Important questions -Differential Calculus

11th Standard

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Maths

Use blue pen Only

Time : 00:50:00 Hrs
Total Marks : 55

    Part A 

    Answer all the questions

    10 x 1 = 10
  1. If y=\({1\over4}u^4,u={2\over 3}x^3+5,\) then \({dy\over dx}\) is

    (a)

    \({1\over 27}x^2(2 x^3+15)^3\)

    (b)

    \({2\over 27}x(2 x^3+5)^3\)

    (c)

    \({2\over 27}x^2(2 x^3+15)^3\)

    (d)

    \(-{2\over 27}x(2 x^3+5)^3\)

  2. If y=cos (sin x2),then \({dy\over dx}\) at x= \(\sqrt{\pi\over 2}\) is

    (a)

    -2

    (b)

    2

    (c)

    \(-2\sqrt{\pi\over 2}\)

    (d)

    0

  3. If y=mx+c and f(0)=\(f'(0)=1\),then f(2) is

    (a)

    1

    (b)

    2

    (c)

    3

    (d)

    -3

  4. If f(x)=x tan-1 x, then f'(1) is

    (a)

    \(1+{\pi\over 4}\)

    (b)

    \({1\over 2}+{\pi\over 4}\)

    (c)

    \({1\over 2}-{\pi\over 4}\)

    (d)

    2

  5. If f(x)=x+2, then f'(f(x)) at x= 4 is

    (a)

    8

    (b)

    1

    (c)

    4

    (d)

    5

  6. If ,then the right hand derivative of f(x) at x = 2 is

    (a)

    0

    (b)

    2

    (c)

    3

    (d)

    4

  7. If g(x)=(x2+2x+3) f(x) and f(0)=5 and \(lim_{x \rightarrow 0}{f(x)-5\over x}=4\),then g'(0) is 

    (a)

    20

    (b)

    14

    (c)

    18

    (d)

    12

  8. If  ,then at x=3, f '(x) is

    (a)

    1

    (b)

    -1

    (c)

    0

    (d)

    does not exist

  9. If is differentiable at x = 1, then

    (a)

    \(a={1\over2},b={-3\over 2}\)

    (b)

    \(a={-1\over2},b={3\over 2}\)

    (c)

    \(a=-{1\over2},b=-{3\over 2}\)

    (d)

    \(a={1\over2},b={3\over 2}\)

  10. The number of points in R in which the function \(f(x)=|x-1|+|x-3|+sin \ x\) is not differentiable, is

    (a)

    3

    (b)

    2

    (c)

    1

    (d)

    4

  11. Part B

    Answer all the questions

    10 x 2 = 20
  12. Show that the greatest integer function \(f(x)=\left\lfloor x \right\rfloor \) is not differentiable at any integer?

  13. Find the derivatives of the following functions using first principle. f(x)=6

  14. Find the derivatives of the following functions using first principle.f(x)=-x2+2

  15. Differentiate the following with respect to x: y=ex+sin x+2

  16. Differentiate the following with respect to x :\(y={log x \ x \over e^x}\)

  17. Differentiate 2x.

  18. Differentiate the following:y = cos (tanx)

  19. Find \({dy\over dx}\) if x=at2 ; y=2at, t\(\neq 0.\)

  20. Find the derivatives of the following y = xcosx

  21. Find the derivatives of the following\(\sqrt{x^2+y^2}=tan^{-1}({y\over x})\)

  22. Part C 

    Answer all the questions

    5 x 3 = 15
  23. Determine whether the following function is differentiable at the indicated values. f(x) = |x| + |x - 1| at x = 0, 1

  24. Show that the following functions are not differentiable at the indicated value of x.
    {\(\begin{matrix} 3x, & x<0 \\ -4x & x\ge 0 \end{matrix};\) x=2

  25. Find the derivatives of the following functions with respect to corresponding independent variables:\(y={ \ x\over sin \ x+ cos \ x}\)

  26. Differentiate y \(=x^{\sqrt{x}}\)

  27. Find f'(x)if f(x)=cos-1(4x3-3x).

  28. Part D 

    Answer all the questions

    2 x 5 = 10
  29. Find the slope of the tangent line to the graph of f(x) = 7x + 5 at any point (x0, f(x0)).

  30. Examine the differentiability of functions in R by drawing the diagrams |sin x|

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