1. Let g(x) = ∫0^x arctan((t^2)/(2)) dt. (a) Find the...

Transcript

00:01 So in this question, we say we want to use a suitable mclaren series to estimate the sign of nine degrees to five decimal place.

00:09 So the first thing i want to have to do is convert from degrees into radians because we know that our trig values only work in radiance in calculus.

00:20 Our mclaren series are only valid when we are in radians.

00:24 So if i have nine degrees, that's going to be equal to nine times pi over 100.

00:31 180 radians, and so that's equivalent to just pi over 20 radians.

00:40 And so really what i'm trying to estimate in this question is the sign of the quantity of pi over 20.

00:49 Now, we should memorize our mclaren series for the sign of x.

00:54 So the sign of x is equal to x minus x cubed over 3 factorial, plus x to the fifth over five factorial minus x to the seventh over seven factorial plus dot dot dot dot which means if i'm looking for the sign of pi over 20 well what's that going to be it's going to be pi over 20 minus pi over 20 quantity cubed over three factorial plus pi over 20 quantity quantity to the fifth over five factorial minus pi over 20 quantity to the seventh over seven factorial plus dot dot dot dot dot.

01:38 So i'm going to head to my calculator and i'm going to say, okay, let's start with pi over 20, okay, as my initial approximation.

01:50 Now from this, i'm going to be subtracting pi over 20, that quantity is being cubed, and then i'm dividing that by three factorial.

02:03 I'm dividing that by three factorial...

Question

1. Let $g(x) = \int_0^x \arctan(\frac{t^2}{2}) dt$. (a) Find the Maclaurin series for $g(x)$. (b) Estimate $g(0.5)$ accurate to within $\pm 5.0 \times 10^{-7}$

Question

1. Let $g(x) = \int_0^x \arctan(\frac{t^2}{2}) dt$. (a) Find the Maclaurin series for $g(x)$. (b) Estimate $g(0.5)$ accurate to within $\pm 5.0 \times 10^{-7}$

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