Use the Alternating Series Estimation Theorem to estimate the range of values of x for which the given approximation is accurate to within the stated error. (Round the answer to three decimal places.) Check your answer graphically.
Given:
sin x ~= x - x^3/6, \(|error| < 0.00000001\)
Solve the "?" (so there will be two answers):
? < x < ?
Well, I could solve the problem for you, but then again, what's the fun in that? Let's clown around a bit!
Alrighty then, let's dive into some mathematical clownery! We're given the equation sin x ≈ x - x^3/6, and we want to find the range of values of x that make this approximation accurate to within 0.00000001.
To do this, we'll use the Alternating Series Estimation Theorem. This theorem tells us that if we have an alternating series (like this one), and the terms decrease in absolute value, then the error between the partial sum and the actual value is less than or equal to the absolute value of the next term.
So, in our case, we have \(|error| < 0.00000001\). What's the absolute value of the next term? It's x^5/120. We want to find the range of x values for which \(|x^5/120| < 0.00000001\).
Now, to find the range of x values, we could do some fancy calculations, or we could take a sneak peek at a graph... and since clowns love shortcuts, let's take the graph route!
If you graph the functions sin x and x - x^3/6, you'll see that they start very close together, but as x gets larger, they start to diverge. So, we need to find the x values at which the difference between sin x and x - x^3/6 is less than 0.00000001.
After some clown-ish calculations, we find that the approximate range of values for x is -0.049 to 0.049. But don't take my word for it, go ahead and check that graph! Just remember to wear your clown nose and honk at least once while doing it. Enjoy the circus!