Exploring FTC Part 2
Our next area of focus is proving why the derivative and integral are inverses of each other. By knowing this, how can we determine the distance a ladybug covers from time a = 0 seconds to some x seconds where x is between a and b? Above is another picture of the FTC part 2. Remember, it's telling us that by taking the derivative of the integral of a function using a dummy variable 't', we can find f(x) if our integral is from a = 0 seconds to some x where a < x < b.
The technology we are going to use to visualize the second part of the FTC will be the Nspire calculator. If you do not have this kind, a TI-83 or equivalent can also work to graph the functions and find the integral and derivative values. We will be using them to find how far the ladybug is able to crawl!
To begin, we will continue using f(x) = x to monitor the ladybug's velocity as it moves along a leaf. Our goal is to prove that the derivative and the integral are in fact inverse operations of each other. Once we find this, we will see what this means and how this allows us to further understand the FTC.
On your calculator, go to the home screen. Under scratchpad, go to 2) Graph > Menu > 3) Graph Entry > 1) Function. Then type in f(x) = x and press enter. You should see the function in blue similar to below! Again, this is the constant velocity of the ladybug as it moves along a leaf.
The technology we are going to use to visualize the second part of the FTC will be the Nspire calculator. If you do not have this kind, a TI-83 or equivalent can also work to graph the functions and find the integral and derivative values. We will be using them to find how far the ladybug is able to crawl!
To begin, we will continue using f(x) = x to monitor the ladybug's velocity as it moves along a leaf. Our goal is to prove that the derivative and the integral are in fact inverse operations of each other. Once we find this, we will see what this means and how this allows us to further understand the FTC.
On your calculator, go to the home screen. Under scratchpad, go to 2) Graph > Menu > 3) Graph Entry > 1) Function. Then type in f(x) = x and press enter. You should see the function in blue similar to below! Again, this is the constant velocity of the ladybug as it moves along a leaf.
Now, we are going to find the area under the curve of this function from a=0 seconds to b=10 seconds. This means we are monitoring the ladybug during the first ten seconds of movement. Before doing this on your calculator, calculate by hand using FTC part 1 what the area should be.
After, follow these instructions to check your answer:
Click on menu again > 6) Analyze Graph > 6) Integral. Now, move your cursor to the origin where x=0 seconds and click down. As you move your cursor to x=10 seconds, you will notice that the area in consideration is encased by dotted black lines. Also, the space under the function is filled in grey. This represents the area under the curve. Once you click on x=10, the area will appear.
Was your answer correct?
After, follow these instructions to check your answer:
Click on menu again > 6) Analyze Graph > 6) Integral. Now, move your cursor to the origin where x=0 seconds and click down. As you move your cursor to x=10 seconds, you will notice that the area in consideration is encased by dotted black lines. Also, the space under the function is filled in grey. This represents the area under the curve. Once you click on x=10, the area will appear.
Was your answer correct?
Next, we are going to look at a graph of values. Click on menu > 7) Table > 1) Split-screen Table. By doing this, we can see the x and y values of the function. Since f(x) = x is very simple, the x and y values will be the same.
If we changed the function to f(x) = x^2, we would see the following values.
If we changed the function to f(x) = x^2, we would see the following values.
For now, we will just continue with f(x) = x. Now, that we have our function, we want to be able to compute the integral of this function from any start to end time. So, make sure that the graph is highlighted, not the table. Add a second function similar to the function in red in the following picture.
As mentioned before, t is your dummy variable because it acts as a placeholder until we substitute x in for t. What do you think this integral is telling us in regards to the ladybug's movement?
It tells us that we are integrating from 0 to whatever x value we want to choose for our new function, f(t), in terms of t. If our function were f(x) = x^2, then f(t) = t^2.
For now, I am going to remove the table from the screen so there is more room. You can do this by going back to 7) Table and click on 2) Remove table.
Next, we are going to take the derivative of f2(x). To do so, add a third function similar to the picture. After we take the derivative of the integral, what did you notice? Think about this for a few seconds before reading on.
The function of f3(x) is in black. How is f3(x) similar to f(x)?
Hopefully, you see that the black line is laying on top of the blue line now. Somehow, by taking the derivative of the integral of a new function, t, we get g(x).
This is exactly what the second part of the FTC is telling us! Here is the formula again:
The function of f3(x) is in black. How is f3(x) similar to f(x)?
Hopefully, you see that the black line is laying on top of the blue line now. Somehow, by taking the derivative of the integral of a new function, t, we get g(x).
This is exactly what the second part of the FTC is telling us! Here is the formula again:
How can we put into words what is happening? Basically, if we take the derivative in respect to x of an integral from 0 to x of a function in terms of t, then we will get that function in terms of x.
Why does this work?
We know from the first part of the FTC that we have the velocity of the ladybug's movement. By knowing a beginning and end point, we can determine the total distance the ladybug crawls. We also know that the derivative is synonymous to finding the slope of the tangent line at a given point. If we work from inward out, we are first finding the integral or area or accumulation of the ladybug's progress as it crawls. Then, we are taking the derivative of this. We know the derivative tells us the rate of change of a function at a specific x.
Thus, aren't the integral and derivative simply opposites? In this case, we can use the term "inverses" such that they "undo" each other. Thus, we used f(t) as a placeholder until we derived the inverse, allowing us to substitute x into the function.
Now, let's explore a little with areas.
Why does this work?
We know from the first part of the FTC that we have the velocity of the ladybug's movement. By knowing a beginning and end point, we can determine the total distance the ladybug crawls. We also know that the derivative is synonymous to finding the slope of the tangent line at a given point. If we work from inward out, we are first finding the integral or area or accumulation of the ladybug's progress as it crawls. Then, we are taking the derivative of this. We know the derivative tells us the rate of change of a function at a specific x.
Thus, aren't the integral and derivative simply opposites? In this case, we can use the term "inverses" such that they "undo" each other. Thus, we used f(t) as a placeholder until we derived the inverse, allowing us to substitute x into the function.
Now, let's explore a little with areas.