Use a linear approximation (or differentials) to estimate the given number:

√99.8

Question

Use a linear approximation (or differentials) to estimate the given number:

√99.8

anonymous · Answer

10*10 is 100; so the root of 99.8 (which is about the root of 100) is about 10.

anonymous · Answer

I'm not sure if that is linear approximation, though.

anonymous · Answer

That was close but it doesn't give me an answer @DanielHendrycks  Thank you either way.

queelius · Answer

We want to find out what f(0.98) is.

To do this, we will solve the linear approximation of this problem near x = 100. Let f(x) = x^(1/2). We can take the value of f(x) at x = 100 then take the slope of the curve (derivative) at that point, and then multiply that slope by the change in x (which is 0.2).

The derivative of f(x) is f'(x) = −1/[2x^(1/2)]. At x = 100, the slope is therefore -1/20. So, f(20) + f'(20)*0.2 = 9.99. So, that's the approximate solution.

anonymous · Answer

thank you :)

queelius · Answer

I did notice two errors in my solution. The derivative is 1/[2x^(1/2)], not -1/[2x^(1/2)]. Also, the change in x is -0.2. So, f(x) + f'(x)*(change in x) -> 10 + (1/20)*(-0.2) = 9.99. It's still the same answer, but the way I got there was incorrect.

anonymous · Answer

use taylor series for one approximation