Successive Differentiation-Previous year questions
Previous Year Questions on Successive Differentiation
Successive Differentiation has been a fundamental concept in mathematics, enabling the computation of higher-order derivatives. Its historical significance lies in its application to solving complex mathematical models and real-world scenarios. The importance of Successive Differentiation is reflected in its wide range of applications in physics, engineering, and economics.
Vidyasagar University
2023-24 (NEP)
- If \(y=\frac{x^{3}}{x^{2}-1} \) then find \(y_{n}\). [2]
- If \(y=x^{n-1}\log{x}\), then prove that \(y_{n}=\frac{(n-1)!}{x} \). [2]
- If \(y=\frac{x^{2}-6}{x^{3}-x^{2}-2x}\), find \(y_{n} \). [2]
- Wrtie the Leibnitz theorem successive derivatives up to \(4^{th}\) order. [2]
- If \(y=\sin{\left(m\cos^{-1}{\sqrt{x}} \right)} \) then prove that \(\lim\limits_{x\to 0}\frac{y_{n+1}}{y_{n}}=\frac{4n^{2}-m^{2}}{4n+2} \). [5]
- If \(y=\sin{\left(m\cos^{-1}{\sqrt{x}} \right)} \) then prove that \(\lim\limits_{x\to 0}\frac{y_{n+1}}{y_{n}}=\frac{4n^{2}-m^{2}}{4n+2} \). [4]
- If \(y=\left(\sin^{-1}{x} \right)^{2} \) show that \(\left(1-x^{2} \right)y_{n-2}\right)-\left(2n+1 \right)xy_{n-1}\right) -n^{2} y_{n}\right)=0\). Also find \(y_{n}(0) \). [6]
- If \(y=e^{ax}\cos^{2}{bx} \), find \(y_{n}\) where \(a,b\gt 0\). [2]
- If \(y=x^{n-1}\log{x} \), then prove that \(y_{n}=\frac{(n-1)!}{x}\). [2]
- Let \(P_{n}=D^{n}\left(x^{n}\log{x} \right) \), hence show that \(P_{n}=n!\left(\log{x}+1+\frac{1}{2}+…+\frac{1}{n} \right) \). [5]
- If \(y=2\cos{x}\left(\sin{x}-\cos{x} \right) \) then find the value of \(\left(b_{20}\right)_{0}\). [2]
- If \(y=\sin{\left(m\cos^{-1}{\sqrt{x}} \right)} \) then prove that \(\lim\limits_{x\to 0}\frac{y_{n+1}}{y_{n}}=\frac{4n^{2}-m^{2}}{4n+2} \). [4]
- Wrtie the Leibnitz theorem successive derivatives up to \(4^{th}\) order. [2]
FAQs
- What is Successive Differentiation?
It is the process of finding higher-order derivatives of a function by repeatedly differentiating it. - Where is Successive Differentiation applied?
It is applied in physics, engineering, and economics for solving dynamic systems and modeling behavior. - What are higher-order derivatives?
These are derivatives obtained by differentiating a function multiple times. - Is Successive Differentiation part of university courses?
Yes, it is included in the curriculum of many universities in Differential Calculus. - What is the significance of higher-order derivatives?
They help in analyzing the concavity, inflection points, and behavior of functions. - What topics are related to Successive Differentiation?
Topics include Linear Algebra, Classical Algebra, and Abstract Algebra. - How is Successive Differentiation used in real life?
It is used in optimizing systems, analyzing motion, and modeling economic trends. - Are there limitations to Successive Differentiation?
Yes, it requires the function to be differentiable multiple times, which may not always be the case. - Where can I find practice questions on Successive Differentiation?
Explore Mathematics Questions or topics like Differential Calculus. - Can Successive Differentiation be automated?
Yes, software tools like Mathematica and MATLAB can perform it efficiently.
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