Question

Find the first Taylor polynomial T1(x) for f(x)=exp(x) based at b=0:

Answer 1

Answer:

f(x) = 1 + x + (x²/2!) + (x³/3!) + ....... = Σ (xⁿ/n!) (Summation from n = 0 to n = ∞)

Step-by-step explanation:

f(x) = eˣ

Expand using first Taylor Polynomial based around b = 0

The Taylor's expansion based around any point b, is given by the infinite series

f(x) = f(b) + xf'(b) + (x²/2!)f"(b) + (x³/3!)f'''(b) + ....= Σ (xⁿfⁿ(b)/n!) (Summation from n = 0 to n = ∞)

Note: f'(x) = (df/dx)

So, expanding f(x) = eˣ based at b=0

f'(x) = eˣ

f"(x) = eˣ

fⁿ(x) = eˣ

And e⁰ = 1

f(x) = 1 + x + (x²/2!) + (x³/3!) + ....... = Σ (xⁿ/n!) (Summation from n = 0 to n = ∞)