NCERT Solutions Class 12 Mathematics Chapter 5 Continuity And Differentiability Download In Pdf
Question 1. Prove that the function f (x) = 5x â€“ 3 is continuous at x = 0, at x = â€“ 3 and at x = 5.
QuestioQuestion Examine the continuity of the function f (x) = 2x2 â€“ 1 at x = 3.
Question 3. Examine the following functions for continuity.
(a) f (x) = x â€“ 5
(b) f (x) = 1 x âˆ’ 5
(c) f (x) = 2 25 5 x x âˆ’ +
(d) f (x) = | x â€“ 5 |
Question 4. Prove that the function f (x) = xn is continuous at x = n, where n is a positive integer.
Question 5. Is the function f defined by , if 1 ( ) 5, if > 1 x x f x x â‰¤ = continuous at x = 0? At x = 1? At x = 2? Find all points of discontinuity of f, where f is defined by
Question 18. For what value of Î» is the function defined by ( 2 2 ), if 0 ( ) 4 1, if 0 f x x x x x x âŽ§âŽªÎ» âˆ’ â‰¤ = + > continuous at x = 0? What about continuity at x = 1?
Question 19. Show that the function defined by g (x) = x â€“ [x] is discontinuous at all integral points. Here [x] denotes the greatest integer less than or equal to x.
Question 20. Is the function defined by f (x) = x2 â€“ sin x + 5 continuous at x = Ï€?
Question 21. Discuss the continuity of the following functions:
(a) f (x) = sin x + cos x
(b) f (x) = sin x â€“ cos x
(c) f (x) = sin x . cos x
Question 22. Discuss the continuity of the cosine, cosecant, secant and cotangent functions.
Question 23. Find all points of discontinuity of f, where sin , if 0 ( ) 1, if 0 x x f x x x x < = + â‰¥
Question 24. Determine if f defined by 2 sin 1 , if 0 ( ) 0, if 0 x x f x x x â‰ = = is a continuous function?
is a continuous function.
Question 31. Show that the function defined by f (x) = cos (x2) is a continuous function.
Question 32. Show that the function defined by f (x) = | cos x | is a continuous function.
Question 33. Examine that sin | x | is a continuous function.
Question 34. Find all the points of discontinuity of f defined by f (x) = | x | â€“ | x + 1 |.
Differentiate the functions with respect to x in Exercises 1 to 8.
Question 1. sin (x2 + 5)
Question 2. cos (sin x)
Question 3. sin (ax + b)
Question 4. sec (tan ( x ))
Question 5. sin ( ) cos ( ) ax b cx d + +
Question 6. cos x3 . sin2 (x5)
Question 7. 2 cot ( x2 )
Question 8. cos( x )
Question 9. Prove that the function f given by f (x) = | x â€“ 1 |, x âˆˆ R is not differentiable at x = 1.
Question 10. Prove that the greatest integer function defined by f (x) = [x], 0 < x < 3 is not differentiable at x = 1 and x = 2.
Differentiate the following w.r.t. x:
Question 1. sin ex x
Question 2. esin 1 x âˆ’
Question 3. ex3
Question 4. sin (tanâ€“1 eâ€“x)
Question 5. log (cos ex)
Question 6. 2 5 ex + ex +... + ex
Question 7. e x , x > 0
Question 8. log (log x), x > 1
Question 9. cos , 0 log x x x >
Question 10. cos (log x + ex), x > 0
Differentiate the functions given in Exercises 1 to 11 w.r.t. x.
Question 1. cos x . cos 2x . cos 3x
Question 2. ( 1)( 2) ( 3)( 4)( 5) x x x x x âˆ’ âˆ’ âˆ’ âˆ’ âˆ’
Question 3. (log x)cos x
Question 4. xx â€“ 2sin x
Question 5. (x + 3)2 . (x + 4)3 . (x + 5)4
Question 6. 1 1 1
Question 7. (log x)x + xlog x
Question 8. (sin x)x + sinâ€“1 x
Question 9. xsin x + (sin x)cos x
Question 10. 2 cos 2 1 1 xx x x x + + âˆ’
Question 11. (x cos x)x + 1 (xsin x)x Find dy dx of the functions given in Exercises 12 to 15.
Question 12. xy + yx = 1
Question 13. yx = xy
Question 14. (cos x)y = (cos y)x
Question 15. xy = e(x â€“ y)
Question 16. Find the derivative of the function given by f (x) = (1 + x) (1 + x2) (1 + x4) (1 + x8) and hence find f â€²(1).
Question 17. Differentiate (x2 â€“ 5x + 8) (x3 + 7x + 9) in three ways mentioned below:
(i) by using product rule
(ii) by expanding the product to obtain a single polynomial.
(iii) by logarithmic differentiation. Do they all give the same answer?
Question 18. If u, v and w are functions of x, then show that d dx (u. v. w) = du dx v. w + u . dv dx . w + u . v dw dx in two ways - first by repeated application of product rule, second by logarithmic differentiation.
If x and y are connected parametrically by the equations given in Exercises 1 to 10, without eliminating the parameter, Find dy dx .
Question 1. x = 2at2, y = at4
Question 2. x = a cos Î¸, y = b cos Î¸
Question 3. x = sin t, y = cos 2t 4. x = 4t, y = 4 t
Question 5. x = cos Î¸ â€“ cos 2Î¸, y = sin Î¸ â€“ sin 2Î¸ 6. x = a (Î¸ â€“ sin Î¸), y = a (1 + cos Î¸)
Question 7. x = sin3 cos 2 t t , cos3 cos2 y t t =
Question 8. cos log tan 2 x = a = a sin t
Question 9. x = a sec Î¸, y = b tan Î¸
Question 10. x = a (cos Î¸ + Î¸ sin Î¸), y = a (sin Î¸ â€“ Î¸ cos Î¸)
Question 11. If x asin 1 t , y acos 1 t , show that dy y dx x
Find the second order derivatives of the functions given in Exercises 1 to 10.
Question1. x2 + 3x +
Question 2. x20
Question 3. x . cos x
Question 4. log x
Question 5. x3 log x
Question 6. ex sin 5x
Question 7. e6x cos 3x
Question 8. tanâ€“1 x
Question 9. log (log x)
Question 10. sin (log x)
Question 11. If y = 5 cos x â€“ 3 sin x, prove that 2 2 d y y 0 dx + =
Question 12. If y = cosâ€“1 x, Find 2 2 d y dx in terms of y alone.
Question 13. If y = 3 cos (log x) + 4 sin (log x), show that x2 y2 + xy1 + y = 0
Question 14. If y = Aemx + Benx, show that 2 2 d y (m n) dy mny 0 dx dx âˆ’ + + =
Question 15. If y = 500e7x + 600eâ€“ 7x, show that 2 2 d y 49y dx =
Question 16. If ey (x + 1) = 1, show that 2 2 2 d y dy dx dx =
Question 17. If y = (tanâ€“1 x)2, show that (x2 + 1)2 y2 + 2x (x2 + 1) y1 = 2
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