(See Steps) (a) Find the second derivatives of f(x)=∫_-∞^x^2 / 2 e^x-t^2 / 2 d t and g(x)=∫_-∞^x^2 / 2 e^-(x^2+1) t^2 d t. (b) Derive
Question: (a) Find the second derivatives of \(f(x)=\int_{-\infty}^{x^{2} / 2} e^{x-t^{2} / 2} d t\) and \(g(x)=\int_{-\infty}^{x^{2} / 2} e^{-\left(x^{2}+1\right) t^{2}} d t\).
(b) Derive the solution of the ordinary differential equation
\[\frac{d^{2} y}{d x^{2}}=f(x), \quad x>0, \quad y(0)=0, \quad \frac{d y}{d x}(0)=0\]in the form
\[y(x)=\int_{0}^{x}(x-t) f(t) d t\](c) Find the three second partial derivatives of \(f(x, y)=e^{-\frac{(x-1)^{2}-(y+1)^{2}}{2}}\).
Deliverable: Word Document 
![[All Steps] A gambler plays a game](/images/solutions/MC-solution-library-37622.jpg "[All Steps] A gambler plays a game in which there is a probability")
[All Steps] A gambler plays a game in which there is a probability
![[Solution] Let X and Y have a](/images/solutions/MC-solution-library-37623.jpg "[Solution] Let X and Y have a bivariate normal distribution. Show that")
[Solution] Let X and Y have a bivariate normal distribution. Show that
Solution: The random variable Y has the standard normal density
![[Solution] A hen lays N eggs, where](/images/solutions/MC-solution-library-37625.jpg "[Solution] A hen lays N eggs, where N has the Poisson distribution")
[Solution] A hen lays N eggs, where N has the Poisson distribution
(See Solution) Consider the partial differential equation (partial u)/(partial
Exponential Function Graph maker - MathCracker.com