Use the graph of f to sketch a graph of f^'' . (Hint: Sketch f^' first.)

Name: Problem 32, Chapter 2: Calculus: Early Transcendental Functions
Uploaded: 2023-08-15T19:06:54-08:00
Duration: 8 min 16 s
Channel: Ceyda Guley
Description: Use the graph of f to sketch a graph of f^'' . (Hint: Sketch f^' first.)

Use the graph of f to sketch a graph of f^'' . (Hint: Sketch...

Key Concepts

Derivative as a Measure of Slope

The first derivative of a function at any point represents the slope of the function at that point. When sketching from the graph of f, you analyze how steeply the function rises or falls to estimate the value of f'. This step is necessary before determining f'', as the behavior of f' (its increases or decreases) directly informs the second derivative.

Second Derivative as the Rate of Change of the First Derivative

The second derivative, f'', is the derivative of the first derivative f'. It indicates how the slope (or steepness) of f is changing. When sketching the graph of f'', one considers where the slope of f' is increasing (f'' > 0) or decreasing (f'' < 0), offering insight into the curvature or concavity of the original function.

Concavity and Graph Behavior

The sign of the second derivative tells us about the concavity of f: if f'' is positive, the function is concave up (shaped like a cup), and if f'' is negative, it is concave down (shaped like a cap). This concept aids in sketching f'' by translating the visual curvature of f into the sign and approximate magnitude of f''.

Inflection Points

Inflection points occur where the concavity of f changes, meaning f'' changes sign, typically passing through zero. Identifying these points on the graph of f can help determine where the graph of f'' intersects the horizontal axis, providing important reference markers for the sketch of f''.

Transcript

00:01 Okay, in this video we have two different graphs of the first parent function and we are asked to draw the graph of the second derivative.

00:12 Our hint is that we should sketch the first derivative first and i concur with that.

00:17 So let's go ahead and start with a.

00:19 So for a, what are we looking for when we're graph sketching? we're looking for a couple different things.

00:25 We're looking at the slope at any given region and how are we going to figure out what that is? well, we need to look at our critical values.

00:36 So critical values are places where the slope is going to be zero.

00:42 So on this graph we can see it's right here.

00:43 You can draw a tangent line if it's flat.

00:45 That's a place where the slope is zero.

00:48 When we're curve sketching, i also like to look at a place where the graph crosses the axis.

00:57 So those are those three points right there.

00:59 So those are going to be our big points.

01:02 So the first derivative is the slope of the line of the parent function.

01:10 So let's just go ahead and label each of these.

01:13 Look at these critical values and wherever the critical values are, we'll just evaluate them from those intervals.

01:20 So from this point up until the critical value, we can see that this slope is increasing.

01:28 From this critical value to this one, we see the slope is decreasing.

01:32 And then from this critical value onward, we see that it's increasing again.

01:36 So what does that mean? well, that means that whenever the slope is increasing, the value of the first derivative is going to be positive.

01:45 And when it's decreasing, it's going to be negative.

01:47 So now we know where on the graph those different areas are going to end up lying.

01:53 So let's go ahead and start curve sketching.

01:57 So we know we're going to start in the positives.

02:00 We're going to start in the positives and we know that this point right here where the slope is zero, that means that the first derivative is going to be zero.

02:08 And that means that this point right here is where it's going to cross the x -axis.

02:14 So we know we're positive, but it's going to have the x -axis crossed at this first point i've drawn an arrow to.

02:22 So what does that mean? well, this is actually something like an x -cubed function.

02:28 We know that this is going to go down in terms of polynomial order.

02:31 So it's probably going to be an x -squared.

02:33 So let's start drawing this parabola.

02:35 So we go like this.

02:36 We hit that point where the slope is zero.

02:41 And then we hit a point where from this interval right here, that is all decreasing slope.

02:50 So the slope is going to be in the negatives.

02:52 The value of the first derivative is negative.

02:55 So we dip underneath that x -axis to be negative.

02:58 Then we come right back up.

03:04 So this is going to be our first derivative graph.

03:06 Now, the reason why this hint is sketch the first derivative is because we can do the same process with this second derivative.

03:18 So let's look at our critical values.

03:21 That's going to be this one right here, where we initially pointed out up here, where it crosses the x -axis.

03:32 That point is going to end up being our one critical value.

03:39 So from this point up until this point, the value of the slope is decreasing.

03:47 And from this point to this point, it's increasing.

03:50 So this is going to be a negative.

03:52 And then this is going to be a positive.

03:53 And because this was an x -squared, our second derivative function is just going to be an x.

03:59 So it's going to be a line.

04:00 So we start in the negatives...

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