Posted by Anonymous on Wednesday, February 18, 2009 at 11:24am.
1. Sketch the curves y=e^x and y=e^2x, using the same axes. The line y=4 intersects the first curve at A and the second curve at B. Calculate the length AB to two decimal places.
2. Find the coordinates of the turning point on the curve y=2e^3x+8e^3x and determine the nature of this turning point.
For 1, I have sketched the curves, but i'm not sure how to find the length. I thought I could do y=y to find the x coordinates (and use those as my limits) for each but I don't know how to find them out with e^x.
For 2, I differentiated it to get y=6e^3x24e^3x and don't know what to do after that.
Please help. Thanks in advance for any help!

Maths  bobpursley, Wednesday, February 18, 2009 at 12:25pm
2. find each x by
4=e^x
ln 4= x so the point is ln4,4
and the other point is
ln2=2x
x=ln4/2 and the boint B is 1/2 ln4,4
Use the distance formula to find the length.

MATH  drwls, Wednesday, February 18, 2009 at 12:27pm
1. Since the y coordinates of A and B are the same, the distance between the points will be the difference in the X values of those two points.
2. The derivative is
y' = 6e^3x 24e^3x
When that equals zero,
e^3x = 4e^3x
e^6x = 4
6x = ln 4 = 1.3863
x = 0.23105 is an extreme point of the function. The value of y there is
2 e^0.693 + 8 e^0.693 = 2*2 + 8/(1/2) = 20
Take the second derivative of y(x) to see if (0.23105,20) is a minumum or maximum. It is going to be positive. You also see that, since y becomes infinite as x gets much larger, it must be a minimum.
y''(x) =
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