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    Green's Theorem on a Polar Rose

    Area using Green's Theorem is given by: \displaystyle \frac{1}{2}\oint_{C}-ydx+xdy One petal is swept out at \theta=\frac{\pi}{3} \displaystyle y=\cos(3t)\cos(t), \;\ x=\cos(3t)\sin(t) \displaystyle dx=\cos(t)\cos(3t)-3\sin(t)\sin(3t) \displaystyle dy=-\sin(t)\cos(3t)-3\cos(t)\sin(3t) So...
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    Need help with powers

    It's asking for the fourth root of 16. What to the 4th power gives you 16?. Here is a small 'trick' that can help find the values of fractional exponents. For example, say you're given \displaystyle 16^{\frac{3}{4}}. What is this equal to?. It is handy to write it in terms of radical...
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    Cooling problem

    Newton's Law of Cooling says \frac{dT}{dt}=k(T-T_{m}) where T_{m} is the ambient temp. In this case, T_{m}=0 So, you have \frac{dT}{dt}=kT Separate variables and integrate: \int\frac{dT}{T}=\int kdt \ln(T)=kt+c T=e^{kt+c}=Ce^{kt} Now, use your initial conditions to find C and k. T(0)=25...
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    Trouble with Comparison test for Series

    This looks like a fun series to evaluate. \displaystyle \sum_{n=1}^{\infty}\frac{nx^{n}}{n+5} You could start with \displaystyle\sum_{n=1}^{\infty}x^{n}=\frac{x}{1-x}, do some differentiating, integrating, etc. hammer it into the required form, then let x=1/2. Knowing that its sum will be...
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    Integration of 1/(a+btanx)

    Here is another way to tackle it. This method can be handy on various integrals of the form \int\frac{a\cdot \cos(x)+b\sin(x)}{c\cdot \cos(x)+e\sin(x))}dx by letting \text{Numer}=A[\text{Denom}]+B\frac{d}{dx}[\text{Denom}]. \int\frac{1}{a+b\tan(x)}dx =\int...
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    Help! stuck on this word problem

    There appears to be some info missing from the problem. How many total tickets were sold in all?. But, let x=# adult tickets sold and let y=# children's tickets sold. Then, the total revenue is 14x+7y=1600 Now, assuming you mistakenly left out the 'number of tickets sold' portion of the...
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    Average velocity functions containing sine

    There's a difference between 35\sin(2)\cdot 0.5=17.5\sin(2)=15.91.. and 35\sin(2(.5))=29.45.... One is 35\sin(2t) and the other 35t\sin(2)
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    Help Solving Equation of Tangent Line (VERY Simple)

    I moved this thread to Intermediate/Advanced Algebra, though it appears to be the beginnings of the definition of a derivative. This is an exercise in simplifying rational expressions. We have \displaystyle \frac{\frac{2}{2+h}-1}{h} Now, as with any fraction one is adding or subtracting, you...
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    Cant figure this out:(( ughh

    This is not calculus but a quadratic equation. Expand (x-11)^{2} and you get a quadratic: x^{2}-22x+129=0 use the quadratic formula to solve. x=\frac{-b\pm\sqrt{b^{2}-4ac}}{2a}
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    Distance of a Projective up a Ramp

    Try starting with the formula: \displaystyle y=\tan(a)x-\frac{g}{2v^{2}\cos^{2}(a)}x^{2} Now, set x=d\cos(b), \;\ y=d\sin(b) \displaystyle d\sin(b)=d\tan(a)\cos(b)-\frac{g}{2v^{2}\cos^{2}(a)}\cdot d^{2}\cos^{2}(b) Now, solve for d and it should simplify down to the needed formula. Note it...
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    Wanting to check my work using the subsitution method.

    Often with substitution, look at what is in parentheses. Use that for the sub. In this case, try letting u=x^{9}+9, \;\ \frac{du}{9}=x^{8}dx This is certainly not always the case, but with this one it looks promising. After making this sub, there is one more obvious sub you can make and you...
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    graphing calculator

    I am not familiar with the 84, but I have an 83 Plus. With it, all you have to do is store 5 in variable X. Then, as mmm said, type in X^2-3X+5 and it will return 15. X is a 2nd function on the 83. You hit ALPHA STO to display X. To store 5, or any other numerical value, in the variable X...
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    Tangent Line for Polar Coordinates

    r=10cos\theta Converting to cartesian: \sqrt{x^{2}+y^{2}}=\frac{10x}{\sqrt{x^{2}+y^{2}}} y=\pm\sqrt{10x-x^{2}} Since x=rcos\theta, \;\ y=rsin\theta, \;\ r=10cos\theta, you can use \theta=\frac{\pi}{6} to find the point of tangency for the line, its slope, and thus the equation.
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    Bernoulli equations getting lost just before they become linear odes

    I will step through a Bernoulli for part a. A Bernoulli DE has the form \frac{dy}{dx}+P(x)y=f(x)y^{n} We use the sub u=y^{1-n} as long as n\neq 0, \;\ n\neq 1 So, since we have n=2, we use y=u^{-1} \frac{dy}{dx}=-u^{-2}\frac{du}{dx}, by the chain rule. Subbing into our given DE we get...
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    domain and range

    y=2+\sqrt{x-1} The domain is all of the values you can plug in for x that return a real value. In other words, the domain is what you put in and the range is what comes out. So, as for the domain, what happens if you sub in a value for x that is less than 1?. What is the smallest value for x...
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    Basic partial differentials

    Yep.
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    Basic partial differentials

    . When you take the derivative of \displaystyle 3x^{\frac{1}{3}}. You multiply the exponent by the constant. Thus, \displaystyle 3\cdot \frac{1}{3}=1. That's what happened to the 3. Then, subtract 1 from the exponent as you done. Getting, \displaystyle...
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    The limit of (1-(2/x))^x as x approaches infinity

    If you're still interested, here is a graph of your function right of the y-axis. I set the viewing window parameters for x from 0 to 200 and for y from 0 to 0.2 Notice that e^{-2} is a horizontal asymptote?. See how your function approaches it as x gets larger and larger. Hence the limit of...
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    The limit of (1-(2/x))^x as x approaches infinity

    This limit is related to the ever famous \displaystyle \lim_{x\to \infty}\left(1+\frac{1}{x}\right)^{x}=e But, instead of having a 1, we have a -2. Which gives e^{-2}=\frac{1}{e^{2}} But, since you are required to use calculator methods, I assume they mean entering in larger and larger x...
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    Generating Functions

    Using the binomial series, note that: \displaystyle\frac{x}{(1-x)^{3}}=\sum_{k=0}^{\infty}\binom{-3}{k}(-1)^{k}x^{k+1} This can also be written as: Note that \displaystyle \frac{x}{(1-x)^{n}}=x+\binom{1+n-1}{1}x^{2}+\binom{2+n-1}{2}x^{3}+\cdot\cdot\cdot + \binom{r+n-1}{r}x^{r+1}+\cdot\cdot\cdot
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