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Unformatted text preview: Series Notice again that the polynomials are not distinct, though in general as the degree increases
so does the accuracy of the estimate.
To illustrate the power of using Taylor polynomials to approximate functions, we can use
a computer to aid us in showing that for f (x) = sin(x) and a = 0, we have
T13,0 (x) = x − x3 +
The next diagram represents a plot of the function sin(x) − T13,0 (x). (This is the error
between the actual value of sin(x) and the approximated value of T13,0 (x). DE Math 128 322 (B. Forrest)2 CHAPTER 4. Sequences and Series 4.5. Introduction to Taylor Series You will notice that the error is very small until x approaches 4 or −4. However, the y -scale
is diﬀerent from that of the x-axis so even near x = 4 or x = −4, the actual error is still
quite small. The diagram suggests that on the slightly more restrictive interval [−π, π ],
T13,0 (x) does an exceptionally good job of approximating sin(x).
To strengthen this point even further, we have provide the plot of the graph of sin(x) − T13,0
on the inter...
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