I&=\displaystyle\int{\frac{dx}{x^2\sqrt{x^2-16}}}

\quad x=4\sec\theta

\quad dx=4\tan \theta\sec \theta

\end{array}$

just seeing if I started with the right x and dx or is there better

Mahalo]]>

$lim_{x\to +\infty}\frac{ \lfloor x\rfloor}{x}$]]>

May I ask how to solve this type of problem?]]>

$$\int\frac{3x^2+x+12}{(x^2+5)(x-3)}

=\frac{A}{(x^2+5)}+\frac{B}{(x-3)}$$

$$3x^2+x+12=A(x-3)+B(x^2+5)$$

x=3 then 27+3+12=14B

3=B

x=0 then

12=-3A+15

1=A

$$\int\frac{1}{(x^2+5)} \, dx

+3\int\frac{1}{(x-3)}\, dx$$

$\displaystyle

\frac{\arctan\left(\frac{x}{\sqrt{5}}\right)}{\sqrt{5}}

+3\ln\left(\left|x-3\right|\right)+C$

maybe??? not sure]]>

Let $(\Omega, p)$ be a discrete probability room with induced probability measure $P$ and let $A, B\subseteq \Omega$ be two events.

I want to show that $P(A\cap B)-P(A)P(B)=P(A^c)P(B)-P(A^c\cap B)$.

For that do we write to what for example $P(A^c\cap B)$ is equal to simplify the expression or which way is the best one?

We have that $(A\cap B)\cap (A^c\cap B)=\emptyset$ and so \begin{align*}&P((A\cap B)\cup (A^c\cap B))=P(A\cap B)+P (A^c\cap B)\\ & \Rightarrow...

Show equality of probabilities]]>

Can you please help me ?

I have tried to do it many times but I end up getting the wrong answer.

Thank you in advance.]]>

I am hoping someone can help me understand a PDE. I am reading a paper and am trying to follow the math. My experience with PDEs is limited though and I am not sure I am understanding it all correctly. I have 3 coupled PDEs, for $n$, $f$ and $c$, that are written in general form, and I would like to write them in 2d (in terms of x and y directions). The equations for $f$ and $c$ are fairly straightforward, but I am having some trouble with the one for $n$:

$$\frac{\partial...

Writing PDE in terms of x and y]]>

Assume a function y = f(x) , differentiable everywhere. Now we have for some Δx

Δy = f(x + Δx) - f(x)

The differential of x, is defined as “dx”, can be any real number, and dx = Δx

The differential of y, is defined by “dy” and

dy = f’(x) dx

Clearly,

Δy ≈ dy, depending on the magnitude of Δx.

In calculus an expression like “dx” usually denotes something infinitesimally small.

Why is it necessary to have dy and dx used as real...

Question about the differential in Calculus]]>

How to find image of $f(x)= x + sinx$ about the given line $y = - x$ .

Similarly can we take image of a function about a function? OR is it necessary about which we take image should be a point, line only?]]>

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For $x,\,y,\,z\ge 0$ and $x+y+z=1$, prove that $0\le xy+yz+zx-2xyz\le\dfrac{7}{27}$.

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Remember to read the POTW submission guidelines to find out how to submit your answers!]]>

How to answer this question? Any math help, hint, or even correct answer will be accepted.]]>

Find values of $r$ of the form $y = e^{rt}$

$y''+y'-6y=0$

$r^2+r-6=(r-2)(r+3)\quad \therefore r=2 \quad r=-3$

well so far

it that all there is to do?

However I didn't see clearly what the purpose of this was or how it is eventually used.]]>

I can see how the correct answer for part b is arrived at by counting the number of combinations for "2 girls" then counting the number of combinations for "3 girls" , and adding these results.

But I've attached my first way of approaching it, which is wrong. Wondering if anyone can tell me why my approach doesn't give the correct answer. I know it's probably something obvious that I'm missing.

Thanks]]>

$y'=-1-2y$

$\begin{array}{ll}

rewrite &y'+2y=-1\\

exp &u(t)=\exp\ds\int 2 \, dx=e^{2t}\\

product &(e^{2t}y)'=-e^{2t}\\

&e^{2t}y=\ds\int -e^{2t} \, dt=\dfrac{-e^(2 t)}{2}+c\\

hence &y(t)=\dfrac{-e^{2t}}{2} + \dfrac{c}{e^{2t}}\\

t \to \infty &=-\dfrac{1}{2}+0\\

so &y \to \dfrac{1}{2}\textit{ as t} \to \infty...

