# Get help with multivariable calculus equations

Recent questions in Multivariable calculus
Multivariable functions

### 1. Given a complex valued function can be written as $$f(z) = w = u(x,y) + iv(x,y)$$, where w is the real part of w and v is the imaginary part of v. Using algebraic manipulation figure out what wu and v is if $$\displaystyle{f{{\left({z}\right)}}}=\overline{{{z}}}{\left({1}-{e}^{{{i}{\left({z}+\overline{{{z}}}\right)}}}\right)}$$ Here $$\displaystyle\overline{{{z}}}$$ denotes the complex conjugate of z. 2. Using the concept of limits figure out what the second derivative of $$\displaystyle{f{{\left({z}\right)}}}={z}{\left({1}-{z}\right)}$$ is. 3. Use the theorems of Limits that have been discussed before to show that $$\displaystyle\lim_{{{z}\rightarrow{1}-{i}}}{\left[{x}+{i}{\left({2}{x}+{y}\right)}\right]}={1}+{i}.{\left[\text{Hint: Use }\ {z}={x}+{i}{y}\right]}$$

Multivariable functions

### Suppose that the random variables X and Y have the joint p.d.f. $$f(x,y)=\begin{cases}kx(x-y),0 (i) Evaluate the constant k. (ii) Find the marinal p.d.f. of the two random variables Multivariable functions ANSWERED asked 2021-09-13 ### Find all the complex roots. Leave your answers in polar form with the argument in degrees. The complex fourth roots of \(\displaystyle{100}+{100}\sqrt{{3}}{i}$$

Multivariable functions

### A polynomial P is given. $$\displaystyle{P}{\left({x}\right)}={x}^{{3}}+{64}$$ a)Find all zeros of P , real and complex.(Enter your answers as a comma-separated list. Enter your answers as a comma-separated list.) $$x=?$$ b) Factor P completely. $$P(z)=?$$

Multivariable functions

### Let $$\displaystyle{Y}_{{1}},{Y}_{{2}},\dot{{s}},{Y}_{{n}}$$ be independent and identically distributed random variables such that for $$\displaystyle{0}{<}{p}{<}{1},{P}{\left({Y}_{{i}}={1}\right)}={p}\ \text{ and }\ {P}{\left({Y}_{{i}}={0}\right)}={q}={1}-{p}$$ (Such random variables are called Bernoulli random variables.) (Maximum Likelihood Estimation ) Please construct the likelihood function for parameter p. (Maximum Likelihood Estimation ) Please obtain the MLE estimator for p.

Multivariable functions

### In each of the following problems , use the information given to determine a. $$(f+g)(-1)$$ b.$$(f-g)(-1)$$ c.$$(fg)(-1)$$ d. $$\displaystyle{\left({\frac{{{f}}}{{{g}}}}\right)}{\left(-{1}\right)}$$ for the following functions, $$f=\left\{(5,2),(0,-1),(-1,3),(-2,4)\right\} \text{ and } g=\left\{(-1,3),(0,5)\right\}$$ $$f=\left\{(3,15),(2,-1),(-1,1)\right\} \text{ and } g(x)=-2$$

Multivariable functions

### Find the indefinite integral by making a change of variables. $$\displaystyle\int{x}^{{2}}\sqrt{{{1}-{x}}}{\left.{d}{x}\right.}$$

Multivariable functions

### Simplify $$\displaystyle{\frac{{{4}{c}+{16}}}{{{5}{c}}}}\div{\frac{{{c}+{4}}}{{{15}{c}^{{3}}}}}$$ and state any restrictions on the variables.

