Interesting Study of a function $y = \exp(- x/14) / 14$ 1)Complete study of the function
-monotony of the function
-concavity
-maximum and minimum
-Zero of the function
-if the function is ascending or descending
-inflection points
2)make the primitive and find the area “under the curve” in a closed “interval” (break)
-Primitive-make the derivative
-Calculate the area analytically and numerically in a closed interval
3)
-average value [0;1])
-expected value
-Variance
-compare values for the exponential distribuition 
 A: More generally, consider the function $f(x;\theta)=(1/\theta)e^{-x/\theta}$, $x \geq 0$, where $\theta > 0$ is fixed, i.e. the density function of the exponential distribution with mean $\theta$. It is strictly decreasing and convex. $
\int {f(x;\theta )dx}  =  - e^{ - x/\theta }  + C$, and $F(x): = \int_0^x {f(u;\theta )du}  = 1 - e^{ - x/\theta } $, for any $x \geq 0$ (distribution function of exponential distribution); in particular $\int_a^b {f(x;\theta )dx}  = F(b) - F(a) = e^{ - a/\theta }  - e^{ - b/\theta } $. The corresponding expectation is $\int_0^\infty  {xf(x;\theta )dx}  = \theta $, and more generally, the corresponding $n$th moment is given, for all $n \in \mathbb{N}$, by
$$
\mu'_n := \int_0^\infty  {x^n f(x;\theta )dx}  = \theta ^n \int_0^\infty  {x^n e^{ - x} dx}  = \theta ^n \Gamma (n + 1) = \theta ^n n!,
$$
where $\Gamma$ denotes the gamma function. In particular, the corresponding variance is given, according to the formula ${\rm Var}(X)={\rm E}(X^2)-{\rm E}^2(X)$, by $\mu'_2 - (\mu'_1)^2 = 2 \theta^2 - \theta^2 = \theta^2$.
