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!
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.
To find the minimum distance between the curves y = e^x and y = ln(x), we can start by considering a point (a, e^a) on the curve y = e^x, where 'a' is a real number.
Since e^x and ln(x) are inverse functions, if a point (a, e^a) lies on the curve y = e^x, then the point (e^a, a) lies on the curve y = ln(x).
Now, let's find the distance between the points (a, e^a) and (e^a, a) using the distance formula:
Distance = √((x2 - x1)^2 + (y2 - y1)^2)
Plugging in the values, we get:
Distance = √((e^a - a)^2 + (a - e^a)^2)
We can simplify this expression by squaring both sides:
Distance^2 = (e^a - a)^2 + (a - e^a)^2
Next, let's simplify the expression inside the parentheses:
Distance^2 = e^(2a) - 2ae^a + a^2 + a^2 - 2ae^a + (e^a)^2
Combining like terms, we have:
Distance^2 = e^(2a) + 2a^2 - 4ae^a
To find the minimum distance, we need to find the minimum value of the expression Distance^2. To do this, we can take the derivative of Distance^2 with respect to 'a' and set it equal to zero.
Differentiating the expression Distance^2 with respect to 'a', we get:
d(Distance^2)/da = 2e^(2a) + 4a - 4e^a
Setting this derivative equal to zero, we have:
2e^(2a) + 4a - 4e^a = 0
Simplifying, we get:
2e^(2a) - 4e^a + 4a = 0
Now, we have an equation that needs to be solved to find the value of 'a' that gives us the minimum distance. Unfortunately, this equation does not have an exact solution that can be found algebraically.
To proceed further, we can use numerical methods such as graphing the equation or using numerical approximation methods like the Newton-Raphson method to find an approximate solution for 'a'. With this value of 'a', we can then calculate the minimum distance between the curves y = e^x and y = ln(x) using the expression Distance.