# Multivariable functions Answers

Multivariable functions

### Write formulas for the indicated partial derivatives for the multivariable function. $$\displaystyle{f{{\left({x},{y}\right)}}}={7}{x}^{{2}}+{9}{x}{y}+{4}{y}^{{3}}$$ a)$$\displaystyle\frac{{\partial{f}}}{{\partial{x}}}$$ b)(delf)/(dely)ZSK c)$$\displaystyle\frac{{\partial{f}}}{{\partial{x}}}{\mid}_{{{y}={9}}}$$

Multivariable functions

### COnsider the multivariable function $$\displaystyle{g{{\left({x},{y}\right)}}}={x}^{{2}}-{3}{y}^{{4}}{x}^{{2}}+{\sin{{\left({x}{y}\right)}}}$$. Find the following partial derivatives: $$\displaystyle{g}_{{x}}.{g}_{{y}},{g}_{{{x}{y}}},{g{{\left(\times\right)}}},{g{{\left({y}{y}\right)}}}$$.

Multivariable functions

### Write first and second partial derivatives $$\displaystyle{g{{\left({r},{t}\right)}}}={t}{\ln{{r}}}+{11}{r}{t}^{{7}}-{5}{\left({8}^{{r}}\right)}-{t}{r}$$ a)$$\displaystyle{g}_{{r}}$$ b)$$\displaystyle{g}_{{{r}{r}}}$$ c)$$\displaystyle{g}_{{{r}{t}}}$$ d)$$\displaystyle{g}_{{t}}$$ e)$$\displaystyle{g}_{{{\mathtt}}}$$

Multivariable functions

### Write formulas for the indicated partial derivatives for the multivariable function. $$\displaystyle{g{{\left({k},{m}\right)}}}={k}^{{3}}{m}^{{6}}−{8}{k}{m}$$ a)$$\displaystyle{g}_{{k}}$$ b)$$\displaystyle{g}_{{m}}$$ c)$$\displaystyle{g}_{{m}}{\mid}_{{{k}={2}}}$$

Multivariable functions

### Write formulas for the indicated partial derivatives for the multivariable function. $$\displaystyle{g{{\left({x},{y},{z}\right)}}}={3.1}{x}^{{2}}{y}{z}^{{2}}+{2.7}{x}^{{y}}+{z}$$ a)$$\displaystyle{g}_{{x}}$$ b)$$\displaystyle{g}_{{y}}$$ c)$$\displaystyle{g}_{{z}}$$

Multivariable functions

### Multivariable optimization. Find the demisions of the rectangular box with largest volume if the total surface are is given as 64 $$\displaystyle{c}{m}^{{2}}$$

Multivariable functions

### Multivariable optimization question. Find three positive real numbers whose sum is one and the sum of their squares is a minimum.

Multivariable functions

### Find the critical points of the following multivariable function, and using the Second Derivative Test to find the maxima and minima of the equation below: $$\displaystyle{a}{\left({x},{y}\right)}=\frac{{{8}{x}-{y}}}{{{e}^{{{x}^{{2}}+{y}^{{2}}}}}}$$

Multivariable functions

### A surface is represented by the following multivariable function, $$\displaystyle{f{{\left({x},{y}\right)}}}={x}^{{3}}+{y}^{{3}}-{3}{x}-{3}{y}+{1}$$ Calculate coordinates of stationary points.

Multivariable functions

### Consider this multivariable function. f(x,y)=xy+2x+y−36 a) What is the value of f(2,−3)? b) Find all x-values such that f (x,x) = 0

Multivariable functions

### Use polar coordinates to find the limit. [Hint: Let $$\displaystyle{x}={r}{\cos{{\quad\text{and}\quad}}}{y}={r}{\sin{}}$$ , and note that (x, y) (0, 0) implies r 0.] $$\displaystyle\lim_{{{\left({x},{y}\right)}\to{\left({0},{0}\right)}}}\frac{{{x}^{{2}}-{y}^{{2}}}}{\sqrt{{{x}^{{2}}+{y}^{{2}}}}}$$

Multivariable functions

### Write first and second partial derivatives $$\displaystyle{f{{\left({x},{y}\right)}}}={2}{x}{y}+{9}{x}^{{2}}{y}^{{3}}+{7}{e}^{{{2}{y}}}+{16}$$ a)$$\displaystyle{f}_{{x}}$$ b)$$\displaystyle{f}_{{\times}}$$ c)$$\displaystyle{f}_{{{x}{y}}}$$ d)$$\displaystyle{f}_{{y}}$$ e)$$\displaystyle{f}_{{{y}{y}}}$$ f)$$\displaystyle{f}_{{{y}{x}}}$$

Multivariable functions

### Nonexistence of a limit Investiage the limit $$\displaystyle\lim_{{{\left({x},{y}\right)}\to{\left({0},{0}\right)}}}\frac{{\left({x}+{y}\right)}^{{2}}}{{{x}^{{2}}+{y}^{{2}}}}$$

Multivariable functions

### What tis the complete domain D and range R of the following multivariable functions: $$\displaystyle{w}={2}{\sin{{x}}}{y}$$

Multivariable functions

### Write formulas for the indicated partial derivatives for the multivariable function. $$\displaystyle{g{{\left({k},{m}\right)}}}={k}^{{4}}{m}^{{5}}−{3}{k}{m}$$ a)$$\displaystyle{g}_{{k}}$$ b)$$\displaystyle{g}_{{m}}$$ c)$$\displaystyle{g}_{{m}}{\mid}_{{{k}={2}}}$$

Multivariable functions

### Average value over a multivariable function using triple integrals. Find the average value of $$\displaystyle{F}{\left({x},{y},{z}\right)}={x}^{{2}}+{y}^{{2}}+{z}^{{2}}$$ over the cube in the first octant bounded bt the coordinate planes and the planes x=5, y=5, and z=5

Multivariable functions

### Let $$\displaystyle{z}{\left({x},{y}\right)}={e}^{{{3}{x}{y}}},{x}{\left({p},{q}\right)}=\frac{{p}}{{q}}{\quad\text{and}\quad}{y}{\left({p},{q}\right)}=\frac{{q}}{{p}}$$ are functions. Use multivariable chain rule of partial derivatives to find (i) $$\displaystyle\frac{{\partial{z}}}{{\partial{p}}}$$ (ii) $$\displaystyle\frac{{\partial{z}}}{{\partial{q}}}$$.

Multivariable functions

### A concert promoter produces two kinds of souvenir shirt, one kind sells for $18 ad the other for$25. The company determines, the total cost, in thousands of dollars, of producting x thousand of the $18 shirt and y thousand of the$25 shirt is given by $$\displaystyle{C}{\left({x},{y}\right)}={4}{x}^{{2}}-{6}{x}{y}+{3}{y}^{{2}}+{20}{x}+{19}{y}-{12}.$$ How many of each type of shirt must be produced and sold in order to maximize profit?

Multivariable functions