Maclaurin polynomial of e to the 2x

Problem

Write the fourth-order Maclaurin polynomial of $f(x) = e^{2x}$. (Hint: expand $e^t$ and substitute $t = 2x$.)

Solution

From the known expansion $e^t = 1 + t + \dfrac{t^2}{2} + \dfrac{t^3}{6} + \dfrac{t^4}{24} + o(t^4)$, substituting $t=2x$: $e^{2x} = 1 + 2x + \frac{(2x)^2}{2} + \frac{(2x)^3}{6} + \frac{(2x)^4}{24} + o(x^4).$ Simplifying the coefficients: $\boxed{T_4(x) = 1 + 2x + 2x^2 + \frac{4}{3}x^3 + \frac{2}{3}x^4}.$

Links

Topics: Derivatives
Concepts: Maclaurin polynomial · Standard expansion
Methods: Taylor polynomial
Skills: Use formulae
Exercise type: Derivative computation