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Produce graphs of $f$ that reveal all the importantaspects of the curve. Estimate the intervals of increase anddecrease and intervals of concavity, and use calculus to findthese intervals exactly.$$f(x)=\frac{1}{x^{8}}-\frac{2 \times 10^{8}}{x^{4}}$$

$f^{\prime \prime}(x)<0$ on $(-\infty,-0.0016),(-0.0016,0),(0,0.0016)$ and $(0.0016, \infty)$

Calculus 1 / AB

Chapter 4

APPLICATIONS OF DIFFERENTIATION

Section 4

Curve Sketching

Derivatives

Differentiation

Applications of the Derivative

Oregon State University

Baylor University

University of Michigan - Ann Arbor

Lectures

03:09

In mathematics, precalculu…

31:55

In mathematics, a function…

07:38

Produce graphs of $ f $ th…

10:46

11:23

07:20

09:48

08:58

09:01

09:17

07:48

08:09

02:54

Find (by hand) the interva…

03:08

Find the intervals on whic…

05:12

Use a computer algebra sys…

03:18

03:32

02:43

Graph the function. Estima…

03:45

05:34

02:42

Increasing and decreasing …

08:57

So for this problem we're going to be Drafting the following, its one over X to the 8th minus minus two times 10 to the 8th. Mhm. And then over X to the 4th. So it's clear that we're gonna have to change our graph eyes a little bit. Um We're gonna go let's see this is changing a lot. Um So we feel that there's a lot going on with this graph. Um So what we'll do is because we can't see all of it at the same time, let's just focus on the different portions of it. So if we look at the first derivative graph here, we see that um at these smaller values, we see that the graph is increasing here and decreasing down here increasing here and decreasing down here, we could also change this to be so the y values will change be a chance and see what that does. It doesn't help that time just because it goes so far. Um We see over here, since the graph is decreasing, the photograph is negative as well. And since the graph here is increasing, we have the derivative drafting positive here. Then looking at the second derivative graph, we see it changes things slightly because in this case is both concave down, concave down. So these have to be negative because these are the second derivative graphs notice the difference that they both are one negative and then the right side positive. When we do this, both sides go negative. That's because both sides are concave down, whereas before this is looking at increase and decrease. So we see these little things um, that allow us to analyze the grass in terms of the derivative graphs, which is a helpful tool to use. It's

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