ATAR Notes: Forum

HSC Stuff => HSC Maths Stuff => HSC Subjects + Help => HSC Mathematics Advanced => Topic started by: Onekeeper on May 25, 2018, 05:33:52 pm

Title: Exponential Functions
Post by: Onekeeper on May 25, 2018, 05:33:52 pm: Hi, Is anyone able to give me a hand solving this question?

Find the stationary point on the curve y=xe^x and determine its nature. Find any points of inflexion and find values of y as x becomes very large or small. Hence sketch the curve.

Thanks in advance!
Title: Re: Exponential Functions
Post by: RuiAce on May 25, 2018, 06:17:11 pm: \begin{align*}y&=xe^x\\ \frac{dy}{dx} &= (1+x)e^x\\ \frac{d^2y}{dx^2} &= (2+x)e^x\end{align*}
$\text{We can infer the stationary point at }x=-1\text{ and prove that}\\ \text{there is an inflexion at }x=-2\text{ the usual way.}$
Then just plug in some large values such as $x = 1000$ onto the calculator and see what you get. (You should find something close to 0 when you plug something like $x=-1000$, which indicates an horizontal asymptote as you go towards the left.) Much of this is very standard and you're expected to know how to do it so if you need further details clarified you should make it clear where you are having trouble.