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The figure shows the graphs of $ f $, $ f' $, and $ f'' $. Identify each curve, and explain your choices.

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03:19

Daniel Jaimes

Calculus 1 / AB

Chapter 2

Limits and Derivatives

Section 8

The Derivative as a Function

Limits

Derivatives

Sharieleen A.

October 27, 2020

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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.

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The figure shows the graph…

The figure shows graphs of…

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So in this function, this problem were given this family of curves here and asked to determine Which one is F F prime, F double prime. And since there's actually four of them here, we could say F triple prime, I suspect. So let's look at it for a minute, remember, But each time we do the derivatives what we're doing, we're graphing the slope of the previous curve. So the F prime is the slopes of the F curve. An F double prime is the slope of the F prime curve and so on. So, let's see here for a minute. If we just start out here with the for a second. So let's look at the slope of what D. Is. So, first of all, the slope, as we start up, this curve is very positive and it slows down and hits zero right here. Well, that's what this green line does and then the curve is negative. The slope is negative all the way down till it hits the inflection point right here. Which is where the slope would be zero or be its most excuse me? It's at its most negative point and then it comes on down and Slows down here. So it approaches zero again, Still negative but approaches zero. So that follows the green curve, doesn't it? And then the slope goes positive and gets greater and greater and greater, which is what this is green curve does. So we know that C is the derivative of D. Okay, that's cool. So let's look now for a second at sea. So, see starts out with this really strong negative slope coming down through here and that slope All right, as we're coming down through here approaches zero as C hits a minimum point right here and then it starts to become positive, first a small parties number and then a great positive number up through there. Well that looks like B, doesn't it? He is a very strong negative number, Comes up towards zero and then goes up becomes very strong positive number. So be we could say B we can say B is the derivative C. Couldn't we? Okay, now, let's look at the beaker for a minute. So B is the slope of it, right? Is very strong, positive, very strong, positive, very strong, positive starts to slow down, doesn't it? It gets zero, then continues to increase as well, but still positive all the way on up. Well, it's not what A is A. Is a is a very positive number of positive, positive positive positive positive gets goes toward zero right here. The same time that B is going toward this inflection point and then coming out of the inflection point, it's still positive, starts out kind of small and goes very large. So that follows the A curve. So that means A is the derivative of B. So all of this together and tells us what tells us that our function F is D f prime. You see, F double prime his B an f triple crime is, hey, so there we go

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