Let

$$w = ( 4 x - 3 y + 2 z ) ^ { 5 }$$

. Find

$$\frac { \partial ^ { 2 } w } { \partial x \partial z }$$

Consider the function $f(x)=3 x-x^{2}-1$. a. Write down $f^{\prime}(x)$. b. Show that at x = 1, $f(x)=f^{\prime}(x)$. c. Find the coordinates of a second point on the curve y = f(x) for which $f(x)=f^{\prime}(x)$.

Let

$$f(x, y)=x e ^ { 2 y }$$

. Find

$$f _ { x y }$$

.

Let

$$w = ( 4 x - 3 y + 2 z ) ^ { 5 }$$

. Find

$$\frac { \partial ^ { 3 } w } { \partial x \partial y \partial z }$$

.

Given $w=e^y\cos x$, find

$$\left. \frac{\delta^3 x}{\delta y^2\delta x} \right|_{(\pi /4,0)}$$

Given z=

$$( 2 x - y ) ^ { 5 }$$

, find

$$\frac { \partial ^ { 3 } z } { \partial y \partial x \partial y }$$

Given

$$z=( 2 x - y ) ^ { 5 }$$

, find

$$\frac { \partial ^ { 3 } z } { \partial x ^ { 2 } \partial y }$$

.

Given $w=e^y\cos x$, find

$$\left. \frac{\delta^3 x}{\delta x^2\delta y} \right|_{(\pi /4,0)}$$

What factors affect how strongly an atom is attracted to its electrons?

Let

$$f(x, y)=x e ^ { 2 y }$$

. Find

$$f _ { y x }$$

.

Let

$$f(x, y)=\sin \left( 3 x ^ { 2 } + 6 y ^ { 2 } \right)$$

. Find

$$f _ { x x }$$

.

(a) Find the required Fourier series for the given function.(b) Sketch the graph of the function to which the series converges over three periods. f(x)=1,0<x<1,0,1<x<2;cosine series, period 4

determine whether the method of separation of variables can be used to replace the given partial differential equation by a pair of ordinary differential equations. If so, find the equations. tuxx+xut=0

Given 3y + x = 30, find the maximum/minimum value of 2xy and determine its nature.