Continuity and Differentiability L12

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राह संघर्ष की जो चलता है,
वो ही संसार को बदलता है,
जिसने रातों से जंग जीती है,
सूर्य बन कर वो ही निकलता है।

Lecture - 12 Chapter 5 Continuity and Differentiability

This lecture is the continuation of the previous lecture based on miscellaneous questions from NCERT Miscellaneous Exercise of Chapter 5 Class 12 Maths Continuity and Differentiability.

Questions discussed in this lecture:

NCERT MISCELLANEOUS EXERCISE

Question 12. Find \( \frac{dy}{dx} \), if \( y = 12(1 – \cos t), x = 10(t – \sin t), \frac{-\pi}{2}<t<\frac{\pi}{2} \)

Question 13. Find \( \frac{dy}{dx} \), if \( y = \sin^{-1}x + \sin^{-1}{\sqrt{1 – x^2}}, 0<x<1 \)

Question 14. If \( x\sqrt{1 + y} + y\sqrt{1 + x} = 0\), for, \( -1<x<1 \), prove that \( \frac{dy}{dx} = – \frac{1}{(1 + x)^2} \)

Question 15. If \( (x – a)^2 + (y – b)^2 = c^2 \), for some c > 0, prove that \( \frac{\left[ 1 + \left( \frac{dy}{dx}\right )^2 \right]^{\frac{3}{2}}}{\frac{d^2y}{dx^2}} \) is a constant independent of a and b.

Question 16. If \( \cos y = x \cos (a + y) \), with \( \cos a ne pm 1 \), prove that \( \frac{dy}{dx} = \frac{\cos^2 (a + y)}{\sin a} \).

Question 17. If \( x = a (\cos t + t \sin t) rm{and} y = a (\sin t – t \cos t)\), find \( \frac{d^2{x}}{dx^2} \).

Question 18. If \( f(x) = | x |^3 \), show that f ”(x) exists for all real x and find it.

Question 19. Using mathematical induction prove that \( \frac{d}{dx}(x^n) = nx^{n – 1}\) for all positive integers n.

Question 20. Using the fact that sin (A + B) = sin A cos B + cos A sin B and the differentiation, obtain the sum formula for cosines.

Question 21. Does there exist a function which is continuous everywhere but not differentiable at exactly two points? Justify your answer.

Question 22. If \( \begin{vmatrix} f(x) & g(x) & h(x) \\ l & m & n \\ a & b & c \end{vmatrix} \) prove that \( \frac{dy}{dx} = \begin{vmatrix} f'(x) & g'(x) & h'(x) \\  l & m & n \\ a & b & c \end{vmatrix} \)

Question 23. If \( y = e^{a \cos^{-1}x}, -1 le x le 1 \), show that \( (1 – x^2) \frac{d^2{y}}{dx^2} – x \frac{dy}{dx} – a^2 y = 0 \)

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