Q. 26 - Calculus

Question

Find the fourth Maclaurin polynomial $P_4\left(x\right)$ for the specified function:

$\sin \left(3x\right)$.

Step-by-Step Solution

Answer

The fourth Maclaurin polynomial is,

$P_4\left(x\right)=3x-\frac{9}{2}{x}^3$.

Step 1. Given Information.

The function is,

$\sin \left(3x\right)$.

Step 2. Describing the polynomial.

Let $f\left(x\right)=\sin \left(3x\right)$.

Since for any function $f$with a derivative of order $4$ at $x=0$, the fourth Maclaurin polynomial is given by,

$P_4\left(x\right)=f\left(0\right)+f\text{'}\left(0\right)x+\frac{f\text{'}\text{'}\left(0\right)}{2!}{x}^2+\frac{f\text{'}\text{'}\text{'}\left(0\right)}{3!}{x}^3+\frac{f\text{'}\text{'}\text{'}\text{'}\left(0\right)}{4!}{x}^4$.

Step 3. Finding the fourth Maclaurin polynomial.

The value of the function at $x=0$ is,

$$\begin{gathered} f\left(0\right)=\sin \left(3.0\right) \\ =\sin \left(0\right) \\ =0 \end{gathered}$$

The derivatives of the function $f\left(x\right)=\sin \left(3x\right)$ are,

$$\begin{gathered} f\text{'}\left(x\right)=\frac{d\left(\sin \left(3x\right)\right)}{dx} \\ =3\cos \left(3x\right) \\ f\text{'}\left(0\right)=3\cos \left(3.0\right) \\ =3\cos \left(0\right) \\ =3 \end{gathered}$$

Also,

$$\begin{gathered} f\text{'}\text{'}\left(x\right)=\frac{d\left(3\cos \left(3x\right)\right)}{dx} \\ =3\frac{d\cos \left(3x\right)}{dx} \\ =-9\sin \left(3x\right) \\ f\text{'}\text{'}\left(0\right)=-9\sin \left(3.0\right) \\ =-9\sin \left(0\right) \\ =0 \end{gathered}$$

Also,

$$\begin{gathered} f\text{'}\text{'}\text{'}\left(x\right)=\frac{d(-9\sin \left(3x\right))}{dx} \\ =-9\frac{d\left(\sin \left(3x\right)\right)}{dx} \\ =-27\cos \left(3x\right) \\ f\text{'}\text{'}\text{'}\left(0\right)=-27\cos \left(3.0\right)) \\ =-27\cos \left(0\right) \\ =-27 \end{gathered}$$

Also,

$$\begin{gathered} f\text{'}\text{'}\text{'}\text{'}\left(x\right)=\frac{d(-27\cos \left(3x\right))}{dx} \\ =-27\frac{d\left(\cos \left(3x\right)\right)}{dx} \\ =81\sin \left(3x\right) \\ f\text{'}\text{'}\text{'}\text{'}\left(0\right)=81\sin \left(3.0\right) \\ =81\sin \left(0\right) \\ =0 \end{gathered}$$

The fourth Maclaurin polynomial is,

$$\begin{gathered} P_4\left(x\right)=0+3x+\frac{0}{2!}{x}^2+\frac{(-27)}{3!}{x}^3+\frac{0}{4!}{x}^4 \\ =3x-\frac{9}{2}{x}^3 \end{gathered}$$