-b.1.1.4 directional field as t \to \infny]]>

a. transforms a vector of dimension n to a vector of dimension m

b. transforms a vector of dimension m to a vector of dimension n

c. a vector of dimension n+m to a vector of dimension m

d. a vector of dimension n+m to a vector of dimension n]]>

The die is rolled.

B be the event blue face lies down, and R be the event a red face lands down

This is represented by the following tree diagram, where p, s, t are probabilities.

[ATTACH type="full" width="238px"...

s.aux.26 our-sided die has three blue faces, and one red face......]]>

]]>

x\approx1.48

\quad x\approx 005$

ok I could only do this with a calculator but need steps]]>

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Evaluate $\dfrac{1}{\sin 6^{\circ}}+\dfrac{1}{\sin 78^{\circ}}-\dfrac{1}{\sin 42^{\circ}}-\dfrac{1}{\sin 66^{\circ}}$.

-----

Remember to read the POTW submission guidelines to find out how to submit your answers!]]>

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Let $P(x)=x^3-2x+1$ and $Q(x)=x^3-4x^2+4x-1$. Show that if $P(r)=0$, then $Q(r^2)=0$.

-----

Remember to read the POTW submission guidelines to find out how to submit your answers!]]>

]]>

Solve for x give exact for\\

$\log{(x-10)}-\log{(x-6)}=\log{2}$

$\begin{array}{rrll}

\textsf{subtraction rule} &\log\left(\dfrac{x-10}{x-6}\right)&=\log{2} \\

\textsf{drop logs} &\dfrac{x-10}{x-6}&=2 \\

&x-10&=2(x-6)=2x-12\\

\textsf{isolate x} &2&=x

\end{array}$

hopefully ok but???

quess we could just put all the logs on r side and set em to zero]]>

]]>

0.123456876…

0.254896487…

0.143256876…

0.758468126…

0.534157162…]]>

(i) Represent this information on a standard normal curve diagram, indicating clearly the area representing 90\%

(ii) Find the value of \textbf{t}. $P(Z\le t) =0.9\quad Z = 1.282\quad t=57+(4.4(1.282))=62.64$ hours

\begin{tikzpicture}[scale=0.6]

%preamble \usepackage{pgfplots}

\newcommand\gauss[2]{1/(#2*sqrt(2*pi))*exp(-((x-#1)^2)/(2*#2^2))} % Gauss function, parameters mu and sigma

\begin{axis}[every axis plot post/.append style={...

(solved) standard normal sscurve P(Z le t) =0.9]]>

Here

I was wondering how to calculate the contents of it

Mainly the first equasion equaling 1 or 0]]>

I am looking at the following exercise:

Find the solution $u(t,x)$ of the problem

$$u_t-u_{xx}=2 \sin{x} \cos{x}+ 3\left( 1-\frac{x}{\pi}\right)t^2, t>0, x \in (0,\pi) \\ u(0,x)=3 \sin{x}, x \in (0,\pi) \\ u(t,0)=t^3, u(t, \pi)=0, t>0$$

At the suggested solution, it is stated that the boundaries are non-zero, so we want to set them equal to $0$.

We set $v=u-\left( 1-\frac{x}{\pi}\right) t^3-\frac{x}{\pi} \cdot 0 \Rightarrow v=u-t^3+t^3 \frac{x}{\pi}$.

Could you explain...

How do we choose this function?]]>

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If $a,\,b$ and $c$ are positive reals such that $abc=1$, prove that $\sqrt{\dfrac{a}{a+8}}+\sqrt{\dfrac{b}{b+8}}+\sqrt{\dfrac{c}{c+8}}\ge 1$.

-----

Remember to read the POTW submission guidelines to find out how to submit your answers!]]>

The die is rolled.

P(R)=\dfrac{1}{4}

This is represented by the following tree diagram, where p, s, t are probabilities.

[ATTACH type="full"...

aux 26 our-sided die has three blue face, and one red face.]]>

I've been trying to find a reasonable way to solve following problem:

N people meet each others in groups of K people such that no one meets more than once.

