Calculus Level 5

Suppose $f(x)$ and $g(x)$ are 2 continuous functions defined for $0 \leq x \leq 1$.

$\displaystyle f(x) = \int_0^1 e^{x + t} . f(t) dt$ and $\displaystyle g(x) = \int_0^1 e^{x + t} . g(t) dt + x$

$A$. The value of $f(1)$ is

1) $0$

2) $1$

3) $e^{-1}$

4) $e$

$B$. The value of $g(0)$ - $f(0)$ is

5) $\frac{2}{3 - e^{2}}$

6) $\frac{2}{ e^{2} - 2}$

7) $\frac{2}{ e^{2} - 1}$

8) $0$

$C$. The value of $\frac{g(o)}{g(2)}$ is

9) $0$

10) $\frac{1}{3}$

11) $\frac{1}{ e^{2}}$

12) $\frac{2}{ e^{2}}$