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CALCULUS AND ANALYSIS
MULTIVARIABLE CALCULUS
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Recent questions in Multivariable calculus
Multivariable functions
ANSWERED
asked 2021-09-19
Prove mathematically (using variables, not numbers) that kx>kz for hydraulic conductivity - topic: Groundwater Hydrology
\(\displaystyle{K}_{{z}}={\frac{{{d}}}{{{\frac{{{d}_{{1}}}}{{{K}_{{1}}}}}+{\frac{{{d}_{{2}}}}{{{K}_{{2}}}}}+\dot{{s}}+{\frac{{{d}_{{n}}}}{{{K}_{{n}}}}}}}}\)
\(\displaystyle{K}_{{x}}={\frac{{{K}_{{1}}{d}_{{1}}+{K}_{{2}}{d}_{{2}}+\dot{{s}}+{K}_{{n}}{d}_{{n}}}}{{{d}}}}\)
Multivariable functions
ANSWERED
asked 2021-09-19
Find the Jacobian of the transformation
\(\displaystyle{x}={u}+{4}{v},{y}={u}^{{2}}-{2}{v}\)
Multivariable functions
ANSWERED
asked 2021-09-18
Describe the procedure for finding critical points of a function in two independent variables.
Multivariable functions
ANSWERED
asked 2021-09-18
Question No.3, Part (A)
i. From the Euler Relatins, decude that
\(\displaystyle{e}^{{-{3}{i}{\frac{{\pi}}{{{4}}}}}}=-{\frac{{\sqrt{{2}}}}{{{2}}}}{\left({1}+{i}\right)}\)
ii. Find the cartesian form of the complex number ,
\(\displaystyle\sqrt{{2}}{e}^{{{i}{\frac{{\pi}}{{{4}}}}}}\)
iii. Find polar and exponential forms of the complex number ,
\(\displaystyle{\frac{{{3}}}{{{2}}}}+{\frac{{{3}\sqrt{{3}}}}{{{2}}}}{i}\)
Multivariable functions
ANSWERED
asked 2021-09-17
There are four washing machines in an apartment complex: A, B, C, D. On any given day the probability that these machines break down is as follows:
P(A) = 0.04, P(B) = 0.01, P(C) = 0.06, P(D) = 0.01 .
Assume that the functionality of each machine is independent of that of others. What is the probability that on a given day at least one machine will be working?
Multivariable functions
ANSWERED
asked 2021-09-17
Explain how to find the degree of a polynomial in two variables.
Multivariable functions
ANSWERED
asked 2021-09-17
Convert the equalities below equalities by adding sack variables. Use
\(\displaystyle{s}_{{1}}\ \text{ and }\ {s}_{{2}}\)
for your slack variables.
\(\displaystyle{3}{x}_{{1}}+{9}{x}_{{2}}\leq{42}\)
converts to () =42
\(\displaystyle{15}{x}_{{1}}+{7}{x}_{{2}}\leq{38}\)
converts to () = 38
Multivariable functions
ANSWERED
asked 2021-09-17
Lynbrook West, an apartment complex, has 100 two-bedroom units. The monthly profit (in dollars) realized from renting x apartments is represented by the following function.
\(\displaystyle{P}{\left({x}\right)}=-{11}{x}^{{2}}+{1830}{x}-{34000}\)
(a) What is the actual profit realized from renting the 41st unit, assuming that 40 units have already been rented?
Multivariable functions
ANSWERED
asked 2021-09-17
Find the complex zeros of the following polynomial function. Write f in factored form.
\(\displaystyle{f{{\left({x}\right)}}}={2}{x}^{{4}}-{5}{x}^{{3}}-{20}{x}^{{2}}+{115}{x}-{52}\)
The complex zeros of f are ?
Multivariable functions
ANSWERED
asked 2021-09-16
Some of the solution sets for quadratic equations in the next sections in this chapter will contain complex numbers such as
\(\displaystyle{\frac{{-{4}+\sqrt{{-{12}}}}}{{{2}}}}\ \text{ and }\ {\frac{{-{4}-\sqrt{{-{12}}}}}{{{2}}}}\)
. We can simplify the first number as follows.
