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Use the methods of this section to sketch the curve $y=x^{3}-3 a^{2} x+2 a^{3},$ where $a$ is a positive constant. What do the members of this family of curves have in common? How do they differ from each other?

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Calculus 1 / AB

Chapter 4

APPLICATIONS OF DIFFERENTIATION

Section 3

Derivatives and the Shapes of Graphs

Derivatives

Differentiation

Applications of the Derivative

Campbell University

Harvey Mudd College

University of Michigan - Ann Arbor

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.

00:34

Use the methods of this se…

0:00

01:22

Graph the curves. Explain …

01:05

The equation $y^{2}=x^{3}-…

for this program. We need toe ride out the first of the directive in a second security at the beginning. So the first of the dura tape off the given function Why is three X squared minus three a square and the safe conduct narrative? It's just six. X eso. Let's first figure out what's the increasing and decreasing Terrible. They just, uh, let white crime because of zero. So obviously we have two solutions XY coast across I minus eight. I remember that we in this program we ah a is a positive constant. So we can see that affording stop intervals minus infinity to minus a minus eight to cross a in the eight to infinity over the first interval, we can see that. Why prime? Yes, positive. So the function is increasing over the second trouble. Why? Prime is negative. So the function istea crazy over the last interval, the white primes positive. So funky is increasing. And from here we see that we have a local um makes him at X equals two minus a. So with value EF minus a if off my, uh why off minus a usually cost to minus a kill minus three a square plus to a square, so it should be pressed. So it's equals to four a square, which is always positive unless a equals 20 We also have a local Merriman at X equals toe A with why a, um you caused zero. I'm not fulfil cavity. We set the second activity because zero So we have X equals zero on if x belongs to next. If infinity 20 white up of prime According to our second narrative, white up a promise nest and zero. So it's conclave down. I, yes, exists inside Ah zero to infinity, who had a prime Miss Griffin zero. So the functions hung about they So I said, we have an inflection point at X equals zero. Ah, where y zero u closed to through up to a que, um this should be Oh, this you Cube. This is not type of here. No, not based on this information we can grab. We can schedule a graph off the function. Okay, so let's see. So we have a low. We have a local maximum at X equals two minus a with value for a cube. So the functionary looks like this if we graph, um, so we have a local makes month at X equals two minus a. We have a local minimum attack seacoast way, and the inflection point is the same as the Y intercept. So this is the inflection point. Um, So the y coordinate here is for a cube. This is to a cube now, So this is a graph. Now, if a is varying, for example, face decreasing, we have another, um, function. Looks like this. So if a is b cuisine, um, the function well, looks like this. Then use this local maximum at X equals two minus a. We're decreased. We would be really cool close to xx iss eso. Conversely, face increasing this point, we will be well going. Goes up.

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