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

Question

Using differentials, find the approximate values of the following:
$\log_\text{e}10.02$ it being given that $\log_\text{e}10=2.3026$

✓

Answer

Consider the function $\text{y}=\text{f} (\text{x})=\log_\text{e}\text{x}$ Let:
x = 10 $\text{x}+\triangle \text{x}=10.02$Then,
$\triangle\text{x} =0.02$ For x $\text{y}\log_\text{e} 10=2.3026$Let:
$\text{dx}=\triangle \text{x}=0.02$Now, $\text{y}=\log_\text {e}\text{x}$
$\Rightarrow\frac {\text{dy}}{\text{dx}}=\frac{1}{\text {x}}$ $\Rightarrow\Big(\frac {\text{dy}}{\text{dx}}\Big)_{\text{x} =10}=\frac{1}{10}$ $\therefore\triangle \text{y}=\text{dy}=\frac{\text{dy}} {\text{dx}}\text{dx}=\frac{1} {10}\times0.02=0.002$ $\Rightarrow\text{y} =0.002$ $\therefore\log_\text {e}10.02=\text{y}+\triangle\text{y} =2.3046$

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 APPLICATION OF DERIVATIVES→APPLICATION OF DERIVATIVES — all question types→Maths STD 12 Science question papers→Gujarat Board STD 12 Science question papers→Gujarat Board question paper generator→

Similar questions

Let $\overrightarrow{\text{a}} = \hat{\text{i}} + 4\hat{\text{j}} +2\hat{\text{k}}, \overrightarrow{\text{b}} = 3\hat{\text{i}} - 2\hat{\text{j}} +7\hat{\text{k}}$ and $\overrightarrow{\text{c}} = 2\hat{\text{i}} - \hat{\text{j}} + 4\hat{\text{k}}$ Find a vector $\overrightarrow{\text{d}}$ which is perpendicular to both $\overrightarrow{\text{a}} \text{and} \overrightarrow{\text{b}}\text{and} \overrightarrow{\text{c}} . \overrightarrow{\text{d}} = 27.$

Given $A=\left[\begin{array}{ccc}1 & -1 & 1 \\ 1 & -2 & -2 \\ 2 & 1 & 3\end{array}\right]$ and $B=\left[\begin{array}{ccc}-4 & 4 & 4 \\ -7 & 1 & 3 \\ 5 & -3 & -1\end{array}\right]$ find $AB$ and use this result in solving the following system of equations. $x - y + z =4 , x - 2y - 2z = 9 , 2x + y + 3z = 1$

Find the general solution of the differential equation
$y ~dx – (x + 2y^2) dy = 0$.

If y(x) is a solution of the different equation $\Big(\frac{2+\sin\text{x}}{1+\text{y}}\Big)\frac{\text{dy}}{\text{dx}}=-\cos\text{x}$ and $\text{y}(0)=1,$ then find the value of $\text{y}\Big(\frac{\pi}{2}\Big).$

Differentiate the following functions from first principles:
$x^2e^x$.

By using the properties of definite integral, evaluate the integral $\int_0^{\frac{\pi}{2}}(2 \log \sin x-\log \sin 2 x) d x$

Prove that the function f given by $\text{f}(\text{x})=\log\cos\text{x}$ is strictly increasing on $\Big(-\frac{\pi}{2},0\Big)$ and strictly decreasing on $\Big(0,\frac{\pi}{2}\Big).$

Evaluate the following integrals:$\int(\log\text{x})^2\cdot\text{x dx}$

Differentiate the following functions with respect to x:
$\frac{\text{e}^\text{x}\log\text{x}}{\text{x}^2}$

Solve the following differential equation:
$\text{2x}^{2}\frac{\text{dy}}{\text{dx}}-\text{2 xy + y}^{2}=0$