Question

Find equations of the line tangent to the curve

y = (x-1) / (x+1)
that are parallel to the line x-2y=2

Answer 1

First thing to do is find the derivative of the curve. Do you know how to do that?

Answer 2

second thing to do is to set the result equal to
\[\frac{1}{2}\] because that is the slope of your given line

Answer 3

idk how 2 find teh derivative

Answer 4

sorry tahts wat confuses me

Answer 5

ok no problem. i will show you

Answer 6

Have you heard of the Quotient Rule??

Answer 7

yes: f'g - fg' /g^2

Answer 8

Almost

Answer 9

\[\frac{g(x)(f'(x)) - f(x)(g'(x))}{g(x)^2}\]

Answer 10

so if you plug in the numbers you get \[\frac{(x+1)(1) - (x-1)(1)}{(x+1)^2}\]

Answer 11

Do you see how i got that?

Answer 12

yes

Answer 13

good

Answer 14

Distribute the negative sign and you get \[\frac{x + 1 - x + 1}{(x+1)^2}\]

Answer 15

dont we need to find the derivative of each of the f and g expressions?

Answer 16

Notice the x's cancel and the ones add up so the derivative is \[\frac{2}{(x+1)^2}\]

Answer 17

I found the derivatives of f and g when i plugged them into the Quotient rule up at the top. The derivative of (x+1) is 1 and the derivative of (x-1) is also 1.

Answer 18

The quotient rule says if you have f(x)/g(x).. first put (g(x))^2 in the denominator.. in the numerator keep g(x) the same and multiply by the derivative of f(x) minus f(x) times the derivative of g(x)

Answer 19

oh ok i see

Answer 20

I apologize, i should of split up the two steps. Do you understand what i did?

Answer 21

yes..then u simply substituted f and g and their derivatives into the equation

Answer 22

right and then simplified.

Answer 23

so now how do we find teh equation of the tangent line

Answer 24

Ok it says it wants it parallel to the line \[x - 2y = 2\] You need to put this in standard form which is y=mx + b. So can you do that?

Answer 25

y = 1/2 x +1

Answer 26

There you go.. Now a line is said to be parallel when two lines have the same slope. So the slope of the tangent line will also be 1/2

Answer 27

so do we pick any b intercept

Answer 28

That is the part that im not sure.. I believe you plug in 1/2 into the derivative and that will be your y intercept.

Answer 29

or set it equal to 1/2.. im not exactly sure.. let me see if i can figure it out

Answer 30

well wat is the equation of the line that is tangent to the original equation...shudn't we find taht out first

Answer 31

Just a sec.. i think i figured it out

Answer 32

no problem

Answer 33

Then again maybe not.. Maybe satelite can help you with the last part.. After you find a point on the graph you use the point-slope formula to find the equation.. but im tired and cant think of how to do that.

Answer 34

its okay..thank you for your help

Answer 35

but can u explain one last thing

Answer 36

Whats that?

Answer 37

how do i find teh equation of the lien tangent to the orig. equation after finding the derivative is equal to 2/ (x+1)^2

Answer 38

You find the slope which in your case is 1/2 and then you find a y value and then use the point-slope formula.

Answer 39

is x = 1? since i equated 2/(x+1)^2 = 1/2