#### Important questions -Differential Calculus

11th Standard

Reg.No. :
•
•
•
•
•
•

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|