Chapter 6, Problem 1RE

Match each expression in column A with an equivalent expression in column B. Assume that $z=f\left(x,y\right).$

$\left[6.2,6.6\right]$

Column A	Column B
$\frac{\partial z}{\partial x}$	a. ${\displaystyle {\int }_2^3\frac{1}{2}xdx}$
	b. $f_{yx}$
	c. $f_{xy}$
	d. ${\displaystyle {\int }_2^3{y}^{3}dy}$
	e. $f_x$
	f. $15x^2{y}^7$
	g. $35x^3{y}^6$
	h. $f_y$

To determine

The correct match for the expression $\frac{\partial z}{\partial x}$ with an equivalent expression in column B.

Answer to Problem 1RE

Solution:

The expression $\frac{\partial z}{\partial x}$ matches with $\underset{\_}{\left(e\right).f_x}$.

Explanation of Solution

Given Information:

The table:

$\begin{array}{ll}\text{A}\hfill & \text{B}\hfill \\ \frac{\delta z}{\delta x}\hfill & {\displaystyle \underset{2}{\overset{3}{\int }}\frac{1}{2}xdx}\hfill \\ \frac{\delta z}{\delta y}\hfill & f_{yx}\hfill \\ \frac{\delta }{\delta x}\left(5x^3{y}^7\right)\hfill & f_{xy}\hfill \\ \frac{\delta z}{\delta y}\left(5x^3{y}^7\right)\hfill & {\displaystyle \underset{2}{\overset{3}{\int }}{y}^{3}dy}\hfill \\ \frac{{\delta }^{2}z}{\delta x\partial y}\hfill & f_x\hfill \\ \frac{{\delta }^{2}z}{\delta y\partial x}\hfill & 15x^2{y}^7\hfill \\ {\displaystyle \underset{2}{\overset{3}{\int }}{\displaystyle \underset{0}{\overset{1}{\int }}2x{y}^{3}dxdy}}\hfill & 35x^3{y}^6\hfill \\ {\displaystyle \underset{2}{\overset{3}{\int }}{\displaystyle \underset{0}{\overset{1}{\int }}2x{y}^{3}dydx}}\hfill & f_y\hfill \end{array}$

Consider the expression,

$\frac{\partial z}{\partial x}$

Now, this expression can also be written as,

$\frac{\partial z}{\partial x}=f_x$

Which matches with the expression (e) in column B.

Hence, the expression $\frac{\partial z}{\partial x}$ matches with option (e) which is $f_x$.