# Thread: Equation of tangent lines, find intersection

1. ## Equation of tangent lines, find intersection

I am having a bit of an issue with this problem.

f(x) = e^(-cx) and g(x) = -e^(-cx) where c is a positive real number.

"Find equations of the tangent lines to the curves f(x) and g(x) at x = 0. Then show they intersect on the x-axis at the point (1/c, 0)."

Okay so for the tangent lines they want me to take the derivatives of f(x) and g(x) right?

Well here they are:
f'(x) = -ce^(-cx)
g'(x) = ce^(-cx)

f'(0) = -c
g'(0) = c

But these are constants! Just flat horizontal lines that won't intersect because they're parallel. So why is the problem asking me to prove they intersect at (1/c, 0)? I know I did something wrong, but what?

Any hints are much appreciated.

2. Originally Posted by Gryllis
I am having a bit of an issue with this problem.

f(x) = e^(-cx) and g(x) = -e^(-cx) where c is a positive real number.

"Find equations of the tangent lines to the curves f(x) and g(x) at x = 0. Then show they intersect on the x-axis at the point (1/c, 0)."

Okay so for the tangent lines they want me to take the derivatives of f(x) and g(x) right?

Well here they are:
f'(x) = -ce^(-cx)
g'(x) = ce^(-cx)

f'(0) = -c
g'(0) = c

But these are constants! Just flat horizontal lines that won't intersect because they're parallel. they ain't parallel and also they are NOT horizontal. Horizontal lines have slope=0. Here the slopes of the two lines are the derivatives you have calculated. observe that the tangent lines considered have DIFFERENT slopes. one has slope c and other has -c. Of course slope of a line will be constant. Don't get confused over that
So why is the problem asking me to prove they intersect at (1/c, 0)? I know I did something wrong, but what?

Any hints are much appreciated.
did this help?

3. Originally Posted by Gryllis
f(x) = e^(-cx) and g(x) = -e^(-cx) where c is a positive real number.
"Find equations of the tangent lines to the curves f(x) and g(x) at x = 0. Then show they intersect on the x-axis at the point (1/c, 0)."
f'(x) = -ce^(-cx) & f'(0) = -c
g'(x) = ce^(-cx) & g'(0) = c
So why is the problem asking me to prove they intersect at (1/c, 0)? I know I did something wrong, but what?
Note that $(0,1)\in f~\&~(0,-1)\in g$.
So the tangent lines are:
$y_f=f'(0)x+f(0)=-cx+1$ and
$y_g=g'(0)x+g(0)=cx-1$.

4. Ooooh. How did I not see that...

Thank-you!