Calc1231W9T1 - Taylor polynomial of a polynomial

Tuesday, 18 September 2018

8:36 PM

Machine generated alternative text: For a differentiable function f, it is possible to find a degree 1 polynomial which shares the 0th and I'st derivatives with f at x = a. Here the O'th derivative means "take the derivative zero times" - this is just the function value at x = a, denoted by f (O) (a) = f(a). To construct such a polynomial for f@) x at x = 3, compute the derivatives we would like the polynomial to have: It is a simple matter to write the degree 1 polynomial which has these derivatives at x where 3 9 6 This line is also know as the [select all that apply]: 41 Taylor Polynomial of degree 1 at x = 3 Cl Taylor Polynomial of degree 3 at Tangent line at x = 3 Cl Taylor Polynomial Cl Taylor Polynomial of degree 1 3. It is a)

 

Machine generated alternative text: Let's take this up a notch and find a polynomial which shares the first three derivatives with a function f@) at x "tangent quadratic" which approximates the function near the point Suppose we have f(x) x and a = 2. First calculate the derivatives we would like the polynomial to have: The desired polynmoial is Which in this case is f(2) (a) 2! a . The process is very similar to the previous question, and results in a 2 y

 

Machine generated alternative text: If a function f(x) has n derivatives at x = a, then it has a "'tangent polynomial" of degree n at x = a. This polynomial is called the Taylor polynomial of degree n at x Pn@) The Taylor polynomial is expressed in terms of powers of @ — a) as pn(x) E k, (x-a a, and denoted This polynomial has the special property that all the first n derivatives of pn@) match the first n derivatives of the function f at a. In other words, for O < k < n (k) f(k) (a) = pn (a). 2 let's find the degree 2 Taylor polynomial P2(x) ata -2 For example; if f@) = 2m + 2m + 3, n First calculate the desired derivatives at x Then apply the formula above to deduce that

 

Machine generated alternative text: Let's find the Taylor polynomial P3 , which is a cubic approximation, to the hyperbola 1 about the point x = 1 The first four derivatives are Hence the approximation is —1 This cubic approximation (blue) and hyperbola (red) are plotted below Notice that the approximation is generally less accurate as we move away from x 8 7 6 5 4 3 2 1.2 16 2

 

Machine generated alternative text: Give an example of a degree three polynomial f which has the degree ftvo Taylor polynomial 2 P2(x) = 1 + 4x Give an example of a degree three polynomial g which has the degree two Taylor polynomial q2(x) 1 +6 (x —

 

 

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