# Equation of tangent lines, find intersection

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• May 1st 2011, 07:50 AM
Gryllis
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.
• May 1st 2011, 08:06 AM
abhishekkgp
Quote:

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?
• May 1st 2011, 08:07 AM
Plato
Quote:

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 $\displaystyle (0,1)\in f~\&~(0,-1)\in g$.
So the tangent lines are:
$\displaystyle y_f=f'(0)x+f(0)=-cx+1$ and
$\displaystyle y_g=g'(0)x+g(0)=cx-1$.
• May 1st 2011, 08:07 AM
Gryllis
Ooooh. How did I not see that...

Thank-you! :)