What It Means To Be 'Differentiable'

What It Means To Be 'Differentiable'
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  • 0:06 Review of Derivatives
  • 0:39 Velocity of a Jet
  • 1:50 Velocity of a UFO
  • 2:27 Velocity of a Superjet
  • 3:36 Differentiable
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Lesson Transcript
Instructor: Eric Garneau
Lots of jets can go from zero to 300 mph quickly, but super-jets can do this instantaneously. In this lesson, learn what that means for differentiability.

Review of Derivatives

Graph showing the velocity of a jet
Jet Velocity Graph

Let's review really fast. The rate of change equals the velocity when we're thinking about it in terms of motion. The velocity and the rate of change are the slope of some position as a function of time. The instantaneous rate of change is your tangent to that line at any given point, and it is also known as the derivative. It's exactly how fast you're going at a single point in time.

Velocity of a Jet

So what does this mean for a UFO, a jet and a superjet? Specifically, does each one of them always have a derivative, or a rate of change or a velocity? First, let's consider the jet. If I look at the jet's altitude as a function of time, I have a smooth graph. The jet takes off, increases its altitude and levels out at some really high-up altitude. At any point in time that jet has a velocity, and I can see that on the graph of h as a function of t. Initially the velocity is zero, my instantaneous velocity is zero, the tangent to this line is zero. Then the jet's gonna slowly start increasing its horizontal velocity, and the tangent to this curve has a higher slope. When it reaches the altitude that it's trying to get to, its horizontal velocity again goes to zero, and you see that up here. But at every point in time it has some velocity, even if that velocity is zero.

The velocity of a UFO
UFO Velocity Graph

Velocity of a UFO

Now let's compare this to a UFO. A UFO might be hanging out near some crops, and then all of a sudden it might jump to a higher altitude. Now, at this jump it has no velocity. I can't calculate an average velocity and have that average velocity converge on some tangent. It just doesn't exist. It doesn't exist because there's a discontinuous position. The graph h as a function of t has a discontinuity in it.

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