Recent content by physics604

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    Integration using substitution

    I'm assuming they did something like this? $$\int \frac{-du}{u+1} = -\frac{1}{1}\ln {|u+1|} +C$$
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    Integration using substitution

    How was she able to ln the instead like that? I thought there was a factor of \frac{1}{a} to deal with?
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    Integration using substitution

    1. $$\int \frac{1}{1+e^x}\,dx$$ Homework Equations Substitution The Attempt at a Solution $$u=1+e^x$$ $$du=e^xdx$$ $$\int \frac{1}{u}\frac{1}{e^x}\,du$$ $$\int \frac{1}{u}\frac{1}{u-1}\,du$$ $$\int \frac{1}{u(u-1)}\,du$$ $$= \ln {|u^2-u|} = \ln {|(1+e^x)-(1+e^x)|} = \ln...
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    Integration using various techniques

    Oh, I got it now. Thanks! My integral would be $$\int e^u\,du$$
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    Integration using various techniques

    ∫ex+exdx=∫exeexdx=∫ueudu Do you mean I should do something like this? $$\int e^x\,dx = \int e^xe^{e^x}\,dx = \int ue^u\,e^x dx$$ But the point of my substitution was to turn everything into u's...
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    Integration using various techniques

    $$du=e^xdx$$ How does this work into the calculation?
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    Integration using various techniques

    1. $$\int e^{x+e^x}\,dx$$ Homework Equations Substitution, integration by parts The Attempt at a Solution $$u=e^x$$ $$\int e^{x+e^x}\,dx = \int e^x e^{e^x}\,dx = \int ue^u\,du$$ $$a=u$$ $$da=1du$$ $$dv=e^udu$$ $$v=e^u$$ $$=ue^u-\int e^u\,du = ue^u-e^u$$ $$=e^x e^{e^x}+e^{e^x} =...
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    Calculus, derivatives (curve sketching 2)

    1. If the function f(x)=x3+a2+bx has the local minimum value at \frac{-2}{9}\sqrt{3}, what are the values of and a and b? Homework Equations $$f'(x)=0$$ The Attempt at a Solution I automatically took the derivative, getting $$f'(x)=3x^2+2ax+b$$ However, I have no idea where to go from...
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    Calculus, derivatives (curve sketching)

    1. Find a cubic function f(x)=ax3+bx2+cx+d that has a local maximum value of 3 at x=-2 and a local minimum value of 0 at x=1. Homework Equations $$f'(x)=0$$ The Attempt at a Solution The first thing I did was taking the derivative of f(x). $$f'(x)=3ax^2+2bx+c$$ I know that you can get the...
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    Using limits to find asymptotes.

    Sorry, I meant all values of x have to be greater than 0. I'm taking Calculus AB in school and we don't cover L'Hopital's Rule in that course. My modified function was the alternate method that my teacher taught us. He said that we should divide everything by the largest degree of x so that we...
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    Using limits to find asymptotes.

    1. $$f(x)=x-\frac{1}{6}x^2-\frac{2}{3} lnx$$ Homework Equations limits The Attempt at a Solution I know there is a vertical asymptote at x=0 because all values of x have to be greater than x. The answer says that there is no horizontal asymptote, but I don't know how it...
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    Derivatives, rates of change (triangle)

    I checked the time derivative of x=tany but I still get $$\frac{dx}{dt}=sec^2θ×\frac{dθ}{dt}×y$$ I've also attached a drawing.
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    Derivatives, rates of change (triangle)

    1. A plane flies horizontally at an altitude of 5 km and passes directly over a tracking telescope on the ground. When the angle of elevation is \frac{\pi}{3}, this angle is decreasing at a rate of -\frac{\pi}{3} rad/min. How fast is the plane traveling at that time? Homework Equations...
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    Derivatives, rates of change (cone)

    Sorry that was just a typo. The work should still follow r=h/2.
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