Essential Mathematics: Taylor polynomials

Stopped at page 136: Taylor Series

7 flashcards · Mathematics · by lakesare

1. Why do we usually prefer radians to degrees when we work with calculus? The graphs of (a) y = sin x (b) y = sin x°.You can see that when x = 0 the gradient of the first graph is much larger than the gradient of the second graph, and similarly the gradients of the two graphs are different at other values of x. So the two functions will have different formulas for their derivatives.

   Standard formulas for the derivatives of trigonometric functions hold only for radians. sin'(x) = cos(x) is a formula for radians only!

2. What's a constant VS linear VS quadratic Taylor polynomial?

   The constant Taylor polynomial about 1 for f(x) = ln x The linear Taylor polynomial about 0 for f(x) = sin x The quadratic Taylor polynomial about 0 for f(x) = ex

3. What's the gist of a Taylor polynomial?

   We want a polynomial (for f(x), about a) satisfying these conditions: p(a) = f(a) p'(a) = f'(a) p''(a) = f''(a)

4. constant, x - linear, x2 - quadratic, x3 - cubic, x4 - _________ x5 - _________

   x4 - quartic x5 - quintic

5. How would you find this general formula of the Taylor polynomial?

   We have a general form of a polynomial: And proceed to simply find all coefficients c-s: p(a) = f(a) → c0 = f(a) p'(a) = f'(a) → c1 = f'(a) p''(a) = f(a) → c2 = f''(a) / 2 p(n)(a) = f(a) → cn = f(n)(a) / n!

6. Imagine the Taylor polynomial p(x) graphically (for f(x), about x = 1).

7. Taylor polynomial of what degree do you have to find in order to approximate f(x) to 5 decimal places? Example of the notation: p2(x) = 1 + 3x + x2

   Calculate approximations of degrees 1, 2, 3, ..., until two successive approximations agree to m + 2 decimal places.