If the y-coordinate at each point of inflection on the curve y=e^(-x^2) is 1/e^a, then a = ?
y = e^-x^2
y" = (4x^2-2)e^-x^2
y"=0 when x = ±√2
y(±√2) = e^-2 = 1/e^2 = 1/e^a
Looks like a=2
Steve is wrong. The second derivative isn't what Steve said was
To find the value of a, we need to first determine the points of inflection on the curve y = e^(-x^2) and then find the corresponding y-coordinate at each of those points.
To find the points of inflection, we need to find the second derivative of y with respect to x and set it equal to zero.
Let's start by finding the first derivative of y:
y = e^(-x^2)
dy/dx = -2x * e^(-x^2) [using the chain rule]
Now, let's find the second derivative:
d^2y/dx^2 = (-2) * e^(-x^2) - 2x * (-2x * e^(-x^2))
= -2e^(-x^2) + 4x^2 * e^(-x^2)
= e^(-x^2) (4x^2 - 2)
Setting the second derivative equal to zero:
0 = e^(-x^2) (4x^2 - 2)
Since e^(-x^2) is always positive, we can ignore it and focus on solving the quadratic equation:
4x^2 - 2 = 0
Solving the equation, we get:
4x^2 = 2
x^2 = 1/2
x = ±√(1/2)
x = ±1/√2
x = ±√2/2
Now, let's find the corresponding y-coordinate at each point of inflection:
For x = √2/2:
y = e^(-(√2/2)^2)
= e^(-1/2)
= 1/e^(1/2)
For x = -√2/2:
y = e^(-(-√2/2)^2)
= e^(-1/2)
= 1/e^(1/2)
Therefore, the y-coordinate at each point of inflection is 1/e^(1/2).
So, a = 1/2.
Hence, a = 1/2.