1. ## Multi-variable Calculus Problems

Hello!

I have a few questions that I am having some trouble figuring out,

1) If $\displaystyle w = x^2+y^2+xyz$ where $\displaystyle x=sin(st), y=cos(s+t), z=e^{st}$, use an 'appropriate' version of the chain rule to find dw/ds

2) Finding the directional derivitive of $\displaystyle f(x,y) = \frac{x+y}{e^x-y}$ at (0,2) in the direction of the unit vector u (1,1,1) which makes an angle of $\displaystyle \theta=3\pi/4$ with the positive x-axis

3) If an ant is climbing a hill whose shape is given by $\displaystyle z= f(x,y) = 7 - \frac{(x^2+y^2)}{4}$, and the ant is at $\displaystyle (x,y) = (1,\sqrt{3})$

a) in which direction does the ant preceed to take the steepest route to the top?
b) if it climbs at that direction, at what angle above the horizontal will it be climbing, initially?

---I think this has to do with the gradient of f?---

and 4) Find equations for the tanget plane and normal lines to $\displaystyle e^z = \sqrt{x^2+y^2+z^2}$ at (1,2,0).

I have tried a few different approaches to 1) and to 3) but am not too sure if they are going in the right direction.

If anyone could suggest anything it would be great!

Thanks so much!

2. Originally Posted by matt.qmar
Hello!

I have a few questions that I am having some trouble figuring out,

1) If $\displaystyle w = x^2+y^2+xyz$ where $\displaystyle x=sin(st), y=cos(s+t), z=e^{st}$, use an 'appropriate' version of the chain rule to find dw/ds

2) Finding the directional derivitive of $\displaystyle f(x,y) = \frac{x+y}{e^x-y}$ at (0,2) in the direction of the unit vector u (1,1,1) which makes an angle of $\displaystyle \theta=3\pi/4$ with the positive x-axis

3) If an ant is climbing a hill whose shape is given by $\displaystyle z= f(x,y) = 7 - \frac{(x^2+y^2)}{4}$, and the ant is at $\displaystyle (x,y) = (1,\sqrt{3})$

a) in which direction does the ant preceed to take the steepest route to the top?
b) if it climbs at that direction, at what angle above the horizontal will it be climbing, initially?

---I think this has to do with the gradient of f?---

and 4) Find equations for the tanget plane and normal lines to $\displaystyle e^z = \sqrt{x^2+y^2+z^2}$ at (1,2,0).

I have tried a few different approaches to 1) and to 3) but am not too sure if they are going in the right direction.

If anyone could suggest anything it would be great!

Thanks so much!
for (4)
Let $\displaystyle F(x,y,z) = e^z - \sqrt{x^2+y^2+z^2}$
Now, find $\displaystyle F_x$,$\displaystyle F_y$ and $\displaystyle F_z$ .. etc etc
Its typical problem.