Approximating Distance with a Second-Degree Taylor Polynomial

In summary, the problem involves a car moving with a speed of 10m/s and an acceleration of 1 m/s^2. The task is to estimate the distance the car will travel in the next second using a second-degree Taylor polynomial. However, the participant in the conversation is unsure of how to apply the polynomial and instead solves the problem using a physics equation. They also inquire if it is possible to approximate a function using a second degree Taylor polynomial given the values of f(0), f'(0), and f''(0).
  • #1
freshman2013
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Homework Statement



A car is moving with speed 10m/s and acceleration 1 m/s^2 at the given instant. Using a second-degree Taylor polynomial, estimate how far the car moves in the next second.

Homework Equations




The Attempt at a Solution


I don't get how you're supposed to apply taylor polynomials to this problem. I actually got the right answer, but that was from plugging in acceleration, velocity, and t=1 second into the physics equation d=(1/2)at^2+v(original)*t.
 
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  • #2
If I have a function f(t), and I tell you what f(0), f'(0) and f''(0) are, can you approximate f(t) with a second degree Taylor polynomial?
 

1. What is a Taylor polynomial?

A Taylor polynomial is a way to approximate a function using a polynomial with a finite number of terms. It is named after the mathematician Brook Taylor.

2. What are the applications of Taylor polynomials?

Taylor polynomials are commonly used in calculus and numerical analysis to approximate the value of a function at a given point. They can also be used to find zeros of a function and to estimate the behavior of a function near a certain point.

3. How do you find the coefficients of a Taylor polynomial?

The coefficients of a Taylor polynomial can be found by using the Taylor series formula, which involves taking derivatives of the original function at a given point. Alternatively, they can be found using a Taylor table, which involves plugging in values of x and the corresponding derivatives into a table to find the coefficients.

4. Can Taylor polynomials be used for non-polynomial functions?

Yes, Taylor polynomials can be used for any differentiable function. However, the accuracy of the approximation depends on the smoothness of the function and the number of terms in the polynomial.

5. How do you determine the degree of a Taylor polynomial?

The degree of a Taylor polynomial is determined by the number of terms in the polynomial, which is also equal to the highest order derivative used in the Taylor series formula. For example, a polynomial with 3 terms would have a degree of 2.

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