I would like to know if there is a way to sketch the graph of $f(x)=x^2 e^x$ without finding the axis intercepts, asymptotes, and differentiating to find the maxima/minima etc.
I was informed that in trying to sketch $f(x)$ it is helpful to sketch the separate functions first.
So the graph of $e^x$ is
and the graph of $x^2$ is
the graph of $f(x)=x^2 e^x$ is
How could I sketch this graph myself just by inspection (without using a computer or scientific calculator)?
On
I think there's good reason to believe the answer is "no."
Not sure how someone might trump my "no" vote but the answer will have to work within those constraints. I'm not optimistic.
Taking it off the graph you could do something weirder like use a 3D graph where the $y$-axis is $f(y)$ and the $x$-axis is $g(x)$ and $z = xy$. Obviously the scales will hide the graphs in this case, but sometimes this is useful; i.e. that's exactly what a log-scale is. Consider using color. (Also try this on the Mathematica SE?)
I think it's possible to get a pretty good idea of what this looks like intuitively. First off we know that $x^2e^x\ge 0$ and that it hits the point $(0,0)$. We also know that $e^x$ dominates $x^2$, so as $x$ becomes large, the shape of the curve will pretty much look like $e^x$, but for smaller values of $x$, the function will grow a bit faster. The horizontal asymptote at $y=0$ of $e^x$ will also remain because of this. In order to form a rough sketch, I would plot a few points when $x=0, 1, 2$ and then interpolate visually from there with a pretty tight horizontal asymptote at $y=0$ as $x\to-\infty$.
We can confirm this intuition by looking at a plot of $e^x$ compared to $x^2e^x$.
For larger $x$ we see the very similar shape.