Question

Describe the slope of the tangent line to the curve defined by x 2 + xy - y 2 = 5 when x = 2.
positive
negative
zero
undefined

Answer 1

find dy\dx and plug in x=2

Answer 2

do implicit differentiation and find y'. put x = 2.

Answer 3

2x+ dy/dx (x*y)dy/dx 2y=0

Answer 4

2x+dy/dx (x*y) -2y=0

Answer 5

I prefer using y' when doing implicit differentiation so it is less clutter than using dy/dx.

Answer 6

did I do that right?

Answer 7

x^2 + xy - y^2 = 5
2x + xy' + y - 2yy' = 0
put x = 2 right away to get rid of the x terms.
4 + 2y' + y - 2yy' = 0
4 + y + y'(2 - 2y) = 0
y' = -(4 + y)/(2 - 2y)
We now need to find y when x = 2.
Put x = 2 in the original equation and solve for y.
Then put those y values in y' and find the slope.

Answer 8

4+2y-y^2=5?

Answer 9

yes.

Answer 10

okay is that all I do?

Answer 11

How do I know if it is positive or negative?

Answer 12

solve the quadratic for y first.

Answer 13

how do I do that?

Answer 14

not familiar with solving quadratic equations?

Answer 15

oh sorry I know how to do that. I am really tired and I am trying to study for a test

Answer 16

I studied like 7 1/2 hours yesterday and I just have everything confused. Sorry about that.

Answer 17

You got 4+2y-y^2=5
4 + 2y - y^2 - 5 = 0
-1 + 2y - y^2 = 0
multiply by -1
y^2 - 2y + 1 = 0
factor
(y - 1)^2 = 0
y = 1. So when x = 2, y = 1. Put y = 1 in y'

y' = -(4 + y)/(2 - 2y)
y' = -(4 + 1) / (2 - 2) = -5 / 0 (division by zero)
So slope is UNDEFINED at x = 2.
That means the tangent to the curve at x = 2 is a vertical line.

Answer 18

Do you use the same rules for any problem like this?

Answer 19

or steps.

Answer 20

If you are given a function or an equation and you are asked to find the slope at a point you will have to find y' or dy/dx and evaluate y' at the given x. But if you do implicit differentiation the y will all be mixed up in the y' equation and so you will have to find the y value at the given x which you can do by putting x in the original equation and solving for x. Then you put that y in y' and you will have your slope.

Answer 21

Thanks so much for helping me!

Answer 22

you are very welcome.