Chapter 1: Q41P (page 32)

$f (x) = e^{x}, a = 3 .$

Short Answer

Expert verified

The first few terms of Taylor’s series expansion are:

$e^{x} = e^{3} \cdot [1 + (x - 3) + \frac{(x - 3)^{2}}{2} + \frac{(x - 3)^{3}}{6} + \dots]$ .

Step by step solution

Given Information

The given function.

$f (x) = e^{x} .$

Definition of Taylor’s series

The Taylor’s series is a power series that yields the expansion of a function in the vicinity of a point if the function is continuous, all of its derivatives exist, and the series converges.

Find the Taylor’s series for ex

Use the Maclaurin series of and replace $x$ by $x - 3$ ,

localid="1657388907873" $e^{x} = {e^{3 + (x - 3}}^{)} e^{x} = e^{3} \cdot e^{(x - 3)} e^{x} = 1 + x + \frac{x^{2}}{2} + \frac{x^{3}}{6} + \dots e^{(x - 3)} = 1 + (x - 3) + \frac{(x - 3)^{2}}{2} + \frac{(x - 3)^{3}}{6} + \dots$

Write first few terms of Taylor’s series about localid="1657388920739" $a = 3$

localid="1657388930350" $e^{x} = e^{3} \cdot e^{(x - 3)} = e^{3} \cdot [1 + (x - 3) + \frac{(x - 3)^{2}}{2} + \frac{(x - 3)^{3}}{6} + \dots]$

Hence, first few terms of Taylor’s series expansion are:

localid="1657388838541" $e^{x} = e^{3} \cdot [1 + (x - 3) + \frac{(x - 3)^{2}}{2} + \frac{(x - 3)^{3}}{6} + \dots]$

Unlock Step-by-Step Solutions & Ace Your Exams!

Full Textbook Solutions
Get detailed explanations and key concepts
Unlimited Al creation
Al flashcards, explanations, exams and more...
Ads-free access
To over 500 millions flashcards
Money-back guarantee
We refund you if you fail your exam.

Start your free trial

Over 30 million students worldwide already upgrade their learning with Vaia!

Recommended explanations on Physics Textbooks

View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

$f (x) = e^{x}, a = 3 .$

Short Answer

Step by step solution

Given Information

Definition of Taylor’s series

Find the Taylor’s series for ex

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Physics Textbooks

Force

Conservation of Energy and Momentum

Thermodynamics

Famous Physicists

Work Energy and Power

Oscillations

Study anywhere. Anytime. Across all devices.

f(x)=ex,a=3.

Short Answer

Step by step solution

Given Information

Definition of Taylor’s series

Find the Taylor’s series for ex

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Physics Textbooks

Force

Conservation of Energy and Momentum

Thermodynamics

Famous Physicists

Work Energy and Power

Oscillations

Study anywhere. Anytime. Across all devices.

$f (x) = e^{x}, a = 3 .$