Labware - MA35 Multivariable Calculus - Single Variable Calculus
 MA35 Labs 1 » Single Variable Calculus Contents1.3 More Derivatives 1.3.1 Second Derivatives 1.3.3 Derivative Test 1.3.4 Taylor Series 1.4 Integration Search

Taylor Series

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Taylor series are polynomials that approximate functions.

The existence of Taylor series is based on the fact that one can use the function value and first derivative, second derivative, third derivative, etc. at some x value x0 to predict what the values of a function f(x) should be for different x values some distance from the point x0.

If we know up to the nth derivative of f(x) as well as the function value at x0, then we can construct a Taylor polynomial of degree n.

The second-order Taylor approximation P2(x) is the parabola that has the same function value and first and second derivatives as f(x) at the point x0. The equation for this approximation is as follows:

T2(x) = f(x0)+(x-x0)f '(x0)+1/2(x-x0)2f ''(x0).

The nth order Taylor approximation is

Σi = 0n[f(i)(x0)/i!]*(x - {x_0})i

Where f(i)(x) indicates the ith derivative of f(x).

Demos

 Taylor Series In this demo you can see first, second, and third order Taylor approximations of functions f(x) at different values x0 of x. Note that accuracy increases as order increases.

Exercises

• 1. For a polynomial function of degree n, what condition is needed for a Taylor approximation to be exactly the same as the function itself?
• 2. Describe the behavior of the second order Taylor approximation for the function f(x) = |x^2 - 1|.