Using calculus, it can be shown that the arctangent function can be approximated by the polynomial arctan x approx x - frac{x^{3}}{3} + frac{x^{5}}{5}

tinfoQ 2020-11-05 Answered
Using calculus, it can be shown that the arctangent function can be approximated by the polynomial
arctan x  x  x33 + x55  x77
where x is in radians.
a) Use a graphing utility to graph the arctangent function and its polynomial approximation in the same viewing window. How do the graphs compare?
b) Study the pattern in the polynomial approximation of the arctangent function and predict the next term. Then repeat part (a). How does the accuracy of the approximation change when an additional term is added?
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Expert Answer

bahaistag
Answered 2020-11-06 Author has 101 answers

Step 1
a)
image
Step 2
a) arctan x  x  x33 + x55  x77 (blue graph)
Between from about x= 1 to 1, the polynomial approximation is close to the graph of y=arctan x.
Step 3
image
Step 4
b) Based on the pattern, the next exponent is 9, denominator is 9, and the term is positive.
arctan x  x  x33 + x55  x77 + x99 (magenta graph)
Now the approximation stays closer to y= arctan x over a sligghtly wider interval.

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