Let 𝑇3 be the Maclaurin polynomial of 𝑓(𝑥)=𝑒^𝑥. Use the Error Bound to find the maximum possible value of |𝑓(1.4)−𝑇3(1.4)|

Answer 1

Hi Hanas,

The Taylor approximation theorem states that for any value of n, if the (n+1)^th derivative of f is continuous, and | f⁽ⁿ⁺¹⁾(t) | ≤ M for all t between 0 and x (for us, x = 1.4) then

| f(x) - T_n(x) | ≤ M/(n+1)! |x|ⁿ⁺¹

so in our case we have f⁽ⁿ⁾(x) = e^x for all n, so in particular f⁽⁴⁾(x) = e^x, and since e^x is increasing, we have that |f⁽⁴⁾(x)| ≤ e^1.4 for all t between 0 and 1.4, which implies that

| f(1.4) - T₃(1.4) | ≤ e^1.4/(3+1)! |1.4|³⁺¹

⁼ e^1.4/24 · 1.4⁴

which gives the maximum error (approximately .649).

I hope that helps!