Question #41757

Let a variable plane $P(t)$ have intercepts $a(t), b(t),$ and $c(t)$ on the $x, y,$ and $z$ axes respectively, for $t \geq 0$, defined by the following integral equations: $a(t) = 1 + \int_0^t \frac{e^{tx}}{1+x^2} dx$ $b(t) = 2 + \int_0^t \cos(t-x) e^{-x^2} dx$ $c(t) = 2 + \int_0^{2t} e^{x^2-t^2} dx$ Let $\Delta(t)$ be the area of the triangular section formed by the intersection of the plane $P(t)$ with the coordinate planes. What is the rate of change of $\Delta(t)$ with respect to $t$ at $t=0$?