Question

Find all values of x at which the tangent line to the curve y=(2x+5)/(x+2) has a y-intercept of 2..Can anybody help me with this question?

Answer 1

Could I help you?

Answer 2

sure^^

Answer 3

Thank you.

Answer 4

What is the meaning of "the tangent line to the curve"?

Answer 5

the tangent line is correspond to the curve.

Answer 6

The tangent line means the line of the first derivative of the function at a certain point!

Answer 7

Can you get the first derivative of this function y=(2x+5)/(x+2)?

Answer 8

@syaza97

Answer 9

i got f'(x)=-1/(x+2)^2

Answer 10

\[y'=\frac{ -1 }{ (x+2)^2 }\]

Answer 11

yes... i also get that answer..

Answer 12

Yes, I know.
Now what is the meaning of "y-intercept of 2"?

Answer 13

the tangent line has a y-iintercept of 2 which mean it pass through pt(0,2)

Answer 14

Great!

so we have y=2
Could you know how to find x values?

Answer 15

y-intercept =2 means the equation of tangent line will be of the form
x/a+y/2=1, where you have to determine a.

Answer 16

the equation you are talking about is not for a line!

\[y'=\frac{ -1 }{ (x+2)^2 }\]

Answer 17

tangent line will obviously have form of a line and a we have to determine from the given equation

Answer 18

@HolsterEmission any idea?

Answer 19

Nothing to add; seems like you guys are on the right track.

Answer 20

may if i can help

Answer 21

To recap, if nothing else...

The tangent line through any point on the curve where the derivative exists, say at \(x=c\), will have slope \(f'(c)\). This line will have the equation \(y-f(c)=f'(c)(x-c)\).

The derivative at such a point is
\[f'(x)=-\frac{1}{(x+2)^2}\implies f'(c)=-\frac{1}{(c+2)^2}\]You know that this line passes through \((0,2)\), so you have
\[y-2=-\dfrac{1}{(c+2)^2}x\implies y=-\frac{1}{(c+2)^2}x+2\]This line also passes through the point \((c,f(c))=\left(c,\dfrac{2c+5}{c+2}\right)\), so the line must be identical to
\[y-\frac{2c+5}{c+2}=-\frac{1}{(c+2)^2}(x-c)\implies y=-\frac{1}{(c+2)^2}x+\frac{c+(2c+5)(c+2)}{(c+2)^2}\]In other words, you have to have
\[2=\frac{2c^2+10c+10}{(c+2)^2}\]Find this value of \(c\). The number of solutions here tells you how many tangent lines there are with an intercept at \((0,2)\).

Answer 22

i already get the answer which is c=-1

Answer 23

Thank you everyone for your help^^