Multivariable functions

### Which of the following is true? $$\displaystyle{\frac{{{\left.{d}{y}\right.}}}{{{\left.{d}{t}\right.}}}}+{\frac{{{y}}}{{{1}-{t}}}}={3}{t}+{4}$$ is best solved with undetermined coefficients. $$\displaystyle{\frac{{{\left.{d}{y}\right.}}}{{{\left.{d}{t}\right.}}}}={\frac{{{t}}}{{{y}-{y}{t}^{{2}}}}}$$ is best solved with separation of variables. $$\displaystyle{\frac{{{\left.{d}{y}\right.}}}{{{\left.{d}{t}\right.}}}}={\frac{{{t}}}{{{y}}}}$$ is best solved with integrating factors. $$\displaystyle{\frac{{{\left.{d}{y}\right.}}}{{{\left.{d}{t}\right.}}}}={2}{y}-{\sin{{\left({5}{t}\right)}}}$$ is best solved with separation of variables. $$\displaystyle{\frac{{{\left.{d}{y}\right.}}}{{{\left.{d}{t}\right.}}}}={\frac{{{t}}}{{{y}-{y}{t}^{{2}}}}}$$ is best solved with undetermined coefficients.

Multivariable functions

### Plot each complex number. Then write the complex number in polar form. You may express the argument in degrees or radians. 57.$$1-i$$ 58.$$\displaystyle-{2}\sqrt{{3}}+{2}{i}$$ 59.$$-3-4i$$

Multivariable functions

### If z and $$\displaystyle\overline{{{z}}}$$ are conjugate complex numbers , find two complex numbers, $$\displaystyle{z}={z}_{{1}}\ \text{ and }\ {z}={z}_{{2}}$$ , that satisfy the equation $$\displaystyle{3}{z}\overline{{{z}}}+{2}{\left({z}-\overline{{{z}}}\right)}={39}+{j}{12}$$

Multivariable functions

### Solve the differential equation ..(b and a are variables) $$\displaystyle{\left({D}^{{2}}+{4}{D}+{3}\right)}{b}={1}+{2}{a}+{3}{a}^{{2}}$$

Multivariable functions

### A value of r close to −1 suggests a strong _______ linear relationship between the variables.

Multivariable functions

### Solve each equation over the set of complex numbers, find the magnitudes of the solutions and draw them in the complex plane. Hint: For some of the equations , to get n roots you must use $$\displaystyle{a}^{{3}}+{b}^{{3}}={\left({a}+{b}\right)}{\left({a}^{{2}}-{a}{b}+{b}^{{2}}\right)}$$ and $$\displaystyle{a}^{{3}}-{b}^{{3}}={\left({a}-{b}\right)}{\left({a}^{{2}}+{a}{b}+{b}^{{2}}\right)}$$ $$\displaystyle{x}^{{6}}-{1}={0}$$

Multivariable functions

### The random variables X and Y have joint density function $$\displaystyle{f{{\left({x},{y}\right)}}}={\frac{{{2}}}{{{3}}}}{\left({x}+{2}{y}\right)},{0}\leq{x}\leq{1},{0}\leq{y}\leq{1}$$ Find the marginal density functions of X and Y. $$\displaystyle{{f}_{{X}}{\left({x}\right)}}=?$$ $$\displaystyle{{f}_{{Y}}{\left({y}\right)}}=?$$ Are X and Y statistically independent random variables?

Multivariable functions

### Write down the definition of the complex conjugate, $$\displaystyle\overline{{{z}}}\ \text{ ,if }\ {z}={x}+{i}{y}$$ where z,y are real numbers. Hence prove that, for any complex numbers w and z, $$\displaystyle\overline{{{w}\overline{{{z}}}}}=\overline{{{w}}}{z}$$

Multivariable functions

### $$\displaystyle{\frac{{{\left.{d}{y}\right.}}}{{{\left.{d}{x}\right.}}}}={x}\sqrt{{{1}-{y}^{{2}}}}$$ solve by seperation of variables.

Multivariable functions

### Consider the system $$\displaystyle\dot{{{x}}}={x}-{y},\dot{{{y}}}={x}+{y}$$ a) Write the system in matrix form and find the eigenvalues and eigenvectors (Note: they will be complex valued) b) Write down the general solution for the system of differential equations using only real valued functions. [Hint : use the equality $$\displaystyle{e}^{{{i}\omega{t}}}={\cos{{\left(\omega{t}\right)}}}+{i}{\sin{{\left(\omega{t}\right)}}}$$ ]

Multivariable functions