Mark's Calc III - Individual question feed

Partial Derivatives

Tiffany — Thu, 17 Jul 2014 11:15:53 -0500

So I'm working on the exam review, and I'm looking at question 3 part a. I run into issues on my partial derivatives when they include e. I ended up with $\frac {\delta f}{\delta x} = e^{xy} + xye^{xy}$ and $\frac {\delta f}{\delta y} = x^2e^{xy}$ I don't feel like thats right though, and I can't get mathematica or wolfram alpha to give me answers. Any help would be greatly appreciated!

Double Integrals

Tiffany — Thu, 17 Jul 2014 10:21:11 -0500

So with the problems he put on the board at the end of class today, I worked out the first double integral and was just hoping someone could verify if I'm doing it right. So he gave us the domain : ![image description](/upfiles/calc3.askbot.com/14056097985795571.gif) and the double integral as: $$\int\int x^2y\delta A$$ for which I came up with the integrals being: $$\int _0 ^2 \int _0 ^{4-x^2} x^2y \delta y \delta x$$ $$=\int _0 ^2 \frac 12x^2y^2\delta x \mid _0 ^{4-x^2}$$ $$=\int _0 ^2 \frac 12 x^2 (4-x^2)^2 \delta x$$ $$= \int _0 ^2 \frac 12 x^2(16-8x^2+x^4)\delta x$$ $$=\int _0 ^2 8x^2 -4x^4 + \frac 12 x^6 \delta x$$ $$=\frac 83 x^3 - \frac 45 x^5 + \frac {1}{14} x^7 \mid _0 ^2$$ $$= \frac 83 (8) - \frac 45(32) + \frac{1}{14}(128)$$ $$= \frac{64}{3} -\frac{128}{5} +\frac{128}{14}$$ This method seems to make sense to me(hoping that my terms of integration are correct), so I'm just hoping that someone else follows this too, or you could explain and easier way to do it. Thanks!

Is this sound reasoning?

SpaceManSpiff — Wed, 16 Jul 2014 21:23:15 -0500

For problem 15 part b. The question asks if $$\int_{0}^{1} \int_{0}^{1} \sin(xy)dxdy$$ is positive or negative. It seems like since both $x$ and $y$ are contained between zero and one that $xy$ will always be between zero and one. This means that $\sin(f(x,y))$ will only have a positive argument in this section of the plot and therefore will only yield positive z values which means the integral is positive. That seems reasonable to me but I'm wrong more often than I'm right when I try to reason this way and also the question just feels like a trick. What do you guys think?

Exam 2 Review Sheet

Anonymous — Wed, 16 Jul 2014 07:09:35 -0500

Number 4, part c, of the exam 2 review sheet asks: Let $f(x,y)=xy^3$ (c) From the point $(2,1)$, is there any direction $u$ so that $D_uf(2,1)=10$? So far I have calculated the gradient to be $$\bigtriangledown f(2,1)=<8,6>$$ The idea I had was to create a dot product of $\bigtriangledown f(2,1)$ and a general $u$ vector, and set it equal to 10. The trouble I'm having is figuring out how to solve the two unknowns because I only have one equation.

Lagrange multipliers and their system of equations

asmith14 — Wed, 16 Jul 2014 18:41:12 -0500

On number 9 from the review sheet the question asks to ﬁnd the extremes of $f(x, y) = 2x + 4y$ subject to $x^2+y^2=20$. When I was setting up my system of equations I got $$2=2\lambda x$$ $$4=2\lambda y$$ $$x^2+y^2=20$$ Then when I set them to 0 I got $$2-2\lambda x=0$$ $$4-2\lambda y=0$$ and when you simplify you are not able to pull out a variable $$2(1-\lambda x)=0$$ $$2(2-\lambda y)=0$$ My question is what do you do if this happens because the examples in my notes just have it where you are able to pull out a variable? *Comment*: Good question!!