Chapter 3: Applications of Differentiation
Section 3.3: Taylor Polynomials
Taylor's formula states that a suitably well-behaved function can be represented as a sum of a polynomial and a "remainder" term. This is made precise in Theorem 3.3.1.
Theorem 3.3.1 Taylor's Formula
The function f and f′,f″,…,fn (its first n derivatives) are continuous in α,β
fn is differentiable in α,β
Both x and a are in α,β
There exists a value c in α,β for which the following representation of f holds.
In Theorem 3.3.1, Pnx is a polynomial of degree n, called a Taylor polynomial.
The symbol fka represents the kth derivative of f evaluated at x=a.
By convention, f0 is taken to mean "no derivative," that is, the function f itself.
The expression for the remainder Rnx is the Lagrange form, but there are at least two other forms, one of which is attributed to Cauchy.
In particular, the first-degree Taylor polynomial is P1x=fa+f′ax−a, and the equation y=P1x is the equation of the tangent line at x=a. Thus, any tool that generates the first-degree Taylor polynomial is a tool that likewise returns the equation of the tangent line!
For fx=ex at x=0, obtain Taylor polynomials of degree 1, 2, and 3.
Graphically compare these polynomials on −4,4.
Use R3x to estimate the largest difference between f and P3 on −1,1.
Find the actual value of the largest difference between f and P3 on −1,1.
With f and P3 as in Example 3.3.1, obtain R3x and use it to estimate the largest difference between f and P3 on −1,1. Hence, obtain the maximum value of R4 = f4c4!x4, which in turn requires finding the maximal value of f4c for −1≤c≤1.
For fx=sinx with a=0 in Taylor's Formula, obtain the general form of Pnx.
At x=3, obtain the equation of the line tangent to the graph of fx=lnx2+3 x+2.
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