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#### Differentials and Partial Derivatives - Two Marks Study Materials

12th Standard EM

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Maths

Time : 01:00:00 Hrs
Total Marks : 30
15 x 2 = 30
1. Use the linear approximation to find approximate values of
${ (123) }^{ \frac { 2 }{ 3 } }$

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

3. A sphere is made of ice having radius 10 cm. Its radius decreases from 10 cm to 9-8 cm. Find approximations for the following:
change in the volume

4. Find ∆f and df for the function f for the indicated values of x, ∆x and compare
f(x) = x2 + 2x + 3; x = -0.5, ∆x = dx = 0.1

5. Let g(x, y) = $\frac { { x }^{ 2 }y }{ { x }^{ 4 }+{ y }^{ 2 } }$ for (x, y) ≠ (0, 0) and f(0, 0) = 0
Show that $\begin{matrix} lim \\ (x,y)\rightarrow (1,2) \end{matrix}$ g(x, y) = 0 along every line y = mx, m ∈ R

6. If w(x, y) = x3 − xy + y2, x, y ∊ R, find the linear approximation for w at (1,−1)

7. If v(x,y) = x2 - xy + $\frac14$ y + 7, x,y ∈ R, find the differential dv.

8. A circular metal plate expands under heating so that its radius increases by 2%. Find the approximate increase in the area of the plate if the radius of the plate before heating is 10cm.

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

10. If u=x2+3xy2+y2, then prove that $\cfrac { { \partial }^{ 2 }u }{ \partial x\partial y } =\cfrac { { \partial }^{ 2 }u }{ \partial y\partial x }$

11. If $u={ e }^{ \frac { x }{ y } }sin\left( \cfrac { x }{ y } \right) +{ e }^{ \frac { y }{ x } }cos\left( \cfrac { y }{ x } \right)$ ,then prove that $x\cfrac { \vartheta u }{ \vartheta x } +y\cfrac { \vartheta u }{ \vartheta y } =0$

12. If w=log(x2+y2),x=cosθ,y=sinθ, find $\cfrac { dw }{ d\theta }$

13. If y = sin x and x changes from $\cfrac { \pi }{ 2 } to\cfrac { 22 }{ 14 }$ what is the approximate change in y.

14. Find the linear approximation to $g(z)=\sqrt [ 4 ]{ zat } z=2$

15. Calculate df for $f=\sqrt { 2x+5 }$ when x = 22 and dx = 3.