Find a formula for a function of the form y=bxe−ax with local maximum at (0.5,0.3679). y=
To find a formula for a function of the form y = bxe^(-ax) with a local maximum at (0.5, 0.3679), we need to determine the values of a and b.
To start, let's differentiate the function y = bxe^(-ax) with respect to x to find its derivative:
dy/dx = b * (1 - ax) * e^(-ax)
To find the local maximum, we need to set the derivative equal to 0 and solve for x:
0 = b * (1 - ax) * e^(-ax)
This equation reduces to:
1 - ax = 0
Solving for x:
x = 1/a
Now, substitute this value of x into the original function to find y:
y = b * (1/a) * e^(-a * (1/a))
Simplifying further:
y = b * e^(-1)
Given that the local maximum occurs at (0.5, 0.3679), we can substitute these values into the equation to solve for b:
0.3679 = b * e^(-1)
Dividing both sides by e^(-1):
0.3679 / e^(-1) = b
Calculating the left side of the equation:
0.3679 ≈ 0.3679(2.7183) ≈ 1.0007
Therefore, b ≈ 1.0007.
Finally, we can substitute the values of a and b back into the original equation to get the formula for the function:
y = 1.0007 * x * e^(-2x)
I don't understand y=bxe−ax
which are variables and which are parameters ?
Is e supposed to be Euler's number?