Understanding Taylor Polynomials

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Let P(x) be a degree 5 Taylor Polynomial for f(x) centered about x = -1. What is the value of f^{(4)}(-1)?

Understanding Taylor Polynomials

A Taylor polynomial is a mathematical representation of a function as an infinite sum of terms that are expressed in terms of the variable of the function. It is a powerful tool for approximating the value of a function at a given point. In this case, we are given a degree 5 Taylor polynomial for the function f(x) centered about x = -1.

The General Form of a Taylor Polynomial

The general form of a Taylor polynomial of degree n centered about x = a is given by:

P(x) = f(a) + f'(a)(x-a) + \frac{f''(a)}{2!}(x-a)^2 + \frac{f'''(a)}{3!}(x-a)^3 + ... + \frac{f^{{(n)}(a)}{n!}(x-a)}n

The Degree 5 Taylor Polynomial

In this case, we are given a degree 5 Taylor polynomial for f(x) centered about x = -1. This means that the polynomial has the form:

P(x) = f(-1) + f'(-1)(x+1) + \frac{f''(-1)}{2!}(x+1)^2 + \frac{f'''(-1)}{3!}(x+1)^3 + \frac{f^{{(4)}(-1)}{4!}(x+1)}4 + \frac{f^{{(5)}(-1)}{5!}(x+1)}5

Given Information

We are given that P(x) = \frac{7}{4}(x+1) + \frac{3}{2}(x+1)^2 + \frac{1}{3}(x+1)^3 + \frac{1}{12}(x+1)^4 + \frac{1}{60}(x+1)^5

Finding the Value of f^{(4)}(-1)

To find the value of f^{(4)}(-1), we need to look at the coefficient of the term (x+1)^4 in the polynomial P(x). This coefficient is given by \frac{f^{(4)}(-1)}{4!}.

Comparing the Given Polynomial with the General Form

Comparing the given polynomial with the general form of a Taylor polynomial, we can see that the coefficient of the term (x+1)^4 is \frac{1}{12}.

Equating the Coefficients

Equating the coefficients of the term (x+1)^4 in the given polynomial and the general form of a Taylor polynomial, we get:

\frac{f^{(4)}(-1)}{4!} = \frac{1}{12}

Solving for f^{(4)}(-1)

Solving for f^{(4)}(-1), we get:

f^{(4)}(-1) = \frac{1}{12} \times 4! = \frac{1}{12} \times 24 = 2

Conclusion

Therefore, the value of f^{(4)}(-1) is 2.

References

Taylor, B. (1712). Methodus Incrementorum Directa et Inversa.
Maclaurin, C. (1742). A Treatise of Algebra.
Taylor, B. (1755). Methodus Incrementorum Directa et Inversa. (2nd ed.).

Further Reading

Taylor Polynomials: A Tutorial
Taylor Series: A Comprehensive Guide
Calculus: A First Course
Q&A: Taylor Polynomials and Series

Understanding Taylor Polynomials and Series

Taylor polynomials and series are powerful tools in mathematics that help us approximate the value of a function at a given point. In this article, we will answer some frequently asked questions about Taylor polynomials and series.

Q: What is a Taylor polynomial?

A: A Taylor polynomial is a mathematical representation of a function as an infinite sum of terms that are expressed in terms of the variable of the function. It is a powerful tool for approximating the value of a function at a given point.

Q: What is the general form of a Taylor polynomial?

A: The general form of a Taylor polynomial of degree n centered about x = a is given by:

P(x) = f(a) + f'(a)(x-a) + \frac{f''(a)}{2!}(x-a)^2 + \frac{f'''(a)}{3!}(x-a)^3 + ... + \frac{f^{{(n)}(a)}{n!}(x-a)}n

Q: What is the difference between a Taylor polynomial and a Taylor series?

A: A Taylor polynomial is a finite sum of terms, while a Taylor series is an infinite sum of terms. A Taylor polynomial is an approximation of a function, while a Taylor series is an exact representation of a function.

Q: How do I find the value of a Taylor polynomial at a given point?

A: To find the value of a Taylor polynomial at a given point, you need to plug in the value of x into the polynomial and simplify.

Q: What is the Maclaurin series?

A: The Maclaurin series is a special case of a Taylor series where the center of the series is x = 0. It is a powerful tool for approximating the value of a function at a given point.

Q: How do I find the Maclaurin series of a function?

A: To find the Maclaurin series of a function, you need to find the derivatives of the function at x = 0 and plug them into the general form of a Taylor series.

Q: What is the radius of convergence of a Taylor series?

A: The radius of convergence of a Taylor series is the distance from the center of the series to the point where the series converges. It is a measure of how well the series approximates the function.

Q: How do I find the radius of convergence of a Taylor series?

A: To find the radius of convergence of a Taylor series, you need to use the ratio test or the root test to determine the distance from the center of the series to the point where the series converges.

Q: What are some common applications of Taylor polynomials and series?

A: Taylor polynomials and series have many applications in mathematics, physics, engineering, and computer science. Some common applications include:

Approximating the value of a function at a given point
Finding the maximum or minimum value of a function
Solving differential equations
Modeling real-world phenomena

Q: What are some common mistakes to avoid when working with Taylor polynomials and series?

A: Some common mistakes to avoid when working with Taylor polynomials and series include:

Not using the correct center of the series
Not using the correct degree of the polynomial
Not simplifying the polynomial correctly
Not using the correct test to determine the radius of convergence

Conclusion

Taylor polynomials and series are powerful tools in mathematics that help us approximate the value of a function at a given point. By understanding the general form of a Taylor polynomial and the Maclaurin series, we can use these tools to solve a wide range of problems in mathematics, physics, engineering, and computer science.

References

Taylor, B. (1712). Methodus Incrementorum Directa et Inversa.
Maclaurin, C. (1742). A Treatise of Algebra.
Taylor, B. (1755). Methodus Incrementorum Directa et Inversa. (2nd ed.).

Further Reading

Taylor Polynomials: A Tutorial
Taylor Series: A Comprehensive Guide
Calculus: A First Course