If I'm not mistaken, this requires two things:

- N must be divisible by K
- N - 1 must be divisible by K - 1.

The amount of money a student in the Accounting Department has in his pocket is a random variable that follows the normal distribution, with an average price of $30$ euros and a variance of $100$.

a) What is the probability that a student has $25$ to $35$ euros in his pocket?

b) If we randomly select $25$ students, then what is the probability that a student has less than $20$ euros in his pocket?

c) How much money does $75\%$ of students have in their pocket?

I...

Probability - Amount of money in pocket]]>

Ex: If the input is 2 25 55, the output is:

value - 25

value - 55

Code: #include <iostream> using namespace std; int main() { int inputCount; int i; cin >> inputCount; for(i=0; inputCount > i | | inputCount < i; ++i){ cin >> inputCount; cout << "value - " << inputCount << endl; } return 0; } |

Read and Format integers]]>

We have the following iteration from Newton's method \begin{align*}x_{k+1}&=x_k-\frac{f(x_k)}{f'(x_k)}=x_k-\frac{x_k^n-a}{nx_k^{n-1}}=\frac{x_k\cdot nx_k^{n-1}-\left (x_k^n-a\right )}{nx_k^{n-1}}=\frac{ nx_k^{n}-x_k^n+a}{nx_k^{n-1}}\\ & =\frac{ (n-1)x_k^{n}+a}{nx_k^{n-1}}\end{align*}

I want to show that for $x_0\geq a^{1/n}$ the method converges monotonically to the root.

So first we have to show that the sequence $(x_k)$ is monotone decreasing, right?

I have done the...

Monotonically convergence to the root]]>

I can't figure this one out. I tried to use truth tables, but never found an equivalence , no matter which of the 5 options I tried.

It is given that $\alpha$ is logically equivalent to $\alpha \rightarrow \sim \beta $ .

Which of the following is a tautology ?

1) $\alpha$

2) $\beta$

3) $\alpha \wedge \beta $

4) $\beta \vee \sim\alpha $

5) $\alpha \leftrightarrow \beta $]]>

(Can someone help me for this)]]>

PS: I JUST NEED THE ANSWER AND SOLUTIONS]]>

a. a football team

b. a

c. one of your classes]]>

Find all complex numbers x which satisfy the given condition

$\begin{array}{rl}

1+x&=\sqrt{10+2x} \\

(1+x)^2&=10+2x\\

1+2x+x^2&=10+2x\\

x^2-9&=0\\

(x-3)(x+3)&=0

\end{array}$

ok looks these are not

The signals are:

a) 𝑠(𝑛1, 𝑛2) = 𝑢(𝑛1, 𝑛2)𝑢(𝑛1, 𝑛2 + 2)

b) 𝑠(𝑛1, 𝑛2) = 𝛿(𝑛1 − 𝑛2 + 1)]]>

Electrical Resistance of a Wire

The electrical resistance of a wire varies directly with the length of the wire and inversely with the square of the diameter of the wire.

If a wire 432 feet long and 4 mm in diameter has a resistance of 1.24 $\Omega$

find the length of a wire of the same material whose resistance is 1.44 $\Omega$ and whose diameter is 3 mm

y varies inversely with x $\quad y=\dfrac{k}{x}$

y varies directly with x $\quad y=kx$

OK not real sure how to set this...

2.5.1 varies directly with the length of the wire and inversely with the square of the diameter of the wire.]]>

Find the domain of $f(x)=\sqrt{x^2-25}$

a. $[x\le-5]U[x\ge 5]$

b. x=5

c. $5 \le x$

d. $x\ne 5$

e. $\textit{all reals}$

well just by observation because of the radical I chose c.

but was wondering if imaginary numbers could be part of the domain alto it is not asked for here

other pre-calc repines form MHB]]>

#12 hope I rewrote the problem ok

Verify that $y_2(t)=t^{-2}\ln t\quad y_1(t)=t^{-2}$ is a solution of $t^2y''+5ty'+4y=0$

think the first steop is to compose a charactistic equation using r]]>

Answers from textbook:

My attempt:

Apologies for the blurry photo.]]>

Looking for a good Topology text]]>

My Working -->

Since $tanx$ and $cotx$ always have the same sign, so this holds true for any value of $x$.]]>