Question

Let f(x)=xeˣ.

(a) Find p4​(x), the 4th Taylor polynomial for f(x), about x=0.
(b) Bound the error in the approximation
f(x)≈p₄(x)
when x∈[−2,2], and when x∈[−21​,21​].

Answer 1

To find the 4th Taylor polynomial for f(x), we evaluate the derivatives of f(x) at x=0 and substitute them into the formula.

Step-by-step explanation:

To find the 4th Taylor polynomial for f(x) about x=0, we need to find the derivatives of f(x) up to the 4th derivative and evaluate them at x=0. The 4th Taylor polynomial is given by:

p₄(x) = f(0) + f'(0)x + (f''(0)x²)/2! + (f'''(0)x³)/3! + (f''''(0)x⁴)/4!

For f(x) = xeˣ, the derivatives are:

f'(x) = eˣ + xeˣ,

f''(x) = 2eˣ + xeˣ,

f'''(x) = 3eˣ + xeˣ,

f''''(x) = 4eˣ + xeˣ.

Evaluating these derivatives at x=0 gives:

f(0) = 0,

f'(0) = 1,

f''(0) = 2,

f'''(0) = 3,

f''''(0) = 4.

Substituting these values into the formula, we get:

p₄(x) = 0 + 1x + (2x²)/2! + (3x³)/3! + (4x⁴)/4! = x + x² + x³/2 + x⁴/3.