Dan Sp.
  • Member for 3 years, 10 months
  • Last seen more than 1 year ago
  • Norther Virginia
How do I find where the graph of parametric equations crosses itself?
2 votes

It crosses itself where, for two different values of t, you get equal x and y values. So letting $t_1 = u$ and $t_2=v$: $$u^3-u+3 = v^3-v+3$$ $$u^2-3 = v^2-3$$ These simplify to: $$u(u^2-1) = v(v^2-1)$...

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Understanding the concept of basis
1 votes

Yes, $dim(S)=2$. Here is a basis for $B$: $\begin{bmatrix}0\\1\\0\end{bmatrix}$, $\begin{bmatrix}2\\0\\-1\end{bmatrix}$ The basis does not span $R^3$ but it is a basis for $S$. To complete $B$ for $...

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For which values of $p$ is the line $y_p(x)$ tangent to $f(x)$
Accepted answer
1 votes

You need more than just the slopes to be equal. At these points, y must also be equal. So you also have: $$px=x^4-x^3$$ $$p=x^3-x^2$$ This, with your equation, is a system of 2 equations and 2 ...

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Gambler's ruin: one player
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1 votes

The Huygens' formulas change the size of the bet as time goes on. This is not the best model for your description. You should look into Random Walk. Your problem is directly related to a 1-D random ...

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Simple Big-O Notation for Function
1 votes

None of this is correct! For a function, Big-O describes how a function grows as it approaches infinity or some value. In your case, as x goes to $0$ so: $$ f(x)= O(1) \text{ as } x \rightarrow 0$$ ...

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Rolling a fair 6-faced die twice, what is the probability that the sum of the two rolls is 6 given that the first one is an even number?
1 votes

First die can be 2, 4, or 6. Second any roll 1-6. So the sample space is $3 \cdot 6= 18$ Only two positive outcomes: {2,4} and {4,2}. Probability $= \frac{2}{18} = \frac{1}{9}$

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What is minimum speed needed to jump over sphere object that has radius R and at distance d?
1 votes

I do not believe your approach is the simplest. The height of the jump is: $$ 2R = \frac{\dot y_0^2}{2g} $$ So: $$ \dot y_0 = \sqrt{Rg} $$ Now take each component separately. First, what is the ...

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Is this statement accurate. And if not what are the chances
1 votes

Seems accurate. Life expectancy in the US is about 80 years. So $330,000,000 / 13,000 / 80 \approx 317$.

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Triple integrals, divergence theorem, and the area of the unit sphere?
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1 votes

Start with $\nabla f = (z+yz)\hat{i} + (-2y+xz)\hat{j}+ (x+xy)\hat{k}$. Lets apply the divergence theorem and see what we have to work with. $$ \iint_S \left[(z+yz)\hat{i} + (-2y+xz)\hat{j}+ (x+xy)\...

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Finding a function with level curves
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0 votes

I would first eliminate the parametric equations. The first one: $$x=\cos(t) + 1$$ $$\cos(t) = x - 1$$ And using the right triangle trick: $$\sin(t)=\sqrt{2x-x^2}$$ So, $y=\sqrt{2x-x^2}-1$ is the ...

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Vectors, calculate distance from these two points
0 votes

Since you said you used the quadratic formula, perhaps you made a math error. As the comment says, please show more of your work. You can learn mathjax here which is similar to latex. Here is the ...

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Integration and Increasing Water Levels
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0 votes

You confused one of your rates. The height of the water is increasing at a rate of 0.2 feet per second. This one would be $$\frac{dh}{dt} = 0.2$$ You equated $0.2$ to $\frac{dV}{dt}$. This is ...

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Find the acute angle between the surfaces $xy^2z = 3x + z^2 $ and $3x^2-y^2+2z=1$ at the point $(1,-2,1)$
0 votes

Did you go ahead and take arccos$\left(\frac{-3}{7\sqrt{6}}\right)$? I get about 100$^o$. So your answer is no. The acute angle is 180 - 100 or about 80$^o$.

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How to decide if a function is increasing (strictly) at point where derivative is zero
0 votes

Keep taking derivatives until the first non-zero derivative at the point of interest. So for $x^3$: $$f'(x)=3x^2$$ $$f''(x)=6x, f''(0) = 0$$ $x^3$ is peculiar, it has an inflection point here so ...

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If $a = b$ then $b=a$... Flipping lhs and rhs of algebraic equations
0 votes

Well, lets say you are evaluating how much money you have vs how much money your friend has. You have 4 - \$5 dollar bills. Your friend has 2 - $10 bills. After thinking about it, you conclude you ...

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If $\exists f''(0)$, show $\lim _{h \rightarrow 0} \frac{f(h)-2f(0)+f(-h)}{h^2}=f''(0)$
0 votes

Use two Taylor series for $f'$ at $0$ using steps $h$ and $-h$. $$f(h) = f(0) + h \cdot f'(0) + \frac{h^2 \cdot f''(0)}{2!} + \frac{h^3 \cdot f'''(0)}{3!} + \frac{h^4 \cdot f''''(0)}{4!} ...$$ $$f(-h)...

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Initial Velocity Vector and Diagram from Parametric Equation
0 votes

Your initial velocity vector is correct but your $v(t)$ is incorrect. It should be: $$v(t)=(0.627273, -1.31655t+ 1.96578)$$ Yes, the velocity in the $y$ direction changes over time. At the initial ...

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Quantum mechanics characteristic polynomial
Accepted answer
0 votes

Hint: apply the quadratic formula. $a = 1$, $b = -3\mu$, and $c = \mu^2$. $\mu$ is in fact, just a number. Your two energy values will just pop out as a function of $\mu$.

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Where’s the error in my factorial proof?
0 votes

On this line: $$E^2 O^2 2 \cdot 4 \cdot 6 \cdots N = E^2 O^2 \cdot (2 \cdot 2) \cdot (2 \cdot 3) \cdot (2 \cdot 4) \cdots (2 \cdot (N/2))$$ I believe you missed the first $2$. It should be: $$E^2 O^2 ...

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finite difference periodic boundary
0 votes

You replace $u_{j-1}$ with $u_n$ and replace $u_{j+1}$ with $u_0$ because your box does not have overlap. i.e. $u_0$ is not the same point as $u_n$. These two points are 'next' to each other across ...

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Figuring out an equation for a tangent line
0 votes

One way to get an equation of a line is with a point and a slope. This is the appropriate way with the equation you give: $y-y_1 = m(x-x_1)$. You indicate the easy way to get the point yourself, plug ...

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Get the probability, with Pascal's triangle.
0 votes

The interpretation is as follows. There is one combination of 5 men. 5 men here. There are 5 combinations of 4 men and 1 woman. +20 men. Thats 25 men and 5 women. If you keep going like this, you ...

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Is a two-part function differentiable if the limit of one part equals the value at the other part?
0 votes

Not necessarily. As a counterexample, consider the function $$y = |x|$$ $y$ is differentiable at $x = 0^-$ and at $x = 0^+$. However, note that $y = 0$ at $x = 0$, but it is not differentiable.

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Calculating Flux
Accepted answer
-1 votes

Use the Divergence Theorem for the whole thing. This converts the surface integral to a volume integral: $$ \iint_T F\cdot \hat{n} dS = \iiint_V (\nabla \cdot F) dV$$ This is much more doable to ...

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