Using differentials, find the approximate values of the following… - Vidyadip

Question

Using differentials, find the approximate values of the following:
$\sin\Big(\frac{22}{14}\Big)$

✓

Answer

Consider the function $\text{y}=\text{f} (\text{x})=\sin\text{x}^\circ$ Let:
$\text{x}=\frac{22}{7}$
$\text{x}+\triangle \text{x}=\frac{22}{14}$ Then,
$\triangle\text{x}= \frac{-22}{14}$For $\text{x}=\pi$
$\text{y}=\sin\Big (\frac{22}{7}\Big)=0$ Let:
$\text{dx}=\triangle \text{x}=\sin\frac{-22}{14}=-\sin\Big (\frac{\pi}{2}\Big)=-1$ Now, $\text{y}=\sin\text {x}$
$\Rightarrow\frac {\text{dy}}{\text{dx}}=\cos\text{x}$ $\Rightarrow\Big(\frac {\text{dy}}{\text{dx}}\Big)_{\text{x}= \frac{22}{7}}=-1$ $\therefore\triangle \text{y}=\text{dy}=\frac{\text{dy}} {\text{dx}}\text{dx}=-1\times(-1)=1$ $\Rightarrow\triangle \text{y} =1$ $\therefore\sin\frac {22}{14}=\text{y}+\triangle\text{y}=1$

Need a full question paper?

Generate a complete, print-ready paper with questions like this in minutes — across 16+ boards, with answer keys.

Start Generating Free

Explore more

More "5 Marks Questions" from Differentials, Errors and Approximations→Differentials, Errors and Approximations — all question types→MATHS STD 12 Science question papers→Rajasthan Board STD 12 Science question papers→Rajasthan Board question paper generator→

Similar questions

A coin is tossed thrice and all the eight outcomes are assumed equally likely. In which of the following cases are the following events A and B are independent?
A = the number of heads is two,
B = the last throw results in head.

find the area of the region in the first quadrant by the $x-$axis, the line $y = x$ and circle $x^2+ y^2 = 32.$

Evaluate the following integrals:
$\int\frac{1}{\sin\text{x}+\sin2\text{x}}\ \text{dx}$

$A$ factory has three machines $X, Y$ and $Z$ producing $1000, 2000$ and $3000$ bolts per day respectively. The machine $X$ produces $1\%$ defective bolts $,Y$ produces $1.5\%$ and $Z$ produces $2\%$ defective bolts. At the end of a day, a bolt is drawn at random and is found to be defective. What is the probability that this defective bolt has been produced by machine $X$?

Verify Rolle's theorem of the following function on the indicated interval
$\text{f}(\text{x})=4^{\sin\text{x}}\text{ on }[0,\pi]$

Evaluate the following integrals:
$\int\text{x}^3\tan^{-1}\text{x dx}$

$\text{A}=\begin{bmatrix}1&-2&0\\ 2&1&3\\ 0&-2&1\end{bmatrix}\text{and }\text{B}=\begin{bmatrix}7&2&-6\\ -2&1&-3\\ -4&2&5\end{bmatrix}$, find AB. Hence, solve the system of equations:
x - 2y = 10, 2x + y + 3z = 8 and -2y + z = 7

Differentiate the following functions with respect to x:
$\tan^{-1}\Big(\frac{\text{a}+\text{bx}}{\text{b}-\text{ax}}\Big)$

If $\text{x}=\text{a}(\cos\theta+\theta\sin\theta),\text{y}=\text{a}(\sin\theta-\theta\cos\theta)$ prove that $\frac{\text{d}^2\text{x}}{\text{d}\theta^2}=\text{a}(\cos\theta-\theta\sin\theta),\frac{\text{d}^2}{\text{d}\theta^2}$ $=\text{a}(\sin\theta-\theta\cos\theta)\ \text{and}\ \frac{\text{d}^2\text{y}}{\text{dx}^2}=\frac{\sec^3\theta}{\text{a}\theta}$

$\begin{vmatrix}1+\text{a}&1&1\\1&1+\text{a}&\text{a}\\1&1&1+\text{a}\end{vmatrix}=\text{a}^3+3\text{a}^2$