Finding a point in common

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October 18th 2007, 12:22 PM
polymerase

Finding a point in common

If the two curves $y=e^{2x}$ and $y=k\sqrt{x}$ (where k is a constant) have exactly one point in common, what much be the value of k?
October 18th 2007, 02:17 PM
angel.white

It looks like
$e^{2x}=k\sqrt{x}$

which means
$\frac{e^{2x}}{\sqrt{x}}=k$

I think you would need to know the point where the two equations intersect to determine what the actual value of k is.
October 18th 2007, 07:15 PM
polymerase

Quote:

Originally Posted by angel.white

It looks like
$e^{2x}=k\sqrt{x}$

which means
$\frac{e^{2x}}{\sqrt{x}}=k$

I think you would need to know the point where the two equations intersect to determine what the actual value of k is.

that is obvious info.....trust me this is not an easy question, my professor is always puts on one really hard question n this is one of them
October 18th 2007, 08:45 PM
Jhevon

Quote:

Originally Posted by polymerase

If the two curves $y=e^{2x}$ and $y=k\sqrt{x}$ (where k is a constant) have exactly one point in common, what much be the value of k?

$k = 2 \sqrt{e}$ is the answer

Hint: Let one of the functions be $y_1$ and the other be $y_2$

we must fulfill two conditions.

(1) we must have $y_1 = y_2$

(2) we must have $y_1' = y_2'$

the first condition is obvious, but can you tell me why we need the second? after that, can you come up with the solution?
October 18th 2007, 10:03 PM
angel.white

Quote:

Originally Posted by Jhevon

we must fulfill two conditions.

(1) we must have $y_1 = y_2$

(2) we must have $y_1' = y_2'$

the first condition is obvious, but can you tell me why we need the second?

I can't, and I really want to know, it may be a misconception on my part, but when I was thinking about it, I figured two different shaped curves could touch the same point, giving each a different tangent line. I figured if they were circles, they would have the same tangents, but since they weren't, they wouldn't.

If I had to guess at an answer, I'd say it's because they don't cross each other, if they could cross each other they could clearly touch the same point with different tangent lines. But, thats just a guess.

Also, how would you solve for x at this point? I can see that the point would be $(\frac{1}{4}, \sqrt{e})$ But I don't know how to solve for it without just being able to "see" the answer.
October 19th 2007, 04:42 AM
topsquark

Quote:

Originally Posted by angel.white

I can't, and I really want to know, it may be a misconception on my part, but when I was thinking about it, I figured two different shaped curves could touch the same point, giving each a different tangent line. I figured if they were circles, they would have the same tangents, but since they weren't, they wouldn't.

If I had to guess at an answer, I'd say it's because they don't cross each other, if they could cross each other they could clearly touch the same point with different tangent lines. But, thats just a guess.

Also, how would you solve for x at this point? I can see that the point would be $(\frac{1}{4}, \sqrt{e})$ But I don't know how to solve for it without just being able to "see" the answer.

The answer is that you typically don't. This one happened to have a nice solution, but if you can't find one, the best you can do is a numeric approximation.

-Dan