Calculus Application Problems: Practice Here

Laplace transforms A powerful tool in solving problems in engineering and physics is the Laplace transform. Given a function f(t), the Laplace transform is a new function F(s) defined by
$F(s)={\int }_0^{\mathrm{\infty }}{e}^{-st}f(t)dt$
where we assume s is a positive real number. For example, to find the Laplace transform of $f(t)=e^{-t}$, the following improper integral is evaluated using integration by parts:
$F(s)={\int }_0^{\mathrm{\infty }}{e}^{-st}{e}^{-t}dt={\int }_0^{\mathrm{\infty }}{e}^{-(s+1)t}dt=\frac{1}{(s+1)}$
Verify the following Laplace transforms, where u is a real number.
$f(t)=1\to F(s)=\frac{1}{s}$

The type of conic sections for the nondegenerate equations given below.

a) ${\displaystyle 6x^2+3x+10y=10y^2+8}$

b) ${\displaystyle 3x^2+18xy=5x+2y+9}$

c) ${\displaystyle 4x^2+8x-5=-y^2+6y+3}$