Find the minimum distance between the curves y=e^x and y=lnx. Hint: Use the fact that e^x and lnx are inverse relationships. I have no idea where to start. Thanks!
calculus - Reiny, Monday, January 18, 2016 at 9:02pm
to get you going:
let P(a , e^a) and Q(b, lnb) be the closest points on their respective graphs
since they are inverses the line PQ must be perpendicular to y = x
but y = x has a slope of 1, so the
sope of PQ = -1
See what you can do with that.
I tried to start with that, and using graphs, I got (0,1) on e^x and (1,0) on lnx, with the distance of 2sqrt2. But I don't know how to do this algebraically.
for y=e^x, the slope of the normal at x=a is -e^-a
for y=lnx, the slope of the normal at x=b is -b.
So, we want to find a and b such that the normals through (a,e^a) and (b,lnb) are the same line. That is,
y-e^a = -e^-a(x-a)
and
y-lnb = -b(x-b)
eliminating y, we then want
-e^-a*x + ae^-a+e^a = -bx+b^2+lnb
For these two polynomials to be identical, we need
-e^-a = -b
ae^-a+e^a = b^2+lnb
The first equation says that b = e^-a. Using that in the 2nd equation, we have
ae^-a + e^a = e^-2a - a
It is easy to see that a solution to this is a=0. That means that the two closest points on the curves are
(0,1) and (1,0)
and the minimum distance is thus √2.
To find the minimum distance between the curves y = e^x and y = ln(x) algebraically, you can follow these steps:
1. Start by assuming that the point P(a, e^a) lies on the curve y = e^x, and the point Q(b, ln(b)) lies on the curve y = ln(x).
2. Since the curves y = e^x and y = ln(x) are inverses of each other, the line segment PQ joining P and Q must be perpendicular to the line y = x. This means the product of the slopes of PQ and y = x is -1.
3. The slope of the line segment PQ, denoted as m, is given by:
m = (ln(b) - e^a) / (b - a)
4. The slope of the line y = x is always 1.
5. So, equating the product of the slopes to -1, we have:
1 * ((ln(b) - e^a) / (b - a)) = -1
6. Simplifying the equation:
ln(b) - e^a = -(b - a)
7. Rearranging the terms:
ln(b) + b = e^a + a
8. We need to minimize the distance between the curves, which is the length of the line segment PQ. The distance formula is given by:
d = √((b - a)^2 + (ln(b) - e^a)^2)
9. To find the minimum distance, we need to minimize the expression for d. We can do this by finding the values of a and b that satisfy the equation ln(b) + b = e^a + a.
10. Unfortunately, there is no algebraic way to solve this equation analytically. However, you can use numerical methods like graphing or iteration to approximate the values of a and b that satisfy the equation.
11. Once you have the values of a and b, substitute them back into the distance formula to find the minimum distance between the curves.
Please note that finding the exact minimum distance using an algebraic approach is not straightforward due to the transcendental nature of the equations involved. Using numerical methods is a more practical way to approximate the minimum distance.
To find the minimum distance between the curves y = e^x and y = ln(x), you can follow these steps:
1. Let P(a, e^a) be a point on the curve y = e^x, and let Q(b, ln(b)) be a point on the curve y = ln(x).
2. Since e^x and ln(x) are inverse functions, the line segment PQ connecting P and Q must be perpendicular to the line y = x. Thus, the slope of the line segment PQ should be -1.
3. Use the slope formula to find the slope of the line segment PQ:
slope of PQ = (ln(b) - e^a) / (b - a)
4. Since the line segment PQ is perpendicular to the line y = x, the slope of PQ should be -1. Therefore, we have:
(ln(b) - e^a) / (b - a) = -1
5. Rearrange the equation to get:
ln(b) - e^a = - (b - a)
6. Now, we need to find the values of a and b that minimize the distance between P and Q. This can be done by minimizing the square of the distance formula:
distance^2 = (b - a)^2 + (ln(b) - e^a)^2
7. Differentiate the distance^2 formula with respect to both a and b, and set the derivatives equal to zero to find the values of a and b that minimize the distance.
8. After finding the values of a and b, substitute them back into the distance formula to calculate the minimum distance.
Please note that algebraically solving the equations may involve some complex calculations.