Question

(a) From the definition (Equation 4.46), construct ${\mathbf{n}}_{\mathbf{1}}\mathbf{\left(}\mathbf{x}\mathbf{\right)}$and${\mathbf{n}}_{\mathbf{2}}\mathbf{\left(}\mathbf{x}\mathbf{\right)}$.

(b) Expand the sines and cosines to obtain approximate formulas for${\mathbf{n}}_{\mathbf{1}}\mathbf{\left(}\mathbf{x}\mathbf{\right)}$androle="math" localid="1656329588644" ${\mathbf{n}}_{\mathbf{2}}\mathbf{\left(}\mathbf{x}\mathbf{\right)}$, valid when$\mathbf{x}\mathbf{\ll }\mathbf{1}\mathbf{.}\mathbf{}$. Confirm that they blow up at the origin.

Short answer

a)

The functions are,

${\mathrm{n}}_1\left(\mathrm{x}\right)=-\frac{\mathrm{cosx}}{{\mathrm{x}}^2}-\frac{\mathrm{sinx}}{\mathrm{x}}$

${\mathrm{n}}_2\left(\mathrm{x}\right)=-(\frac{3}{{\mathrm{x}}^3}-\frac{1}{\mathrm{x}})\mathrm{cosx}-\frac{3}{{\mathrm{x}}^2}\mathrm{sinx}$

b)

The new functions are,

$$\begin{gathered} {\mathrm{n}}_1\left(\mathrm{x}\right)\approx -\frac{1}{{\mathrm{x}}^2} \\ {\mathrm{n}}_2\left(\mathrm{x}\right)\approx -\frac{3}{{\mathrm{x}}^3} \end{gathered}$$

Step-by-step solution

Concept used

The general expression for the function n_i(x) is given by,

${\mathrm{n}}_{\mathrm{i}}\left(\mathrm{x}\right)\equiv -(-\mathrm{x}{)}^{\mathrm{i}}{\left(\frac{1}{\mathrm{x}}\frac{\mathrm{d}}{\mathrm{dx}}\right)}^{\mathrm{i}}\frac{\mathrm{cosx}}{\mathrm{x}}$

Construct  n1(x) and  n2(x)

We need to construct ${\mathrm{n}}_1\left(\mathrm{x}\right)$and ${\mathrm{n}}_2\left(\mathrm{x}\right)$, as:

$$\begin{gathered} {\mathrm{n}}_1\left(\mathrm{x}\right)=-(-\mathrm{x})\frac{1}{\mathrm{x}}\frac{\mathrm{d}}{\mathrm{dx}}\left(\frac{\mathrm{cosx}}{\mathrm{x}}\right) \\ =-\frac{\mathrm{cosx}}{{\mathrm{x}}^2}-\frac{\sin \mathrm{x}}{\mathrm{x}} \\ {\mathrm{n}}_1\left(\mathrm{x}\right)=-\frac{\mathrm{cosx}}{{\mathrm{x}}^2}-\frac{\sin \mathrm{x}}{\mathrm{x}} \end{gathered}$$

Similarly solving for n₂(x),

$$\begin{gathered} {\mathrm{n}}_2\left(\mathrm{x}\right)=-{(-\mathrm{x})}^2{\left(\frac{1}{\mathrm{x}}\frac{\mathrm{d}}{\mathrm{dx}}\right)}^2\frac{\mathrm{cosx}}{\mathrm{x}} \\ =-{\mathrm{x}}^2\left(\frac{1}{\mathrm{x}}\frac{\mathrm{d}}{\mathrm{dx}}\right)\left[\frac{1}{\mathrm{x}}\frac{\mathrm{d}}{\mathrm{dx}}\left(\frac{\mathrm{cosx}}{\mathrm{x}}\right)\right] \\ =-\mathrm{x}\frac{\mathrm{d}}{\mathrm{dx}}\left(\frac{1}{\mathrm{x}}·\frac{-\mathrm{xsin}\mathrm{x}-\cos \mathrm{x}}{{\mathrm{x}}^2}\right) \\ =\mathrm{x}\frac{\mathrm{d}}{\mathrm{dx}}\left(\frac{\sin \mathrm{x}}{{\mathrm{x}}^2}+\frac{\cos \mathrm{x}}{{\mathrm{x}}^3}\right) \end{gathered}$$

Further solving above equation,

$$\begin{gathered} {\mathrm{n}}_2\left(\mathrm{x}\right)=\mathrm{x}\left(\frac{{\mathrm{x}}^2}{\mathrm{cosx}-2\mathrm{xsinx}}+\frac{-{\mathrm{x}}^3\mathrm{sinx}-3{\mathrm{x}}^2\mathrm{cosx}}{{\mathrm{x}}^4}\right) \\ {\mathrm{n}}_2\left(\mathrm{x}\right)=\frac{\mathrm{cosx}}{\mathrm{x}}-2\frac{{\mathrm{x}}^2}{{\mathrm{x}}^2}-\frac{\mathrm{sinx}}{{\mathrm{x}}^2}-\frac{3\mathrm{cosx}}{{\mathrm{x}}^3} \\ {\mathrm{n}}_2\left(\mathrm{x}\right)=-\left(\frac{3}{{\mathrm{x}}^3}-\frac{1}{\mathrm{x}}\right)\mathrm{cosx}-\frac{3}{{\mathrm{x}}^2}\mathrm{sinx} \\ {\mathrm{n}}_2\left(\mathrm{x}\right)=-\left(\frac{3}{{\mathrm{x}}^3}-\frac{1}{\mathrm{x}}\right)\mathrm{cosx}-\frac{3}{{\mathrm{x}}^2}\mathrm{sinx} \end{gathered}$$

Thus, the function n₁(x), and n₂(x) are ${\mathrm{n}}_1\left(\mathrm{x}\right)=-\frac{\mathrm{cosx}}{{\mathrm{x}}^2}-\frac{\mathrm{sinx}}{\mathrm{x}}$and ${\mathrm{n}}_2\left(\mathrm{x}\right)=-\frac{3}{{\mathrm{x}}^3}-\frac{1}{\mathrm{x}}\mathrm{cosx}-\frac{3}{{\mathrm{x}}^2}\mathrm{sinx}$respectively.

Calculate approximate values of the sine and cosine function

We can approximate the sine and cosine functions as $\mathrm{sinx}\approx \mathrm{x}\mathrm{and}\mathrm{cosx}\approx 1,$so:

${\mathrm{n}}_1\left(\mathrm{x}\right)\approx -\frac{1}{{\mathrm{x}}^2}+\frac{1}{\mathrm{x}}$

But since is x very small then $1/{\mathrm{x}}^2$is much larger than 1, thus we can neglect 1, so get:

${\mathrm{n}}_1\left(\mathrm{x}\right)\approx -\frac{1}{{\mathrm{x}}^2}$

And for we have:

$$\begin{gathered} {\mathrm{n}}_2\left(\mathrm{x}\right)\approx -\left(\frac{3}{{\mathrm{x}}^3}-\frac{1}{\mathrm{x}}\right)-\frac{3}{{\mathrm{x}}^2}\mathrm{x} \\ {\mathrm{n}}_2\left(\mathrm{x}\right)\approx -\frac{3}{{\mathrm{x}}^3} \end{gathered}$$

Thus, approximate formulas for n₁(x), and n₂(x) are ${\mathrm{n}}_1\left(\mathrm{x}\right)\approx -\frac{1}{{\mathrm{x}}^2}$and ${\mathrm{n}}_2\left(\mathrm{x}\right)\approx -\frac{3}{{\mathrm{x}}^2}$respectively.