Question

find the slope of the tangent to the curve of the intersection of the surface 4z= sqrt(25x^2+9y^2-225) and the plane y=1 at the point (3,1,3/4)

Answer 1

If I can get answers to all these questions that would be great

Answer 2

That seems to be a midterm exam.

Answer 3

This doesn't seem very honest.

Answer 4

myininaya this was an old midterm exam. I'm studying for my final. 5 questions from this midterm will be on the exam.

Answer 5

Oh okay. I just wanted to make sure.

Answer 6

@myininaya so can you help me out?

Answer 7

I can probably help with one or two
but I suggest asking one question at a time.

Answer 8

The first question just wants you to find f_x and f_y

Answer 9

\[\text{ to find } f_x \text{ we need to treat y like it is a constant }\]
I will do this
but I will leave finding f_y to you
\[f=(\sin(4xy^2-y))^2 \\ f_x=2(\sin(4xy^2-y))^{2-1}\cos(4x^2-y) (8x-0)\]
when it came to differentiating y I treated it like a constant

Answer 10

but as you see I used chain rule like twice

Answer 11

now the second order partial derivatives are the ones like
f_xx ,f_yy, f_xy, and f_yx

Answer 12

I will find one of those for you for the next function
and leave the other 3 for you

Answer 13

\[f=\cos(xy^2) \\ f_y=(x \cdot 2y)(-\sin(xy^2))=-2xysin(xy^2) \text{ treated x like constant } \\ \text{ \to find } f_{yx} \text{ we need to use product rule } \\ f_{yx}=(-2xy)_xsin(xy^2)+-2xy(\sin(xy^2))_x \\ =-2ysin(xy^2)-2xy \cdot y^2 \cos(xy^2)\]

Answer 14

so notice I have to find first order partial to find second order partial

Answer 15

are there any questions to what I have done?

Answer 16

I could tell you how to do number 3
if you come back...
but I will start you off if you do and I have left
\[\text{ find } f_x \text{ and } f_y \text{ where } f(x,y)=z \\ \text{ then your normal vector would be } (f_x(3,1),f_y(3,1),-1) \\ \text{ your tangent plane should be } \\ f_x(3,1) \cdot (x-3)+f_y(3,1) \cdot (x-1)-1(z-\frac{3}{4}) =0 \\ \text{ if I'm not mistaken }\]

Answer 17

4) see if you can get a different path to show a different limit from another path