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

Question

Using differentials, find the approximate values of the following:
$\cos\Big(\frac{11\pi}{36}\Big)$

✓

Answer

Consider the function $\text{y}=\text{f} (\text{x})=\cos\text{x}$ Let:
$\text{x}=\frac{\pi}{3}$ $\text{x}+\triangle \text{x}=\frac{11\pi}{36}$ Then,
$\triangle\text{x}= \frac{-\pi}{36}=-5^\circ$ For $\text{x}=\frac{\pi} {3}$
$\text{y}=\cos\Big (\frac{\pi}{3}\Big)=0.5$ Let:
$\text{dx}=\triangle \text{x}=-\sin5^\circ=-0.08716$ Now $\text{y}=\cos\text {x}$
$\Rightarrow\frac {\text{dy}}{\text{dx}}=-\sin\text{x}$ $\Rightarrow\Big(\frac {\text{dy}}{\text{dx}}\Big)_{\text{x}= \frac{\pi}{3}}=0.86603$ $\therefore\triangle \text{y}=\text{dy}=\frac{\text{dy}} {\text{dx}}\text{dx}=-0.86603\times(- 0.08716)=0.075575$ $\Rightarrow\triangle \text{y} =0.075575$ $\therefore\cos\frac {11\pi}{36}=\text{y}+\triangle\text{y} =0.5+0.075570=0.575575$

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

$\begin{vmatrix}1&\text{a}&\text{a}^2\\\text{a}^2&1&\text{a}\\\text{a}&\text{a}^2&1\end{vmatrix}=(\text{a}^3-1)^2$

Show that the relation $R$ in the set $A=\{1,2,3,4,5\}$ given by $R=\{(a, b):|a-b|$ is even $\}$, is an equivalence relation. Show that all the elements of $\{1,3,5\}$ are related to each other and all the elements of $\{2,4\}$ are related to each other. But no element of $\{1,3,5\}$ is related to any element of $\{2,4\}$.

Prove that the curves $y^2 = 4x$ and $x^2 + y^2 - 6x + 1 = 0$ touch each other at the point $(1, 2).$

Find the shortest distance between the lines $\frac{\text{x}-2}{-1}=\frac{\text{y}-5}{2}=\frac{\text{z}-0}{3}$ and $\frac{\text{x}-0}{2}=\frac{\text{y}+5}{-1}=\frac{\text{z}-1}{2}.$

Evaluate the following integrals:
$\int^\limits{\frac{\pi}{2}}_02\sin\text{x }\cos\text{x}\tan^{-1}(\sin\text{x})\text{dx}$

A box contains 100 tickets, each bearing one of the numbers from 1 to 100. If 5 tickets are drawn successively with replacement from the box, find the probability that all the tickets bear numbers divisible by 10.

Find the equation of the perpendicular drawn from the point P(2, 4, -1) to the line $\frac{\text{x}+5}{1}=\frac{\text{y}+3}{4}=\frac{\text{z}-6}{-9}.$ Also, write down the coordinates of the foot of the perpendicular from P.

Evaluate the following integrals:
$\int\text{x}\cos^3\text{x dx}$

If A = $\left[\begin{array}{lll} {3} & {\sqrt{3}} & {2} \\ {4} & {2} & {0} \end{array}\right] \text { and } B=\left[\begin{array}{rrr} {2} & {-1} & {2} \\ {1} & {2} & {4} \end{array}\right]$ verify that

(A′)′ = A
(A + B)′ = A′ + B′
(kB)′ = kB′, where k is any constant.

Find the equations of the tangent and the normal to the following curves at the indicated points.
$\text{xy}=\text{c}^2\text{ at }\big(\text{ct},\frac{\text{c}}{\text{t}}\big)$