considere the function f(x)=4/1+x^2 a) determine a power series representation f
ID: 3344039 • Letter: C
Question
considere the function f(x)=4/1+x^2
a) determine a power series representation for f(x) using the pseudogeometric approach, also determine the interval of convergence.
b) integrate the series you got in part(a) to obtain a power series representation for the function g(x)=4 arctan(x)
c)substitute x=1 into the series in (b) and simplify the result to obtain an infinite series that convergence to PIE. ( this is called Liebniz formula for PIE)
Explanation / Answer
We see that: f(x) = x^4 - 98x^2 + 10 f'(x) = 4x^3 - 196x f''(x) = 12x^2 - 196. Setting f'(x) = 0 yields: 4x^3 - 196 = 0 ==> 4x(x + 7)(x - 7) = 0 ==> 4x = 0, x + 7 = 0, and x - 7 = 0 ==> x = 0, x = -7, and x = 7. We see that: f''(0) = 12(0)^2 - 196 = -196 < 0 f''(7) = 12(7)^2 - 196 = 392 > 0 f''(-7) = 12(-7)^2 - 196 = 392 > 0. This implies that x = 7 and x = -7 are minimums and x = 0 is a maximum. However, since 4x^3 - 196x > 0 ==> 4x(x + 7)(x - 7) > 0 on (-7, 0) U (7, infinity), we see that f(x) is an increasing function on (-7, 0) U (7, infinity). This implies that f(15) > f(0). Also, since f(7) = f(-7) are the minimum value occurs at both x = 7 and x = -7, we see that the minimum value is f(7) = f(-7) = -2391 and the maximum value is f(15) = 28585. I hope this helps!Related Questions
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