How to Estimate Function Values Using Linearization

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  • 0:05 Importance of Linearization
  • 2:40 Using Linearization to…
  • 5:35 Examples of Linearization
  • 10:12 Lesson Summary
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Lesson Transcript
Instructor: Zach Pino
Sometimes landing on Mars isn't that easy. You might need to use linearization to estimate if you will crash into the planet, or miss it entirely. Learn how to do just that in this lesson.

Importance of Linearization

At 1 minute to impact, the estimated velocity is under the critical velocity of the space craft
Mars Landing Linearization

Urgent message from Space Command: 'INTEGRAL 1 Spacecraft landing on Mars in 10 minutes! Need to abort if too fast. Help needed ASAP!' From the President? We better figure this one out soon. Okay, here's what's going on: Our spacecraft, INTEGRAL 1, is approaching the surface of Mars. It needs to be traveling at under 100 miles per hour (roughly 1.7 miles per minute) in order to land safely. Right now, it's going 8 miles per minute, which is closer to 500 miles per hour. It's going to land in 10 minutes. We need to know if it's going to land safely.

So how are we going to do this? We need to use what we know about linearization to find out if this craft is going to slow down in time before it hits the surface of Mars. It's 10 minutes until impact, velocity is 8 miles per minute and it's decelerating at 0.3 miles per minute squared. Let's calculate this out. Linearization. We're going to estimate the velocity at some future time by looking at the current velocity and the deceleration. Remember that the deceleration (or acceleration depending on which way you're going) is the derivative of the velocity. So in this case, our velocity, we're going to call f(x), and our acceleration is f`(x), where our variable x is time. So f in 10 minutes - our current time (x) is zero, delta x is going to be 10 - is equal to our current velocity, which is 8, plus the amount of time we have (this delta x, 10), times our deceleration, which is negative 0.3 miles per minute squared. It's negative because we are slowing down. When I plug that in, I find that in 10 minutes, if we keep up at this rate, we're going to be going 5 miles per minute, which is much higher than our critical velocity of 1.7 miles per minute. Do we call off the mission? Well, 10 minutes is kind of a long time. Maybe things will get better. We can call it off up to the very last minute. So let's wait.

Using Linearization to Estimate Speed

It's been 5 minutes since we last tried to see if we were going to crash. We have 5 minutes until impact. Currently, our spaceship is going 6 miles per minute and it's decelerating at 0.5 miles per minute squared. Let's use linearization to predict how fast it will be going 5 minutes from now. So we're 5 minutes in - where delta x is 5 minutes and linearization says that our speed is 6 miles per minute - plus 5 minutes, times our deceleration (-0.5). Plugging this in, I have 6 - 2.5, which is 3.5 miles per minute. That's still much faster than our critical velocity of 1.7 miles per minute. But we are getting closer, so maybe if we wait just another couple of minutes, maybe we'll slow down and not have to abort this mission to Mars.

Linearization for the example sin(x) near x=0
Linearization example II

All right, it's 3 minutes to impact. Our velocity is 5 miles per minute, but we're decelerating now at 1 mile per minute squared. If we plug this into our linearization, we estimate that our velocity at 10 minutes is going to be approximately our current velocity (5 miles per minute), plus 3 minutes to impact, times our deceleration of -1, 5 + 3(- 1) = 2. Okay, so our estimated velocity is 2 miles per minute. At this point, we still have to call off the mission, but we can wait just another couple of minutes to be sure we really have to call off the mission because we're almost slow enough. Let's see what happens.

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