Book cover for Calculus and Its Applications

Calculus and Its Applications

Larry J. Goldstein, David C. Lay, David I. Schneider

ISBN #9780134437774

14th Edition

3,033 Questions

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63,186 Students Helped

Homework Questions

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Summary
Learning Objectives
Key Concepts
Example Problems
Explanations
Common Mistakes

Summary

This section introduces Taylor polynomials as a powerful tool for approximating functions near a point, alongside Taylor series for representing functions as infinite sums. It also covers the iterative Newton–Raphson algorithm for finding roots, and discusses the analysis of infinite series and series with positive terms for numerical computation. Together, these methods are fundamental in simplifying complex functions and have broad applications in calculus, differential equations, and probability theory.

Learning Objectives

1

Explain the concept of Taylor polynomials and how they approximate functions near a point.

2

Demonstrate the application of the Newton–Raphson algorithm for finding roots of functions.

3

Analyze the properties of infinite series and series with positive terms in numerical computations.

4

Derive Taylor series for common functions such as e^x, ln x, sin x, cos x, and tan x.

Key Concepts

CONCEPT

DEFINITION

Taylor Polynomial

A polynomial used to approximate a function around a specific point by matching derivatives up to a certain order.

Taylor Series

An infinite sum of terms calculated from the values of a function's derivatives at a single point, representing the function in a region around that point.

Newton–Raphson Algorithm

An iterative numerical method for finding approximations to the roots (or zeroes) of a real-valued function.

Infinite Series

A sum of an infinite sequence of terms, whose convergence properties are critical for accurately representing functions.

Series with Positive Terms

A type of infinite series where each term is positive, often simplifying convergence analysis.

Example Problems

Example 1

Determine the third Taylor polynomial of the given function at $x=0.$ $$f(x)=\sin x$$

Example 2

Determine the third Taylor polynomial of the given function at $x=0.$ $$f(x)=e^{-x / 2}$$

Example 3

Determine the third Taylor polynomial of the given function at $x=0.$ $$f(x)=5 e^{2 x}$$

Example 4

Determine the third Taylor polynomial of the given function at $x=0.$ $$f(x)=\cos (\pi-5 x)$$

Example 5

Determine the third Taylor polynomial of the given function at $x=0.$ $$f(x)=\sqrt{4 x+1}$$
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Step-by-Step Explanations

QUESTION

How do you construct a Taylor polynomial of order n for a function f(x) centered at a point a?

STEP-BY-STEP ANSWER:

Step 1: Calculate the value of the function at the point a, which is f(a).
Step 2: Compute the first derivative f'(a) and multiply by (x - a).
Step 3: Compute the second derivative f''(a), divide by 2!, and multiply by (x - a)^2.
Step 4: Continue this process up to the nth derivative f^(n)(a) divided by n! and multiplied by (x - a)^n.
Step 5: Sum all these terms to form the Taylor polynomial: P_n(x) = f(a) + f'(a)(x-a) + f''(a)(x-a)^2/2! + ... + f^(n)(a)(x-a)^n/n!.
Final Answer: The Taylor polynomial approximating f(x) near x = a is given by P_n(x) as defined above.

Taylor Polynomial

QUESTION

How can the Newton–Raphson algorithm be used to find a root of a function f(x)?

STEP-BY-STEP ANSWER:

Step 1: Choose an initial guess x_0 that is close to the presumed root.
Step 2: Evaluate the function f(x_0) and its derivative f'(x_0).
Step 3: Use the iteration formula: x_{n+1} = x_n - f(x_n)/f'(x_n).
Step 4: Repeat the iteration until the difference |x_{n+1} - x_n| is less than a desired tolerance.
Final Answer: x_{n+1} obtained after sufficient iterations approximates the root of f(x).

Newton–Raphson Algorithm

QUESTION

What is the process to determine if an infinite series converges and how can it be used to compute function values?

STEP-BY-STEP ANSWER:

Step 1: Identify the form of the series and write down its general term a_n.
Step 2: Apply convergence tests (e.g., ratio test, comparison test) to ensure the series converges.
Step 3: If the series converges, sum the series either exactly or approximate the sum by taking a sufficient number of terms.
Step 4: Use the series representation to compute function values when direct computation is challenging.
Final Answer: Convergence tests confirm the validity of summing the series, and the sum approximates the function's value.

Infinite Series

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Common Mistakes

  • Failing to check if the series converges before using the Taylor series approximation.
  • Mixing up the order of derivatives or miscalculating factorial terms in Taylor polynomials.
  • Applying the Newton–Raphson algorithm without a good initial guess, leading to divergence or incorrect results.
  • Assuming a finite Taylor polynomial gives an exact value rather than an approximation.