11.6#5

I am also stuck on part 2 of this one

*******************************************

You're almost right: you just made one little sign error.  You correctly realize that the maximum rate of change is the same as the magnitude of the gradient, and I *think* you correctly realize that the direction of the maximum increase is the unit vector in the direction of the gradient, but it looks like you incorrectly compute that the partial derivative ∂(y/z)/∂z = y/z^2 to get the gradient as
i + j + 5 k instead of the correct ∂(y/z)/∂z = - y/z^2 to get the correct gradient i + j - 5 k

10.3#8

Hi Dr. Taylor,

Will you remind me again what the notation means for E and F?

Please and thank you.

**********************************************

for example f_{xy} means ∂/∂x ∂f/∂y or the derivative with respect to x of the derivative with respect to y of f(x,y). In other words, first take the derivative of f with respect to y, then take the derivative of that with respect to x.

Lecture Notes 2/20/17 through 2/24/17 (complete)

Lecture Notes 2/20/17

Lecture Notes 2/22/17

Lecture Notes 2/24/17

11.2 #3

Hey Dr. Taylor, 

I did ok on WW 11.2 but I was wondering if you have any resources like khan academy videos that I could view to ensure I really know the material. I had a hard time with when y=mx or y=x with the limits and I am not sure what to look up to clarify it.

I'll insert a screenshot of one of the problems for reference. 

Thank you,

*****************************

I don't know of a specific Khan Academy or other online video that deals with this, but if anyone comes up with it I'll post the link.  This specific problem is actually not that bad though, once you apply the idea of substitution: for example if y=mx, then you replace y by mx in the above limit to get rid of the y-variable, from which you get:

11.2#6

Hello professor Taylor, 

I am sending you the picture of my work for problem 6 of 11.2 webwork.  I have simpliefed the expression after using polar coordinates but I don't know what next after there!  Need your help!

Thanks, 

***************************************

What you do after is think this way:  

"Well, now I am taking the limit of a function that is the product of two factors, r and some junk that depends on θ.  The r factor is interesting because it just measures how far (x,y) is from (0,0), and since (x,y)-->(0,0) that means that the limit of r is just zero.  At the same time, the junk involving θ, although it doesn't really have a limit, it is staying bounded above and below: since -1 ≤ cos(θ) ≤ 1 and -1 ≤ sin(θ) ≤ 1 we get   

-10 ≤ cos^3(θ) + 9 sin^3(θ) ≤ 10.

This means that  -10 r ≤ r(cos^3(θ) + 9 sin^3(θ) )≤ 10 r.   Since both 10 r and -10 r have zero as a limit, the limit of my function must be zero too!"

President's day

On Fri, Feb 17, 2017 at 5:28 PM,  ********* wrote:

Hello Mr. Taylor,

I have noticed that Monday is President's Day and wondering if there is still class or not. The syllabus says there is, but I just want to make sure.

Sincerely,

*********

*****************************

Yes, class the same as always on Monday

Lecture Notes 2/10/17 through 2/17/17 completed)

Lecture Notes 2/10/17

Lecture Notes 2/13/17

Lecture Notes 2/15/17

Lecture Notes 2/17/17