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6 min read•february 15, 2024
Beth
Beth
In previous guides, we learned all about making conclusions regarding the behavior of a function based on the behavior of its derivatives such as whether the function is increasing or decreasing at a point, concave up or concave at a point, and more! While we mostly focused on algebraically determining the behavior of functions, we can also determine information graphically! The key features of the graphs of , , and are all related to one another. 🔑
Let’s dive into how we can do that!
Given the graphs of , , and or some combination of the three, we can determine information about another much as we did so algebraically. The knowledge you learned in our previous Unit 5 subtopic guides can be carried over to this subtopic—instead of using the equations for , , and , you can look at (one of) their graphs and see where the -axis is crossed or where the graph is positive or negative, increasing or decreasing, etc, to infer information about the other graphs.
Here’s a quick summary of what you’ve learned so far in this unit about trends and concavity:
Let’s apply this information to the following graph of a function, .
Taking a look at this graph, we can describe between each interval:
What about ? Let’s take a look at the concavity of :
Based on where the graph of the function changes direction and concavity, we can also interpret maximums, minimums, x-intercepts, and points of inflection of the graphs of the first and second derivatives.
If we boil this down to two key concepts, realize that:
This may seem like a lot, but once you see it in action, it’ll make more sense! ⬇️
Here’s a relatively easy example! The derivative of the differentiable function , , is graphed.
What can we tell about at the point based on the graph of its derivative ?
By looking at the graph of , we see that crosses the x-axis at the point of interest . It is negative before and positive after . This means that is decreasing before the point and increasing after it, indicating that the point is a relative minimum on the graph of .
The justification we used above to determine the answer is essentially just applying the First Derivative Test but in graphical form! Here’s a quick look at and so you can really see their relationship:
Now, let’s take another look at the example before, but focus on relative extrema and points of inflection.
You’ll notice the following:
Before you move on to taking a look at graphs yourself, take a look at the following graph of and think about:
At , has a relative maximum. This tells us that will have an x-intercept at this point and change from positive to negative!
At , changes from being concave down to concave up. This tells us that will have a relative minimum at this point and has an x-intercept, changing from negative to positive.
Take a look at in blue and in green! You can see exactly these trends.
And here’s a graph with added as well, denoted in purple.
Now it’s time for you to do some practice on your own! These won’t be as tough, they will more generally test your knowledge of these trends.
Question 1:
The second derivative of the differentiable function , , is graphed.
Given that , what can we tell about at the point based on the graph of its second derivative ?
Question 2:
The second derivative of the differentiable function , , is graphed.
What can we tell about at the point based on the graph of its second derivative ?
Question 1:
Answer: has a relative minimum at the point .
Solution:
By looking at the graph of , we see that is positive at the point of interest . This means that is concave up at the point. Combined with the fact that , we can apply the Second Derivative Test to conclude that has a minimum at .
Question 2:
Answer: is concave down at the point .
Solution:
By looking at the graph of , we infer that is negative at the point of interest . This means that is concave down at the point.
Woah! You made it to the end of this guide. To practice with some of this material, we recommend getting into Desmos and graphing a function, its first derivative, and its second derivative to see the features of each. Good luck! 🍀
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6 min read•february 15, 2024
Beth
Beth
In previous guides, we learned all about making conclusions regarding the behavior of a function based on the behavior of its derivatives such as whether the function is increasing or decreasing at a point, concave up or concave at a point, and more! While we mostly focused on algebraically determining the behavior of functions, we can also determine information graphically! The key features of the graphs of , , and are all related to one another. 🔑
Let’s dive into how we can do that!
Given the graphs of , , and or some combination of the three, we can determine information about another much as we did so algebraically. The knowledge you learned in our previous Unit 5 subtopic guides can be carried over to this subtopic—instead of using the equations for , , and , you can look at (one of) their graphs and see where the -axis is crossed or where the graph is positive or negative, increasing or decreasing, etc, to infer information about the other graphs.
Here’s a quick summary of what you’ve learned so far in this unit about trends and concavity:
Let’s apply this information to the following graph of a function, .
Taking a look at this graph, we can describe between each interval:
What about ? Let’s take a look at the concavity of :
Based on where the graph of the function changes direction and concavity, we can also interpret maximums, minimums, x-intercepts, and points of inflection of the graphs of the first and second derivatives.
If we boil this down to two key concepts, realize that:
This may seem like a lot, but once you see it in action, it’ll make more sense! ⬇️
Here’s a relatively easy example! The derivative of the differentiable function , , is graphed.
What can we tell about at the point based on the graph of its derivative ?
By looking at the graph of , we see that crosses the x-axis at the point of interest . It is negative before and positive after . This means that is decreasing before the point and increasing after it, indicating that the point is a relative minimum on the graph of .
The justification we used above to determine the answer is essentially just applying the First Derivative Test but in graphical form! Here’s a quick look at and so you can really see their relationship:
Now, let’s take another look at the example before, but focus on relative extrema and points of inflection.
You’ll notice the following:
Before you move on to taking a look at graphs yourself, take a look at the following graph of and think about:
At , has a relative maximum. This tells us that will have an x-intercept at this point and change from positive to negative!
At , changes from being concave down to concave up. This tells us that will have a relative minimum at this point and has an x-intercept, changing from negative to positive.
Take a look at in blue and in green! You can see exactly these trends.
And here’s a graph with added as well, denoted in purple.
Now it’s time for you to do some practice on your own! These won’t be as tough, they will more generally test your knowledge of these trends.
Question 1:
The second derivative of the differentiable function , , is graphed.
Given that , what can we tell about at the point based on the graph of its second derivative ?
Question 2:
The second derivative of the differentiable function , , is graphed.
What can we tell about at the point based on the graph of its second derivative ?
Question 1:
Answer: has a relative minimum at the point .
Solution:
By looking at the graph of , we see that is positive at the point of interest . This means that is concave up at the point. Combined with the fact that , we can apply the Second Derivative Test to conclude that has a minimum at .
Question 2:
Answer: is concave down at the point .
Solution:
By looking at the graph of , we infer that is negative at the point of interest . This means that is concave down at the point.
Woah! You made it to the end of this guide. To practice with some of this material, we recommend getting into Desmos and graphing a function, its first derivative, and its second derivative to see the features of each. Good luck! 🍀
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