Accumulation and Rates
At this point, you hopefully understand the idea of accumulation of area under the curve. If not, refer back to the Riemann sums tab for the explanation. Now, we need to tie this idea of accumulation to The Fundamental Theorem of Calculus. You understand that given a graph, we can estimate the area from point a to point b by dividing that area under the curve by even amounts (subintervals), but why does this help us find the total distance that a ladybug travels? We can think of the accumulation of rectangles or distance covered as taking the integral of the velocity that the ladybug crawls. The accumulation of the distance covered is equal to the integral from a to b of the steps of x, where steps of x is the velocity the ladybug is crawling.
So what does accumulation have to do with integration?
From what we've done so far, each time we accumulate, we are adding up one rectangle at a time. This gives us the area under the curve between specific time intervals. This area in turn tells us the distance that the ladybug travels. So when we have the derivative (mentioned below), we know the velocity the ladybug is traveling. After we integrate, we are told the position of the ladybug. Since we are always given the time intervals for when the ladybug travels, we use this to find the total distance covered!
From what we've done so far, each time we accumulate, we are adding up one rectangle at a time. This gives us the area under the curve between specific time intervals. This area in turn tells us the distance that the ladybug travels. So when we have the derivative (mentioned below), we know the velocity the ladybug is traveling. After we integrate, we are told the position of the ladybug. Since we are always given the time intervals for when the ladybug travels, we use this to find the total distance covered!
Rates of Change:
Up until now we have had a constant velocity, and this means that our accumulation has been steady. Why is this?
It is because our rate of change has stayed the same. From previous algebra classes you may have heard of the rate of change being described as "rise over run." This is referring to the average rate of change (AROC) because it is over a time interval based off of the formula (see image right).
This idea of rate of change comes from us specifically changing one variable and watching how this affects another variable. When we are are talking about derivatives, the rate of change refers to the instantaneous rate of change or the slope at a specific point. Remember that through this exercise, our derivative is the velocity that the ladybug is crawling!
If you think you're ready to learn about the second part of the FTC, click here!
Up until now we have had a constant velocity, and this means that our accumulation has been steady. Why is this?
It is because our rate of change has stayed the same. From previous algebra classes you may have heard of the rate of change being described as "rise over run." This is referring to the average rate of change (AROC) because it is over a time interval based off of the formula (see image right).
This idea of rate of change comes from us specifically changing one variable and watching how this affects another variable. When we are are talking about derivatives, the rate of change refers to the instantaneous rate of change or the slope at a specific point. Remember that through this exercise, our derivative is the velocity that the ladybug is crawling!
If you think you're ready to learn about the second part of the FTC, click here!