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

Name: Problem 31, Chapter 2: Calculus: Early Transcendental Functions
Uploaded: 2023-08-15T19:37:28-08:00
Duration: 4 min 52 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

Points of Inflection

Points of inflection are where the function changes its concavity from up to down or vice versa. These points are often characterized by the second derivative changing its sign, and they are important markers on the graph that help in accurately sketching the second derivative.

First Derivative

The first derivative of a function represents the slope of the tangent line at any given point on the graph. It is the instantaneous rate of change of the function, and sketching it involves estimating these slopes from the original graph. By interpreting how steeply the function f rises or falls, one can construct a rough graph of f’ that reflects these changes.

Second Derivative

The second derivative is the derivative of the first derivative and provides information about the rate of change of the slope of the function. It describes the curvature or concavity of f. When f'' is positive, the function is concave up (curving upward), and when f'' is negative, the function is concave down (curving downward).

Concavity

Concavity relates to the bending of the function’s graph. The second derivative aids in determining the intervals where the function is concave up or concave down. A positive second derivative indicates that the graph is curving upward, while a negative second derivative indicates a downward curvature.

Transcript

00:01 Okay, so in this question we are given the parent function graphs and asked to draw the second derivative graphs.

00:07 Our hint is to go through the first derivative graph first, so we're gonna go ahead and do that.

00:13 So let's identify our critical values.

00:14 That is where the slope is equal to zero.

00:17 So we have one, two, three right there.

00:22 And on the second function we have this one right here.

00:27 So what does that mean? well these are places where the function is going to cross the x axis on our first derivative.

00:39 So from this point to this point the slope of the line is decreasing.

00:47 From here to here we have an increasing.

00:50 Here to here we have a decreasing.

00:52 And from here to here we have an increasing.

00:55 So that means on our first derivative graph we're gonna go from negative to positive to negative to positive.

01:02 And since this is an x to the fourth function, we can kind of estimate this is gonna look a little bit like an x to the third.

01:12 So let's go ahead and draw that.

01:14 So initially it's negative.

01:15 We end up right about here where that critical value is.

01:20 We inch into the positive a bit and then after that we return back down for that other critical point.

01:33 We dip once more, hit that other side, and then go up increasing.

01:42 So then we're gonna do the same thing here.

01:44 Our critical values right here and right here is gonna look a little bit like an x to the x squared function right there.

01:52 From this point to this point we are increasing.

01:55 From this point to this point we are decreasing.

01:59 And from this point to this point we are increasing again.

02:02 So we're gonna go ahead and have the parabola be right about there...

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