All Flashcards
Why are Maclaurin series important?
They provide a foundation for constructing Taylor series for other functions and are frequently encountered.
How are Taylor series and polynomial approximations related?
Taylor series use polynomial approximations to represent functions as infinite series.
What is the significance of the center 'a' in a Taylor series?
The Taylor series approximates the function best near the center 'a'.
What is the relationship between the Maclaurin series of sin(x), cos(x) and ?
The Maclaurin series for contains all the terms from the Maclaurin series of sin(x) and cos(x).
How do you find the Taylor series for cos(3x) centered at x=0?
- Recall the Maclaurin series for cos(x). 2. Substitute 3x for x in the series. 3. Simplify the expression.
How do you find a Taylor series centered at x=a?
- Find several derivatives of f(x). 2. Evaluate the derivatives at x=a. 3. Identify a pattern. 4. Write the Taylor series using the formula.
How do you find the first four terms of a Taylor series?
- Find the general Taylor series. 2. Plug in n=0, 1, 2, and 3 into the series. 3. Simplify each term.
What are the differences between Taylor and Maclaurin Series?
Taylor Series: Centered at any point 'a'. Maclaurin Series: Centered specifically at x=0.
What are the differences between Taylor Series and Taylor Polynomials?
Taylor Series: Infinite sum of terms. Taylor Polynomials: Finite sum of terms (partial sum).