What is the general formula for a Taylor Polynomial?
$f(x)=f(a)+f'(a)(x-a)+\frac{f''(a)}{2!}(x-a)^2+\frac{f'''(a)}{3!}(x-a)^3+...+\frac{f^{(n)}(a)}{n!}(x-a)^n$
What is the formula for the remainder $R_n(x)$ (Lagrange Error Bound)?
$R_n(x) = \dfrac{f^{(n+1)}(c)}{(n+1)!}(x-a)^{n+1}$
How is a function $f(x)$ related to its Taylor polynomial $P_n(x)$ and remainder $R_n(x)$?
$f(x) = P_n(x) + R_n(x)$
What is the Maclaurin series for $e^x$?
$e^x = 1 + x + \frac{x^2}{2!} + \frac{x^3}{3!} + ...$
What is the Maclaurin series for sin(x)?
$sin(x) = x - \frac{x^3}{3!} + \frac{x^5}{5!} - \frac{x^7}{7!} + ...$
What is the Maclaurin series for cos(x)?
$cos(x) = 1 - \frac{x^2}{2!} + \frac{x^4}{4!} - \frac{x^6}{6!} + ...$
What is the Maclaurin series for $\frac{1}{1-x}$?
$\frac{1}{1-x} = 1 + x + x^2 + x^3 + ...$
What is the Maclaurin series for ln(1+x)?
$ln(1+x) = x - \frac{x^2}{2} + \frac{x^3}{3} - \frac{x^4}{4} + ...$
How to calculate the error bound for approximating $f(x)$ with $P_n(x)$?
$|R_n(x)| \le \frac{M}{(n+1)!}|x-a|^{n+1}$, where M is the max of $|f^{(n+1)}(z)|$ on the interval between a and x.
What is the formula for the third-degree Taylor polynomial $P_3(x)$?
$P_3(x) = f(a) + f'(a)(x-a) + \frac{f''(a)}{2!}(x-a)^2 + \frac{f'''(a)}{3!}(x-a)^3$
What does Taylor's Theorem state?
Provides a way to approximate the value of a function at a point using its derivatives at another point.
What is the application of Taylor's Theorem?
Approximating function values, estimating errors in approximation, and defining functions.
What is the relationship between Taylor's Theorem and Lagrange Error Bound?
Lagrange Error Bound is a direct result of Taylor's Theorem, providing an upper bound for the error in Taylor approximations.
What does the Lagrange Error Bound theorem state?
The error in approximating f(x) by its nth-degree Taylor polynomial is bounded by the (n+1)th term of the Taylor series, with the (n+1)th derivative evaluated at some point between the center and x.
How does Taylor's Theorem relate to polynomial approximations?
Taylor's Theorem provides the theoretical basis for approximating functions with polynomials.
What does Taylor's Theorem guarantee?
The existence of a point 'c' in the interval (a, x) such that the remainder term can be expressed in terms of the (n+1)th derivative at 'c'.
How is Taylor's Theorem used in numerical analysis?
It is used to develop numerical methods for approximating solutions to equations and integrals.
What are the conditions for Taylor's Theorem to be applicable?
The function must be n+1 times differentiable on an interval containing the point of approximation and the center.
How does Taylor's Theorem provide a bound on the error of polynomial approximations?
By providing a formula for the remainder term, which represents the difference between the actual function value and the polynomial approximation.
What are the key components of Taylor's Theorem?
The function value at the center, derivatives at the center, the point of approximation, and the remainder term.
How to find the Lagrange Error Bound for approximating cos(0.1) using a 2nd-degree Maclaurin polynomial?
1. Find the 3rd derivative of cos(x). 2. Find the max value of |f'''(z)| on [0, 0.1]. 3. Apply the Lagrange Error Bound formula.
Steps to approximate $e^{0.5}$ using a 3rd-degree Maclaurin polynomial and find the error bound.
1. Write the 3rd-degree Maclaurin polynomial for $e^x$. 2. Evaluate at x=0.5. 3. Find the 4th derivative. 4. Find max of |f''''(z)| on [0, 0.5]. 5. Apply the error bound formula.
How to determine the degree of Taylor polynomial needed for a specific error bound?
1. Write the Lagrange Error Bound formula. 2. Set the error bound less than the desired value. 3. Solve for n (the degree).
How do you find the maximum value of $f^{(n+1)}(c)$ on the interval [a, x]?
1. Find the (n+1)th derivative of f(x). 2. Determine critical points. 3. Evaluate the derivative at endpoints and critical points. 4. Choose the largest absolute value.
What are the steps to find the Lagrange Error Bound for approximating ln(1.1) using a 2nd-degree Taylor polynomial centered at 1?
1. Find the 3rd derivative of ln(x). 2. Find the max value of |f'''(z)| on [1, 1.1]. 3. Apply the Lagrange Error Bound formula.
How do you solve for the Lagrange Error Bound?
1. Find the (n+1)th derivative of the function. 2. Find the maximum value of the (n+1)th derivative on the interval. 3. Plug the values into the Lagrange Error Bound formula.
How do you write a Taylor polynomial?
1. Find the derivatives of the function. 2. Evaluate the derivatives at the center. 3. Plug the values into the Taylor polynomial formula.
How do you approximate a function using a Taylor polynomial?
1. Write the Taylor polynomial. 2. Plug in the value of x. 3. Evaluate the polynomial.
Steps to calculate the Lagrange Error Bound for approximating sin(0.5) using a 4th-degree Maclaurin polynomial?
1. Find the 5th derivative of sin(x). 2. Find the max value of |f'''''(z)| on [0, 0.5]. 3. Apply the Lagrange Error Bound formula.
How to approximate a value using a Taylor polynomial?
1. Construct the Taylor polynomial. 2. Substitute the value into the Taylor polynomial. 3. Calculate the result.
