Slopes and Rate of Change

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  • 0:06 Rate of Change Review
  • 0:24 Rate of Change Example
  • 1:19 Examples of Changes in…
  • 2:42 Lesson Summary
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Lesson Transcript
Instructor: Jeff Calareso

Jeff teaches high school English, math and other subjects. He has a master's degree in writing and literature.

If you throw a ball straight up, there will be a point when it stops moving for an instant before coming back down. Consider this as we study the rate of change of human cannonballs in this lesson.

Rate of Change Review

X is changing as a function of time linearly, so the velocity is constant
Velocity Curves Example 1

Let's review. The rate of change is how one variable changes as a function of another variable. We can see this as how, for example, my location changes as a function of time. This is like a velocity.

Rate of Change in the Human Cannonball

So what happens in the case of the human cannonball? The human cannonball is launched at 35 mph. His rate of change at launch is 35 mph. When he reaches the apex, the top of his flight, his velocity is actually 0 mph. His rate of change is 0. When he starts plummeting back toward Earth, his rate of change is negative. His velocity is negative. So what happens if I zoom in really close to a point on the curve of his height as a function of time? If I look really closely, I can see that the slope at any given point on this curve is the tangent at that point. This tells us that the rate of change is not only equal to the velocity, it's also equal to the tangent to the curve.

The second set of curves shows velocity increasing over time
Velocity Curves Example 2

Examples of Changes in Velocity

Let's look at a couple of examples. If your position, x, is changing as a function of time linearly, so there's a straight line on this position-time graph, then your velocity is always constant. Say here that your position is changing as a function of time at about 35 mph. The slope on this x-t graph is 35. Your velocity is 35 mph.

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