated the Taylor series for f(x)f(x) near a=4a=4 ?

1. Let f (x) = e-7. We're going to calculate the Taylor series for this function near a = 0.
a. Calculate f (0), ƒ' (0), f" (0), and f(3) (0) . What patterns do you notice here?
b. Write down the degree 3 Taylor polynomial approximation for f (x) near a = 0.
c. Make a graph that shows the function and the degree 3 Taylor polynomial. For what interval of x-values is this a relatively good approximation? For
which x-values is this not a very good approximation?
d. Based on the patterns you observed in (a), find a general formula for the k-th derivative of this function when evaluated at 0.
f(k) (0) =?
e. Write the Taylor series fo f (x) near a = 0, using sigma notation.
f. What is the interval of convergence for this series?
g. What would be different about your result if we instead calculated the Taylor series for f (x) near a = 4?

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