Question

Find a Taylor polynomial of degree 3 centered at x=0 that can be used to approximate the the function e x

. You should reduce the factorials if there are any. P 3x Use your answer above to approximate e1 . Round to 4 decimal places. e1 =

Answer 1

The Taylor polynomial of degree 3 for the function e^x centered at x=0 is T_3(x) = 1 + x + x^2/2! + x^3/3!. Plugging in x=1, we approximate e^1 as 2.6667, rounded to four decimal places.

Step-by-step explanation:

To find a Taylor polynomial of degree 3 for the function ex centered at x=0, we use the formula for Taylor series:

`Tn(x) = f(0) + f'(0)x + (f''(0)x2)/(2!) + (f'''(0)x3)/(3!) + \cdots`

For ex, the derivatives at x=0 are all equal to 1. Thus, the Taylor polynomial of degree 3 is:

`T3(x) = 1 + x + (x2)/(2!) + (x3)/(3!)`

To approximate e1, we plug in x=1 into T3(x):

`T3(1) = 1 + 1 + (1)/(2!) + (1)/(3!) = 1 + 1 + 0.5 + 0.1667= 2.6667`

Therefore, e1 is approximately 2.6667 when rounded to four decimal places.