What is f ''(x) and f ''''(x) of f(x)= e^-x^2
f"(x) = -e^-x^2(2x)
F""(x)= -2e^-x^2 + 2xe^x^2
and also what is
integral [0 to 2] f(x) dx
To calculate the second derivative, f ''(x), of f(x) = e^(-x^2), we can use the chain rule and the derivative of the exponent function.
First, we find the first derivative, f'(x):
f'(x) = d/dx(e^(-x^2))
To find the derivative of the exponent function, we use the chain rule:
d/dx(e^(-x^2)) = e^(-x^2)(-2x)
Now, to find the second derivative, f ''(x):
f ''(x) = d/dx(f'(x))
Using the chain rule again, we find:
d/dx(e^(-x^2)(-2x)) = -e^(-x^2)(2x) + e^(-x^2)(-2)
So, f ''(x) = -e^(-x^2)(2x).
To find the fourth derivative, f ''''(x), we need to differentiate f''(x) with respect to x:
f ''''(x) = d/dx(f ''(x))
Differentiating f ''(x) gives us:
d/dx(-e^(-x^2)(2x)) = -2e^(-x^2) + 2xe^(-x^2)
So, f ''''(x) = -2e^(-x^2) + 2xe^(-x^2).
Now, let's calculate the integral of f(x) from 0 to 2:
∫[0 to 2] f(x) dx
To integrate f(x) = e^(-x^2), we can use the indefinite integral of the Gaussian function:
∫ e^(-x^2) dx = √π * erf(x) / 2 + C
Where erf(x) is the error function.
To evaluate the definite integral, we substitute the upper and lower limits of integration into the indefinite integral:
∫[0 to 2] e^(-x^2) dx = (√π * erf(2) / 2) - (√π * erf(0) / 2)
We can use a numerical approximation or a specialized software or calculator to approximate the value of the integral.