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Trace or copy the graph of the given function $ f $. (Assume that the axes have equal scales.) Then use the method of Example 1 to sketch the graph of $ f' $ below it.

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03:04

Daniel Jaimes

Calculus 1 / AB

Chapter 2

Limits and Derivatives

Section 8

The Derivative as a Function

Limits

Derivatives

Missouri State University

Baylor University

University of Nottingham

Boston College

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.

02:41

Trace or copy the graph of…

06:01

02:42

03:21

0:00

So in this problem were given a graph of a function F. We were asked to trace or copy it. So, first of all, let's get our graph up here. There's X. Here's why. And our graph goes up to some maximum and then comes back down like this. Okay. They were asked to use the method of example 1 to sketch the graph of f prime. This is F F prime, blow it again. We'll have X and Y here. Okay? Yeah. The method says to pick points on here and look at the slopes at those points and use those to determine where we are on the derivative curve. We can see, first of all, if we do appoint a here that this tangent line, right is nearly vertical. So that means that we are at some very positive slope here. Okay? And then if we pick a point be right here on the top of the curve, that's a horizontal, that's a zero slope. So we know that he's going to be here. We know that A. Is up here somewhere for that value of X. Then we pick a grab a point over here, point C. And look at that tangent line that's negative, right? As this is a decreasing curve. And that's at the point where it's the steepest negative. And so that means we're going to have some point C. Down here happening to us, we are negative and we are the steepest negative and notice that this trends on toward zero. So this is C. Okay, so what happens? Well from here, I trend up towards zero from B to C. I got more and more negative like that and from A to B. I came from very positive down to zero. And so there is my graph for the derivative of F.

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