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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.$$f(x)=1+\frac{1}{x}+\frac{8}{x^{2}}+\frac{1}{x^{3}}$$

$f(-12-\sqrt{138})=0.972, f(-12+\sqrt{138})=60.36$

Calculus 1 / AB

Chapter 4

APPLICATIONS OF DIFFERENTIATION

Section 4

Curve Sketching

Derivatives

Differentiation

Applications of the Derivative

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Lectures

04:40

In mathematics, a derivative is a measure of how a function changes as its input changes. Loosely speaking, a derivative can be thought of as how much one quantity is changing in response to changes in some other quantity; for example, the derivative of the position of a moving object with respect to time is the object's velocity. The concept of a derivative developed as a way to measure the steepness of a curve; the concept was ultimately generalized and now "derivative" is often used to refer to the relationship between two variables, independent and dependent, and to various related notions, such as the differential.

44:57

In mathematics, a differentiation rule is a rule for computing the derivative of a function in one variable. Many differentiation rules can be expressed as a product rule.

10:46

Produce graphs of $ f $ th…

07:38

02:21

Produce graphs of $f$ that…

09:01

11:23

01:08

01:09

So for this problem we're going to be analyzing the graph that's given to us, especially in the context of derivative functions. So the graph that were given is one plus one over X plus eight over X squared Plus one over X. Cute! Now looking at this graph has a lot of different turns to it. But we're gonna first look at it in terms of the first road of graphs that's going to increase decrease um And areas of um extreme A. So here we see that the graph is increasing the full time in this portion of the graph, which is why in this graph for here it's always positive. The derivative graphs positive. Then we see that it breaks. But then the graph is still increasing until it reaches that critical point which provides increasing until it reaches that critical point. Then it decreases obviously until it reaches this value, it's going to break and still increase our still decreased as we see here. Um and then it's going to reach this critical point again, a local minimum which is where it's going to increase until it reaches another critical value. This local maximum and then it's gonna decrease again um and it's going to stabilize but it's going to continue to decrease. So that's how we see it in terms of the first derivative draft, then the second derivative graph which deals with con cavity. We see this graph is concave up fully, which is why this graph is going to be concave up as well. Then we see this portion of the graph is con came down, which is why we get negative values. Um here once we reach that value it can become cave down. Then um it goes concave up right here. Which is why it's positive over the course of this interval until goes concave down again. Which is why it's negative over the course of this interval and then it goes concave up again, which is why it remains positive through the rest of the um the graph all the way to infinity. That's how we can analyze graphs in the context of um the actual graph itself. And this should actually be um slightly different. So this is going to change the graph but we analyze it in the same way using the derivatives and we realize how changes in the values of the graph foot change the derivatives, which means it's changing its increased, decreased con cavity, um and so on and so forth.

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