272. Compute the approximate value: 1) $\sqrt[4]{17} ; 2$ ) $\operatorname{arc} \operatorname{tg} 0.98$ ; 3) $\sin 29^{\circ}$.
Solution. If it is required to calculate $f\left(x_{1}\right)$ and it is easier to calculate $f\left(x_{0}\right)$ and $f^{\prime}\left(x_{0}\right)$, then for a sufficiently small absolute value of the difference $x_{1}-x_{0}=d x$, the increment of the function can be replaced by its differential $f\left(x_{1}\right)-f\left(x_{0}\right) \approx f^{\prime}\left(x_{0}\right) d x$ and from this, the approximate value of the desired quantity can be found using the formula

$$
f\left(x_{1}\right) \approx f\left(x_{0}\right)+f^{\prime}\left(x_{0}\right) d x
$$

1) We will consider $\sqrt[4]{17}$ as a particular value of the function $f(x)=\sqrt[4]{x}$ at $x=17=x_{1}$. Let $x_{0}=16$, then $f\left(x_{0}\right)=\sqrt[4]{16}=2$, $f^{\prime}\left(x_{0}\right)=\left.\frac{1}{4} x^{-\frac{3}{4}}\right|_{x=16}=\frac{1}{\sqrt[4]{16^{3}}}=\frac{1}{32}, \quad d x=x_{1}-x_{0}=1$.

Substituting into formula (b), we get

$$
\sqrt[4]{17} \approx f\left(x_{0}\right)+f^{\prime}\left(x_{0}\right) d x=2+\frac{1}{32} \cdot 1=\frac{65}{32} \approx 2.031
$$

2) Let $\operatorname{arctg} 0.98$ be a particular value of the function $y=\operatorname{arctg} x$ at $x=0.98=x_{1}$. Let $x_{0}=1$, then $y\left(x_{0}\right)=\frac{\pi}{4}, y^{\prime}\left(x_{0}\right)=$ $=\left.\frac{1}{1+x^{2}}\right|_{x=1}=\frac{1}{2}, \quad d x=x_{1}-x_{0}=-0.02$.

Using formula (b), we find:

$$
\operatorname{arctg} 0.98 \approx y\left(x_{0}\right)+y^{\prime}\left(x_{0}\right) d x=\frac{\pi}{4}+\frac{1}{2}(-0.02) \approx 0.7754
$$

3) Assuming that $\sin 29^{\circ}$ is a particular value of the function $y=$ $=\sin x$ at $x=\frac{\pi}{180} \cdot 29=x_{1}$ and that $x_{0}=\frac{\pi}{180} \cdot 30=\frac{\pi}{6}$, we get

$$
\begin{gathered}
y\left(x_{0}\right)=\sin \frac{\pi}{6}=\frac{1}{2} ; \quad y^{\prime}\left(x_{0}\right)=\left.\cos x\right|_{x=\frac{\pi}{6}}=\frac{\sqrt{3}}{2} \\
d x=x_{1}-x_{0}=\frac{29 \pi}{180}-\frac{\pi}{6}=-\frac{\pi}{180}
\end{gathered}
$$

$\sin 29^{\circ} \approx y\left(x_{0}\right)+y^{\prime}\left(x_{0}\right) d x=\frac{1}{2}+\frac{\sqrt{3}}{2}\left(-\frac{\pi}{180}\right) \approx 0.4848$.

Find the differentials of the functions: