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If $ f(x) = 2x^2 - x^3 $, find $ f'(x) $, $ f''(x) $, $ f'''(x) $, and $ f^{(4)}(x) $. Graph $ f $, $ f' $, $ f'' $, and $ f''' $ on a common screen. Are the graphs consistent with the geometric interpretations of these derivatives?

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$4 x-3 x^{2}$ ; $4-6 x$ ; -6 ; 0

10:18

Daniel Jaimes

Calculus 1 / AB

Chapter 2

Limits and Derivatives

Section 8

The Derivative as a Function

Limits

Derivatives

Missouri State University

Oregon State University

University of Michigan - Ann Arbor

University of Nottingham

Lectures

04:40

In mathematics, the limit of a function is the value that the function gets very close to as the input approaches some value. Thus, it is referred to as the function value or output value.

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.

0:00

If $ f(x) = 2x^2 - x^3 $, …

07:07

If $f(x)=2 x^{2}-x^{3},$ f…

01:21

If f(x) = 3x2 _ x3 , find …

01:35

f(x) = 5x^(2) -x^(3), …

05:13

If the graphs of two diffe…

we have a function F of X equals two X squared minus extra third. We're going to find the first four derivatives of dysfunction. So F prime of X. F double prime of X. F triple prime of X. And even uh uh the fourth derivative of F a bex. And then we want to graph uh dysfunction along with its first three derivatives and look at the grass and try to interpret uh you know geometrically what what the grass are the different order derivatives and function are telling us Well, F of X equals two X squared minus execute. So F prime of X first derivative of F. Well the derivative of two X squared will be four X. And subtract. And then the derivative of X cubed is three X squared. So that's F prime of X. F double prime of X is obtained. The second derivative is obtained by differentiating the first derivative. So that prime of X is for x minus three X squared. So F double prime of X is going to be the derivative of four x minus three X squared Derivative of four x is four. Subtract The derivative of three times x square is six sex. The third derivative will be gotten by differentiating the second derivative. So the third derivative of f is going to be the derivative of four derivative of a constant is zero uh minus And then the derivative of six X is simply 60. Subtract six is negative six. So uh the third derivative of f is negative six. And if we want to find 1/4 derivative of X. I'm sorry to fourth derivative of F A bex, you simply differentiate the third derivative to third derivative of F A Becks was negative six. So, if we take the derivative of negative six, that's a constant. The derivative of a constant is zero. So the fourth derivative of F of X zero. So here you have the first four derivatives of F of X. Now we're going to graph F of X along with its first three derivatives uh on the graphing calculator. So here we are using a Desmond's graphing calculator here. Right now showing is the graph of F A bec. So this red curb is the graph of F. Quebec's uh and one at a time we're not gonna grab all these at at the same time. Uh You don't want to it will be very confusing and you really don't need them all at the same time. You compare Uh one of these derivatives with the original function one at time. So let's look at the graph of F of X. Let's zoom in just a little bit. Now you can see that ffx has a minimum at this point local minimum. And the graph of F of X has local maximum at that point, a function will have a local minimum and a local maximum when the first derivative equals zero. So, what I'm going to do next is I'm gonna graph uh f prime of X, which is this function right here. And we're going to look at when this uh F prime of X function equal zero. All right, so, the blue curve is F prime of X. The red curve is the original function F of X. So red is ffx, blue is F prime of X. Remember we said this point here on the red graph was a local minimum. If a function has a local minimum, the first derivative is zero. Well, the blue graph is the graph of the first derivative F prime of X. Here. You can see that F prime of X is equal to zero at this very same point. Uh So when X is zero, uh The function the red curve has a local minimum when x zero, F prime of X equals zero. Now the same thing happens when the graph the function has a local maximum. This is a local maximum on the curve. Ffx red curve is ffx. Now when x is 1.333, our function is having a local maximum. Well, functions have graphs, functions have local maximums and minimums when the first derivative. Okay, when the first derivative is zero. So, if our red function F of x has a local maximum, when x is 1.333, then f prime of X should equal zero at the same X value 1.333. Here, you can see when X is 1.333. The first derivative f prime of X is zero. Next let's go back to looking at uh the original graph of F of X. Try to recall what an inflection point is an inflection point is. If you look at this portion of the graph of F of X, it's concave up. If you look at this portion of the graph of F of X, it's concave down while there is a point called an inflection point where the con cavity changes from being concave up to now being concave down. So the inflection point is going to be somewhere around here. Well uh the graph will have an inflection point when the second derivative equals zero. So now we're going to graft the second derivative and see when it equals zero. So this green line is the graph of the second derivative. When is this green line equaling zero when it crosses the X axis. So When X is .667, ah Our 2nd derivative function Is equal to zero. So for x equal .667, Our 2nd derivative is zero. Well When your second derivative is zero, you have an inflection point. Now, I don't want you to look at this point here. I'm not trying to confuse you. Okay, but when X is .667, we're going to have an inflection point. So if we want to have an idea where that inflection point is go from this uh value where X is 0.667. here's X zero here, X is one here excess 10.667. If you go straight up until you hit the curve at this point is your inflection point. So we used the second derivative function. Uh we wanted to know when the second derivative was equal to zero. The second derivative equals zero. When X is 00.667, I'm going to get rid of that, I'm going to get rid of this and I'm going to graph the line x equals .667. I when X was 0.667 the second derivative was equal to zero. That means when X is .667. Since the second derivative was zero at X equals 0.667. That means when X equals 0.667. We have an inflection point on the graph of our original function F. A bex. So for X equals 0.667. The second derivative is zero, meaning for X equals 00.667. You have an inflection point. So now that you see where this inflection point is on the graph of F of X. You can see that it's concave up until it hits that inflection point. Then it becomes concave down

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