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}

Question

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} - \frac{x^{7}}{7}

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?

Answer 1

Step 1
a)

Step 2
a) $\arctan x \approx x - \frac{x^{3}}{3} + \frac{x^{5}}{5} - \frac{x^{7}}{7}$ (blue graph)
Between from about $x = - 1$ to 1, the polynomial approximation is close to the graph of $y = \arctan x .$
Step 3

Step 4
b) Based on the pattern, the next exponent is 9, denominator is 9, and the term is positive.
$\arctan x \approx x - \frac{x^{3}}{3} + \frac{x^{5}}{5} - \frac{x^{7}}{7} + \frac{x^{9}}{9}$ (magenta graph)
Now the approximation stays closer to $y = \arctan x$ over a sligghtly wider interval.

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}

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