Calculus- Early Transcendentals, 2021a

5.4. Linear and Higher Order Approximations 187
and in general
T n (x)=1 + x + x2
2! + x3 xn
+ ···+
3! n! .
Finally we can approximate e = f (1) by simply calculating T n (1). A few values are:
T 1 (1)=1 + 1 = 2
T 2 (1)=1 + 1 + 12
2! = 2.5
T 4 (1)=1 + 1 + 12
2! + 13
3! = 2.6
T 8 (1)=2.71825396825
T 20 (1)=2.71828182845
We can continue this way for larger values of n, butT 20 (1) is already a pretty good approximation of
e, and we took only 20 terms! ♣
Exercises for 5.4.3
Exercise 5.4.9 Find the 5 th degree Taylor polynomial for f (x)=sinx around a = 0.
(a) Use this Taylor polynomial to approximate sin(0.1).
(b) Use a calculator to find sin(0.1). How does this compare to our approximation in part (a)?
Exercise 5.4.10 Suppose that f ′′ exists and is continuous on [1,2]. Suppose also that | f ′′ (x)|≤ 1 4
for all
xin(1,2). Prove that if we use the linearization y = L(x) of y = f (x) at x = 1 as an approximation of
y = f (x) near x = 1, then our estimated value of f (1.2) is guaranteed to have an accuracy of at least 0.01,
i.e., our estimate will lie within 0.01 units of the true value.
Exercise 5.4.11 Find the 3 rd degree Taylor polynomial for f (x)= 1
approximation would not be useful for calculating f (5).
Exercise 5.4.12 Consider f (x)=lnx around a = 1.
(a) Find a general formula for f (n) (x) for n ≥ 1.
(b) Find a general formula for the Taylor Polynomial, T n (x).
1−x
−1 around a = 0. Explain why this
5.4.4 Newton’s Method
A well known numeric method is Newton’s Method (also sometimes referred to as Newton-Raphson’s
Method), named after Isaac Newton and Joseph Raphson. This method is used to find roots, or x-intercepts,
of a function. While we may be able to find the roots of a polynomial which we can easily factor, we saw
in the previous chapter on Limits, that for example the function e x + x = 0 has a solution (i.e. root, or
x-intercept) at x ≈−0.56714. By the Intermediate Value Theorem we know that the function e x + x = 0
does have a solution. We cannot here simply solve for such a root algebraically, but we can use a numerical