\(\displaystyle{\frac{{-{4}+\sqrt{{-{12}}}}}{{{2}}}}\)
\(\displaystyle={\frac{{-{4}+{i}\sqrt{{{12}}}}}{{{2}}}}\)
\(\displaystyle={\frac{{-{4}+{2}{i}\sqrt{{3}}}}{{{2}}}}\)
\(\displaystyle={\frac{{{2}{\left(-{2}+{i}\sqrt{{3}}\right)}}}{{{2}}}}\)
\(\displaystyle=-{2}+{i}\sqrt{{3}}\)
Simplify each of the following complex numbers.
\(\displaystyle{\frac{{{10}+\sqrt{{-{45}}}}}{{{4}}}}\)
Multivariable functions
ANSWERED
asked 2021-09-16
Find the complex zeros of the following polynomial function. Write f in factored form.
\(\displaystyle{f{{\left({x}\right)}}}={x}^{{3}}-{12}{x}^{{2}}+{49}{x}-{58}\)
The complex zeros of f are =?
Multivariable functions
ANSWERED
asked 2021-09-16
Let
\(\displaystyle\alpha={a}+{b}{i}\ \text{ and }\ \beta={c}+{d}{i}\)
be complex scalars and let A and B be matrices with complex entries.
(a) Show that
\(\displaystyle\overline{{\alpha+\beta}}=\overline{{\alpha}}+\overline{{\beta}}\ \text{ and }\ \overline{{\alpha\beta}}=\overline{{\alpha}}\overline{{\beta}}\)
(b) Show that the (i,j) entries of \overline{AB} \text{ and } \bar{A}\bar{B} are equal and hence that
\(\displaystyle\overline{{{A}{B}}}=\overline{{{A}}}\overline{{{B}}}\)
Multivariable functions
ANSWERED
asked 2021-09-15
Solve the system of equation by the method of your choice.
\(\displaystyle{x}^{{2}}+{\left({y}-{9}\right)}^{{2}}={49}\)
\(\displaystyle{x}^{{2}}-{7}{y}=-{14}\)
Multivariable functions
ANSWERED
asked 2021-09-15
Solve the given differential equation by separation of variables.
\(\displaystyle{\left({e}^{{y}}+{1}\right)}^{{2}}{e}^{{-{y}}}{\left.{d}{x}\right.}+{\left({e}^{{x}}+{1}\right)}^{{6}}{e}^{{-{x}}}{\left.{d}{y}\right.}={0}\)
Multivariable functions
ANSWERED
asked 2021-09-15
Evaluate the integral by making an appropriate change of variables.
Multivariable functions
ANSWERED
asked 2021-09-15
Suppose X is a random variable such that E(3X-7)=8 and
\(\displaystyle{E}{\left({\frac{{{X}^{{2}}}}{{{2}}}}\right)}={19}\)
.
What is Var(70-2X)?
Multivariable functions
ANSWERED
asked 2021-09-15
Solve for i in terms of other variables:
\(\displaystyle{F}={C}{\left[{\frac{{{\left({1}+{i}\right)}^{{n}}-{1}}}{{{i}}}}\right]}\)
Multivariable functions
ANSWERED
asked 2021-09-15
A supplement was given to patients to lower their blood pressure. The blood pressure before and after the supplement was recorded.
What kind of variables are Begin and End and what kind of scale are they following? Multiple choice. Choose one answer for scale and one answer for variables.
Note: the scale is identified as either nominal, ordinal, interval, or ratio (Choose the correct answer). Please explain why you chose the answer.
Variables are defined as either numerical (count/discrete or decimals) OR categorical (ordinal or nominal) (Choose the correct answer). Please explain why you chose the answer.
Multivariable functions
ANSWERED
asked 2021-09-15
3) Answer the following questions considering the complex functions given below.
a) Using the definition of complex derivative, evaluate f(z) expression using derivative operation based on limiting case as
\(\displaystyle\lim_{{\triangle{z}\rightarrow{0}}}\)
a.1
\(\displaystyle{f{{\left({z}\right)}}}={\frac{{{1}}}{{{z}^{{2}}}}},{\left({z}\ne{q}{0}\right)}\)
Multivariable functions
ANSWERED
asked 2021-09-14
Solve the differential equation. (x and y are variables)
\(\displaystyle{\left({D}^{{2}}+{4}{D}\right)}{y}={1}+{2}{x}+{3}{x}^{{2}}\)
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