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Differentials and Partial Derivatives Model Question Paper

12th Standard EM

Reg.No. :
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Maths

Time : 01:00:00 Hrs
Total Marks : 40
5 x 1 = 5
1. If w (x, y) = xy, x > 0, then $\frac { \partial w }{ \partial x }$ is equal to

(a)

xy log x

(b)

y log x

(c)

yxy-1

(d)

x log y

2. If f (x, y) = exy then $\frac { { \partial }^{ 2 }f }{ \partial x\partial y }$ is equal to

(a)

xyexy

(b)

(1 +xy)exy

(c)

(1 +y)exy

(d)

(1 + x)exy

3. The change in the surface area S = 6x2 of a cube when the edge length varies from xo to xo+ dx is

(a)

12 xo+dx

(b)

12xo dx

(c)

6xo dx

(d)

6xo+ dx

4. If u = log $\sqrt { { x }^{ 2 }+{ y }^{ 2 } }$, then $\frac { { \partial }^{ 2 }u }{ \partial { x }^{ 2 } } +\frac { { \partial }^{ 2 }u }{ { \partial y }^{ 2 } }$ is

(a)

$\sqrt { { x }^{ 2 }+{ y }^{ 2 } }$

(b)

0

(c)

u

(d)

2u

5. If u = xy + yx then ux + uy at x = y = 1 is

(a)

0

(b)

2

(c)

1

(d)

6. 5 x 2 = 10
7. Find a linear approximation for the following functions at the indicated points.
g(x) = $g(x)=\sqrt { { x }^{ 2 }+9 } ,{ x }_{ 0 }=-4$

8. Find a linear approximation for the following functions at the indicated points.
${ h }({ x })=\frac { x }{ 1+x } =\frac { 1 }{ 2 }$

9. The time T, taken for a complete oscillation of a single pendulum with length l, is given by the equation T = 2ㅠ$\sqrt { \frac { 1 }{ g } }$, where g is a constant. Find the approximate percentage error in the calculated value of T corresponding to an error of 2 percent in the value of 1

10. If f (x, y) = 2x3 - 11x2y + 3y3, prove that $x\frac { \partial f }{ \partial x } +y\frac { \partial f }{ \partial y } =3f$

11. If of f(x, y) = x2 + y3 + 2xy2 find fxx, fyy, fxy and fyx.

12. 5 x 3 = 15
13. f(x,y) = $\frac { xy }{ { x }^{ 2 }+{ y }^{ 2 } }$, (x,y) ≠ (0,0) and f (0,0) = 0 Show that f is not continuous at f, -(0,0) and continuous at all other points of R2

14. Consider g(x,y) = $\frac { 2{ x }^{ 2 }y }{ { x }^{ 2 }+{ y }^{ 2 } }$, if (x,y) ≠ (0,0) and g(0,0) = 0 Show that g is continuous on R2

15. Let U(x, y, z) = x2 − xy + 3 sin z, x, y, z ∈ R Find the linear approximation for U at (2,−1,0).

16. Find the approximate value of f (3.02) where f(x) = 3x2 + 5x +3.

17. Using differentials find the approximate value of tan 46° if it is given that 10 = 0.01745 radians

18. 2 x 5 = 10
19. Let (x, y) = e-2y cos(2x) for all (x, y) ∈ R2. Prove that u is a harmonic function in R2.

20. If V = log r and r2 = x2 +y2 + z2, then prove that $\frac { { \partial }^{ 2 }V }{ \partial { x }^{ 2 } } +\frac { { \partial }^{ 2 }V }{ \partial { y }^{ 2 } } +\frac { { \partial }^{ 2 } }{ \partial { z }^{ 2 } } =\frac { 1 }{ { r }^{ 2 } }$