Question 1

In this question, you will be investigating the family of functions of the form $f(x) = x^{n}e^{-x}$.

Consider the family of functions $f_{n}(x) = x^{n}e^{-x}$, where $x \geq 0$ and $n \in \mathbb{Z}^{+}$.

When $n = 1$, the function $f_{1}(x) = xe^{-x}$ where $x \geq 0$.

Topic – SL:2.6

(a) Sketch the graph of $y = f_{1}(x)$, stating the coordinates of the local maximum point.

Topic – AHL:5.15

(b) Show that the area of the region bounded by the graph $y = f_{1}(x)$, the x-axis and the line $x = b$, where $b > 0$, is given by $\frac{e^{b}-b-1}{e^{b}}$.

You may assume that the total area, $A_{n}$, of the region between the graph $y = f_{n}(x)$ and the x-axis can be written as $A_{n} = \int_{0}^{\infty}f_{n}(x)dx$ and is given by $\lim_{b \rightarrow \infty} \int_{0}^{b}f_{n}(x)dx$.

Topic – AHL:5.17

(c) (i) Use l’Hôpital’s rule to find $\lim_{b \rightarrow \infty} \frac{e^{b}-b-1}{e^{b}}$. You may assume that the condition for applying l’Hôpital’s rule has been met.

Topic – AHL:5.15

(ii) Hence write down the value of $A_{1}$.

You are given that $A_{2} = 2$ and $A_{3} = 6$.

(d) Use your graphic display calculator, and an appropriate value for the upper limit, to determine the value of

Topic – SL:5.14

(i) $A_{4}$;

Topic – SL:5.14

(ii) $A_{5}$.

Topic – AHL:1.3

(e) Suggest an expression for $A_{n}$ in terms of $n$, where $n \in \mathbb{Z}^{+}$.

Topic – AHL:1.7

(f) Use mathematical induction to prove your conjecture from part (e). You may assume that, for any value of $m$, $\lim_{x \rightarrow \infty} x^{m}e^{-x} = 0$.