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Chapter 1: Problem 3

Find the \(36^{\text {th }}\) Maclaurin Polynomial for \(f(x)=e^{x}\)

Short Answer

Expert verified
The \(36^{\text{th}}\) Maclaurin Polynomial for \(e^x\) is \(P_{35}(x) = 1 + x + \frac{x^2}{2!} + \, ... \, + \frac{x^{35}}{35!}\).

Step by step solution

01

Understand the Maclaurin Series

The Maclaurin series for a function \( f(x) \) is a Taylor series expansion about \( x = 0 \). For \( f(x) = e^x \), it is given by: \[ e^x = 1 + x + \frac{x^2}{2!} + \frac{x^3}{3!} + \cdots + \frac{x^n}{n!} + \cdots \]
02

Identify the Polynomial Order

We are asked to find the \(36^{\text{th}}\) Maclaurin Polynomial. This means we need to find the polynomial that includes terms up to \(x^{35}\), since the \(0^{\text{th}}\) term is the constant term \(1\).
03

Write Down the Polynomial Terms

Using the formula for each term in the series: \( \frac{x^n}{n!} \), we write the polynomial up to the term \( \frac{x^{35}}{35!} \). The polynomial would thus be: \[ P_{35}(x) = 1 + x + \frac{x^2}{2!} + \frac{x^3}{3!} + \cdots + \frac{x^{35}}{35!} \]
04

Verify the Polynomial

Ensure all terms from the constant term up through the \(35^{\text{th}}\) power are included. The Maclaurin polynomial \( P_{35}(x) \) should match all the calculated terms below the \(36^{\text{th}}\) derivative level.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Taylor series
The Taylor series is a vital concept in calculus, serving as a tool for approximating complex functions with simpler polynomials. It represents a function as an infinite sum of terms calculated from the values of its derivatives at a single point. Typically, the series expands around a specific point, denoted as \( x = a \). If the point is \( x = 0 \), this special case is known as the Maclaurin series.
  • The Taylor series allows us to represent functions like \( \sin(x) \), \( \cos(x) \), and \( e^x \) in polynomial form.
  • This simplification can help with evaluating functions and solving equations more easily.
The approximation improves as we include more terms, closely matching the behavior of the original function over a specific interval.
Polynomial approximation
Polynomial approximation forms a core aspect of the Taylor series, focusing on expressing a function as a sum of polynomial terms. By utilizing the values of a function's derivatives, we can construct increasingly accurate models of the function's behavior.
  • Each term in the polynomial is derived from the function's derivatives at a single point, which contribute to the shape and accuracy of the approximation.
  • The order of the polynomial indicates the highest degree term used in the approximation; the higher this order, the more accurate the approximation.
An accurate polynomial approximation is especially useful for calculations and simulations in fields where exact functions are complex or require significant computational effort.
Exponential function
The exponential function, denoted by \( e^x \), is one of the most prominent mathematical functions due to its unique properties and frequent appearance in various scientific fields. It is the base exponential expression where \( e \) is Euler's number, approximately 2.71828.
  • One distinctive feature of \( e^x \) is that its derivative and integral are the same as the function itself, making it vital in calculus and differential equations.
  • The Maclaurin series for the exponential function provides a straightforward way to approximate \( e^x \) using only polynomial terms:
\[e^x = 1 + x + \frac{x^2}{2!} + \frac{x^3}{3!} + \cdots \] This series is essential for simplifying calculations involving exponential growth or decay.
Calculus
Calculus, often regarded as the mathematics of change, underpins a significant portion of modern mathematical analysis and applications. It includes two primary branches: differential calculus and integral calculus.
  • Differential calculus focuses on the idea of instantaneous rates of change, represented by derivatives.
  • Integral calculus, on the other hand, deals with the accumulation of quantities, represented by integrals.
Using calculus, mathematicians and scientists can analyze and describe changes related to functions, such as the slope of a curve, area under a curve, and optimization problems. Techniques from calculus are pivotal in deriving Taylor and Maclaurin series, as they heavily rely on derivatives to create polynomial approximations.

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Most popular questions from this chapter

Show that $$ (\sin h)(1-\cos h)=0+O\left(h^{3}\right) $$

Some linearly convergent sequences and their limits are given. Find the (fastest) rate of convergence of the form \(O\left(\frac{1}{n^{p}}\right)\) or \(O\left(\frac{1}{a^{n}}\right)\) for each. If this is not possible, suggest a reasonable rate of convergence. (a) \(6, \frac{6}{7}, \frac{6}{49}, \frac{6}{343}, \frac{6}{2401}, \ldots \rightarrow 0\) (b) \(\left\langle\frac{11 n-2}{n+3}\right\rangle \rightarrow 11\) (c) \(\left\langle\frac{\sin n}{\sqrt{n}}\right\rangle \rightarrow 0\), (d) \(\left\langle\frac{4}{10^{n}+35 n+9}\right\rangle \rightarrow 0^{[\text {[s] }}\) (e) \(\left\langle\frac{1}{n}+\frac{1}{n^{2}}\right\rangle \rightarrow 0\) (f) \(\left\langle\frac{4}{10^{n}-35 n-9}\right\rangle \rightarrow 0^{[s]}\) (g) \(\left\langle\frac{2 n}{\sqrt{n^{2}+3 n}}\right\rangle \rightarrow 2^{[A]}\) (h) \(\left\langle\frac{5^{n}-2}{5^{n}+3}\right\rangle \rightarrow 1\) (i) \(\langle\sqrt{n+47}-\sqrt{n}\rangle \rightarrow 0^{[A]}\) (j) \(\left\langle\frac{n^{2}}{3 n^{2}+1}\right\rangle \rightarrow \frac{1}{3}\) (k) \(\left\langle\frac{\pi}{e^{n}-\pi^{n}}\right\rangle \rightarrow 0\) (1) \(\left\langle\frac{n^{2}}{2^{n}}\right\rangle \rightarrow 0^{[\mathbb{S}]}\) (m) \(\left\langle\frac{7+\cos (5 n)}{n^{3}+1}\right\rangle \rightarrow 0\) (n) \(\left\langle\frac{8 n^{2}}{3 n^{2}+12}+\frac{n}{3 n+10}\right\rangle \rightarrow 3\) (o) \(\left\langle\frac{2 n^{2}+3 n}{1-n^{2}}\right\rangle \rightarrow-2^{[A]}\) (p) \(\left\langle\frac{3 n^{5}-5 n}{1-n^{5}}\right\rangle \rightarrow-3\)

Compute the absolute and relative errors in the approximation of \(\pi\) by \(3 .\)

Write an Octave program (.m file) that uses a loop. an array, and the disp() command to find the values of \(f(n)=\frac{2 n}{\sqrt{n^{2}+3 n}}\) for \(n=0,2,5,10,100,1000,20000\)

Write a for loop that outputs the sequence of numbers. (a) 7,8,9,10,11,12,13,14,15 (b) 20,19,18,17,16,15,14,13 (c) 12,12.333,12.667,13,13.333,13.667,14 (d) 1,9,25,49,81,121,169,225,289,361,441 (e) 1, .5, .25, .125, .0625, .03125, .015625
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