# Taylor Polynomial

stunner5000pt
Find the thrid taylor polynomial P3(x) for the function $f(x) = \sqrt{x+1}$ about a=0. Approximate f(0.5) using P3(x) and find actual error

thus Maclaurin series

$$f(x) = f(0) + f'(0)x + \frac{f''(0)}{2} x^2 + \frac{f^{3}(0)}{6} x^3$$

$$f(x) = x + \frac{1}{2} x - \frac{1}{8} x^2 + \frac{3}{48} x^3$$
am i right so far?
To approximate f(0.5) i simply put x=0.5 in the above equation?
How do i fin the actual error, though?
DO i have to use the remainder in this? Please help!

Thank you

## Answers and Replies

Staff Emeritus
Science Advisor
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stunner5000pt said:
am i right so far?

All but the first term is right. $f(0)\neq x$

To approximate f(0.5) i simply put x=0.5 in the above equation?

After you fix it, yes.

How do i fin the actual error, though?

Plug x=0.5 into f(x) on a calculator, and subtract your result from it. You won't exactly get the "actual" error because your calculator approximates, too. But it will be a very good estimate.

DO i have to use the remainder in this? Please help!

That depends on what is asked for. The remainder doesn't give you the actual error, but rather the maximum of the actual error. So unless you were asked to put bounds on the error, I would think that you would not have to use the remainder.

stunner5000pt
$$f(x) = 1 + \frac{1}{2} x - \frac{1}{8} x^2 + \frac{3}{48} x^3$$

i see the problem, its fixed now

im being cautious so I am goingto put hte upper limits

$$R_{4} = \frac{15}{384} (c+1)^{\frac{-7}{2}} x^4$$

so the error must be lesser than or equal to this R4 value. THat c value lies between 0.5 and x?

Is this right?

stunner5000pt
is this how one would solve for the maximum possible error as stated in the above post? Please do advise

Thank you for your help and input