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I don't understand how they got to the f'(1) equation on the second part of the solution. Oh and on the first part of the solution, f'(x) is there because of implicit differentiation, correct?

Thanks again.

the f'(1) equation comes directly from the first equation you are given for f'(x), with the substitution x=1

when d/dx was applied to both sides, that was using implicit differentiation. as a result, you differentiated [f(x)]2 which needed the chain rule

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the f'(1) equation comes directly from the first equation you are given for f'(x), with the substitution x=1

when d/dx was applied to both sides, that was using implicit differentiation. as a result, you differentiated [f(x)]2 which needed the chain rule

Yeah I noticed that when I took a second look at it in the morning, doh! Thanks again though.

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Multivariable Calculus, chapter of triple integrals

Help please! :(

6nxrhw.png

you have two spheres in space, a small one on top of a large one where they overlap slightly (the smaller one sinks into the larger). the volume of each sphere alone is easy, but the volume of the entire snowman is the sum of each sphere minus the volume that they overlap.

if you are using Cartesian coordinates, then what you want is to find the equation of the circle where the two spheres intersect (this is key). this will tell you the numbers for the bounds of integration in either x or y, and then the function that forms the bounds for the other variable. Once you have that, your bounds for z are easy, just integrate from the bottom of the "head" to the top of the "body".

you can also switch to spherical or cylindrical coordinates (infact cylindrical might be best, theta does a full rotation, r goes from zero to the circle where the spheres intersect and z goes from the bottom to the top as before).

Try that first, hopefully that's enough, but if you need the solution:

the circle is x2+y2=3, so your x bounds would be +/- sqrt(3), y bounds are +/- sqrt(3-x2), and the bounds for z would be 4-sqrt(4-x2-y2) and sqrt(12-x2-y2)

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Can anyone give a good explanation on calculating limits?

well sure, if you have about a month to talk about all of the different types of limits and the various different limit laws at your disposal. I'll say this, the most important type of limits in Calculus are difference quotients: limh->0 (f(x+h)-f(x))/h. The trick here is that you want to just be able to plug in the number zero for h, but doing so now would cause this function to become 0/0, which is undefined, so you must first do some simplification.

For example, lets find limh->0 (f(x+h)-f(x))/h for the function f(x)=x2:

limh->0 (f(x+h)-f(x))/h = limh->0 ((x+h)2-x2)/h

=limh->0 (x2+2xh+h2-x2)/h

=limh->0 (2xh+h2)/h

=limh->0 (2x+h)

=2x+0

=2x

What is happening here is that the original expression ((x+h)2-x2)/h is a function of h that is undefined precisely where we are interested at looking: h=0. We get around that by saying "well if h=/=0, then this function looks exactly like 2x+h, and for this function at h=0 we get the expression 2x"

example: lim x->2 (x^(1/2)-2)/(x-4)

This function however is defined and continuous at the point x=2, so in this case we just plug that number in:

limx->2 (sqrt(x)-2)/(x-4) = (sqrt(2)-2)/(2-4)

=(sqrt(2)-2)/(-2)

=1-1/sqrt(2)

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