Question

Give a bound on the maximum possible error (i.e. remainder) when eˣ is approximated with its 4 th Taylor polynomial centered at a=0 on the interval [−0.5,0.5].

Answer 1

To find the maximum possible error when approximating e^x with its 4th Taylor polynomial, we can use the remainder term of the Taylor series.

Step-by-step explanation:

The maximum possible error or remainder when approximating eˣ with its 4th Taylor polynomial centered at a=0 on the interval [-0.5,0.5] can be found using the remainder term of the Taylor series. The remainder term for the 4th degree polynomial is given by:

R₄(x) = f⁽⁴⁾(c)(x-a)⁴ / 4!

Where c is some value between a and x, and 4! represents the factorial of 4. To find the maximum possible error, we need to find the maximum value of the absolute value of the 4th derivative over the interval [-0.5,0.5].