(a) Graph the function g(x)=e^x-3 x^2 in the viewing...

(a) Graph the function $g(x)=e^x-3 x^2$ in the viewing rectangle $[-1,4]$ by $[-8,8]$. (b) Using the graph in part (a) to estimate slopes, make a rough sketch, by hand, of the graph of $g^{\prime}$. (See Example 2.8.1.) (c) Calculate $g^{\prime}(x)$ and use this expression to graph $g^{\prime}$. Compare with your sketch in part (b).

(a) Graph the function $g(x)=e^x-3 x^2$ in the viewing rectangle $[-1,4]$ by $[-8,8]$.
(b) Using the graph in part (a) to estimate slopes, make a rough sketch, by hand, of the graph of $g^{\prime}$. (See Example 2.8.1.)
(c) Calculate $g^{\prime}(x)$ and use this expression to graph $g^{\prime}$. Compare with your sketch in part (b).

Transcript

00:01 Okay, so we have the function, e of the x minus 3x squared, and we want to graph it on the following window, which is negative 1 to 4 on the x -axis and negative 8 to 8 on the y -axis.

00:11 So here is that graph in green.

00:14 And what we want to do is approximate the derivative of g of x by sketching out its graph first.

00:22 Then we're going to take the derivative and they confirm the graph we just predicted.

00:27 So something we're going to look at right away is if any tangent lines, to this graph g of x are going to be horizontal tangent lines, which means that the derivative will be zero, which means that when we graph it, it's going to cross the x -axis at that point.

00:43 So we can see that there are two local maximas and minimas.

00:49 So we see that there is one here, approximately here.

00:55 That means there'll be a slope of zero at that point.

00:57 And then if we draw another tangent line, we'll see that there'll be a zero point at around here.

01:03 So at these two points, this means if we draw the derivative, let's say in red, that's you have to cross the x -axis at approximately those points.

01:12 Okay? the next thing we want to do is just look at the slopes of these sections that we made with the graph.

01:19 So we can say before this zero slope point, this can be zone one, then zone two here, then zone three right here.

01:29 So if we look at the first zone or chunk first, we can see that this is a positive slope.

01:38 Right is a positive slope approaching a zero slope so this means that our graph will look something like this because again we're it is still a positive slope because we're talking about g prime of x but as to approach that zero because eventually it's going to transition to a negative slope so if we draw in a way that it's a decreasing positive answer to a zero point then will be good there as an approximation.

02:09 And then zone two, we see that this becomes a negative slope.

02:13 So we're going to be below the x -axis.

02:16 We can draw it to look like something like this.

02:20 We're going to kind of comes down.

02:22 Let's draw a little bit better.

02:25 Let's say we have this...

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