Question

Calc. help please?

Answer 1

For the curve given by 4x2 + y2 = 48 + 2xy show that dy/dx = (y - 4x)/(y-x)

Answer 2

ok do u know what a dy/dx is

Answer 3

What I've done so far is isolate one y. y = (-4x^2 + 48)/y + 2x

Then I'd differentiate each the numerator and the denominator and the 2x?

Answer 4

don't isolate y

Answer 5

take the derivative using the chain rule, and also the product rule when needed

Answer 6

8x + 2yy' = 2xy' + 2y

y' = (8x - 2y)/(2x - 2y) = (4x - y)/(x - y) = (y - 4x)/(y - x)

(b.) The tangent to the curve is horizontal when its slope is 0.

dy/dx = y' = 0

y - 4x = 0 --> y = 4x

Plugging this back into the curve equation:

4x^2 + 16x^2 = 48 + 2x(4x)

12x^2 = 48

x^2 = 4

x = 2 (Proves that there is a point P with x-coordinate 2 at which the line tangent to the curve at P is horizontal)

Corresponding y-coordinate is 8.

(c.) dy/dx = (y - 4x)/(y - x)

d²y/dx² = [(y - x)*(y' - 4) + (4x - y)(y' - 1)]/(y - x)^2

d²y/dx² = [(y - 4y) + (4x - y)(-3x/(y - x))]/[(y - x)^2

P = (2, 8)

Pick a point to the left, (1,7):

y'(1,7) = (7 - 4)/(7 - 1) = positive

Pick a point to the right, (3,9):

y'(3,9) = (9 - 12)/(9 - 3) = negative

therefore, it's a max.

Answer 7

i have no idea what that answer above is, or where it came from, but it is not the answer to your question

Answer 8

I think he/she's encountered this set of problems before.

Answer 9

it's an example

Answer 10

\[ 4x^2 + y^2 = 48 + 2xy\] is your equation
the derivative of \(4x^2\) is easy that is \(8x\) right?

Answer 11

i obviously can't tell him the answer

Answer 12

So do implicit differentiation and then isolate y'

Answer 13

right

Answer 14

the derivative of \(y^2\) is \(2yy'\) by the chain rule
is that clear?

Answer 15

Yes

Answer 16

ok and the derivative of 48 is 0

Answer 17

Right because it's constant

Answer 18

now how about the derivative of \(xy\)?

Answer 19

that is the only (marginally) tricky part, as it requires the product rule

Answer 20

xy' + y

Answer 21

ok good so you got it

Answer 22

\[8x+2yy'=xy'+y\] solve for \(y'\)

Answer 23

that is straight up algebra

Answer 24

put everything with \(y'\) on one side, everything else on the other, factor out the \(y'\) then divide

Answer 25

Yeah, I guess I was just unclear on the steps a bit. Thanks guys!

Answer 26

yw , hope you got it now

Answer 27

if not i can walk you through it, but it is all algebra from here on in

Answer 28

Yeah I think I've got it.

Answer 29

kk