Use the graph off to obtain the graph of f^' .

Name: Problem 28, Chapter 2: Calculus, Early Transcendentals
Uploaded: 2021-07-28T18:41:26-08:00
Duration: 6 min 57 s
Channel: Eric Mockensturm
Description: Use the graph off to obtain the graph of f^' .

Key Concepts

Graphical Analysis of Derivatives

Graphical analysis involves studying the behavior of the original function’s graph—identifying intervals where the function is increasing or decreasing, as well as locating local maximum and minimum points—to determine corresponding features in the derivative. The derivative is positive when the function is rising and negative when it is falling.

Relationship Between Function Behavior and Its Derivative

There is a direct connection between the behavior of a function and the behavior of its derivative. For instance, points where the function has a horizontal tangent (peaks or valleys) correspond to zeros of the derivative, while points of inflection lead to changes in the concavity of the function that are reflected in the derivative’s behavior. Understanding these relationships helps in accurately constructing the derivative’s graph from the original function.

Derivative as the Slope of Tangent Lines

The derivative of a function at a point represents the slope of the tangent line to the graph at that point. This concept is crucial because it translates the idea of instantaneous rate of change into a geometric interpretation, allowing one to graph the derivative by assessing the steepness of the original function at various points.

Transcript

00:01 And these problems, they show us a graph of a function of x, in the blue.

00:08 So i use blue to basically reproduce what the graph is in the book.

00:15 Then they ask us to graph the derivative.

00:17 Well, here the function is a line, so we know the derivative is going to be a constant.

00:22 And we just got to figure out where, you know, what that constant should be.

00:26 And it looks like basically the line has a slope of, has a slope of one.

00:34 Well, no.

00:35 Is it one? anyway, the line has a slope that's positive, so we know that the constant should be positive.

00:46 And so it looks like it goes through one zero.

00:50 The slope looks like it's about a half.

00:53 It gets a lot of slope of about a half.

00:56 So this should, this value should go through about basically close to one half.

01:00 So if this is one, then this is a half.

01:03 So the derivative should just be one half.

01:06 Now in the next problem, they again give us a line.

01:13 And i, again, i messed that one up.

01:18 This should be down here.

01:22 So they give us a line, right? yeah, this line here.

01:27 It has a negative slope.

01:28 So that's why it should be down here.

01:32 And we have, let's see here.

01:35 If we look at the slope, it goes through, it looks like minus 1 ,0, and 0 minus 2.

01:43 So it looks like it has a slope of minus 2.

01:46 So our constant would be minus 2.

01:51 Now the next one, let's see here.

01:55 If we have, we're given this curve in blue, so it goes through minus two zero, zero two, and two zero.

02:10 So this parabola, so it's a parabola, assuming it's a parabola, it looks like a parabola, then the derivative would be a line.

02:22 And so if we look at what the slope of this line should be, we know it, we know it should pass through the origin, the slope is zero when x is zero so we know it should pass through that point and so if we're at two here then x squared is and i why am i mixing these up anyway um yeah let's see here i need to um let's see here this this should have a negative slope because the um the slope is always is always positive to the right.

03:04 It's always positive over here and negative over here.

03:07 So this, the slope is wrong.

03:09 So it should be like this.

03:11 So let's fix that.

03:12 Let's see what would be the easiest way to fix that.

03:16 Oh, i made two lines there.

03:25 So that would be one.

03:34 And, let's match the slopes.

03:40 Like that, maybe.

03:43 All right...