MHT CET · Maths · Application of Derivatives
The approximate value of \(\tan ^{-1}(0.999)\) is (use \(\pi=3.1415\) )
- A 0.7843
- B 0.7849
- C 0.7847
- D 0.7851
Answer & Solution
Correct Answer
(B) 0.7849
Step-by-step Solution
Detailed explanation
\(\begin{aligned}
& \text { Let } \mathrm{f}(x)=\tan ^{-1} x \\
\therefore \quad & \mathrm{f}^{\prime}(x)=\frac{1}{1+x^2}
\end{aligned}\)
Here, \(\mathrm{a}=1\) and \(\mathrm{h}=-0.001\)
\(\begin{aligned}
\therefore \quad f(a+h) \approx f(a) & +h f^{\prime}(a) \\
\therefore \quad \tan ^{-1}(0.999) & \approx \frac{\pi}{4}+\frac{1}{1+1}(-0.001) \\
& \approx \frac{\pi}{4}-\frac{0.001}{2} \\
& \approx \frac{\pi}{4}-0.0005 \\
& \approx \frac{3.1415}{4}-0.0005 \\
& \approx 0.7849
\end{aligned}\)
& \text { Let } \mathrm{f}(x)=\tan ^{-1} x \\
\therefore \quad & \mathrm{f}^{\prime}(x)=\frac{1}{1+x^2}
\end{aligned}\)
Here, \(\mathrm{a}=1\) and \(\mathrm{h}=-0.001\)
\(\begin{aligned}
\therefore \quad f(a+h) \approx f(a) & +h f^{\prime}(a) \\
\therefore \quad \tan ^{-1}(0.999) & \approx \frac{\pi}{4}+\frac{1}{1+1}(-0.001) \\
& \approx \frac{\pi}{4}-\frac{0.001}{2} \\
& \approx \frac{\pi}{4}-0.0005 \\
& \approx \frac{3.1415}{4}-0.0005 \\
& \approx 0.7849
\end{aligned}\)
See the Complete Solution
Get step-by-step explanations for this and 2.5 Lakh+ more JEE, NEET & CET questions.
- Unlock all solutions
- Practice the full chapter
- Track accuracy across PYQs
4.8 rated on Google Play · 14,000+ reviews
More questions from Maths
- An ice ball melts at the rate which is proportional to the amount of ice at that instant. Half of the quantity of ice melts in 15 minutes. \(x_0\) is the initial quantity of ice. If after 30 minutes the amount of ice left is \(k x_0\), then the value of \(k\) isMHT CET 2024 Easy
- If the slope of one of the lines given by is two times the other thenMHT CET 2018 Medium
- If \(\tan A=\frac{1}{2}, \tan B=\frac{1}{3}\) then \(\tan (A+2 B)\) has the valueMHT CET 2022 Easy
- A random variable has the following probability distribution
The then value of \(p\) isMHT CET 2024 Easy - \(\lim _{y \rightarrow 0} \frac{\sqrt{1+\sqrt{1+y^4}}-\sqrt{2}}{y^4}=\)MHT CET 2024 Medium
- If \(\int \frac{\mathrm{d} x}{1+3 \sin ^2 x}=\frac{1}{2} \tan ^{-1}(\mathrm{f}(x))+\mathrm{c}\), where c is -a constant of integration, then \(\mathrm{f}(x)\) is equal toMHT CET 2024 Medium
More PYQs from MHT CET
- 10 is divided into two parts such that the sum of double of the first and square of the other is minimum, then the numbers are respectivelyMHT CET 2021 Medium
- When an electron in a hydrogen atom jumps from the third orbit to the second orbit, it emits a photon of wavelength \({\lambda} \lambda^{\prime}\). When it jumps from the fourth orbit to third orbit, the wavelength emitted by the photon will beMHT CET 2020 Medium
- The derivative of \(\log |x|\) isMHT CET 2007 Easy
- Which among the following elements is used in devising photo electric cells?MHT CET 2022 Easy
- The value of \(\tan \left(2 \tan ^{-1}\left(\frac{\sqrt{5}-1}{2}\right)\right)\) isMHT CET 2024 Medium
- The maximum value of the objective function \(Z=3 x+2 y\) for linear constraints \(x+y \leq 7\) \(2 x+3 y \leq 16, x \geq 0, y \geq 0\) isMHT CET 2010 Medium