Linearization of Functions

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  • 0:06 Estimating Distances and Time
  • 1:22 Understanding Linearization
  • 2:20 Using Linearization to…
  • 6:13 Rewriting Linearization
  • 9:10 Lesson Summary
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Lesson Transcript
Instructor: Sarah Wright
Over the river and through the woods to Grandmother's house we go ... Are we there yet? In this lesson, apply linearization to estimate when we will finally get to Grandma's house!

Estimating Distances and Time

Using linearization for the grandmother example
Linearization of Functions

Let's say that one afternoon, you decide to visit your grandmother, who lives 10 miles away from you. So you head off, and you go over the river and through the woods, and after half an hour you see a sign, and it says, 'Grandma's house: 1 mile.' You glance at your watch and think, 'Wow! Thirty minutes for 9 miles? That means my average rate of change, my average velocity, is only 18 miles per hour (mph)! But now I'm going 30 mph.' So then you think, 'Well, if I'm going 30 mph now, how long is it going to take me to get to Grandma's house from here? At 30 mph, 1 mile will take me 2 minutes.' And away you go. Let's take a look at what you just did. You took your current position, which is 1 mile away from Grandma's house, 30 minutes in, and you extrapolated to find out where you would be if you kept going your same speed for another minute. So you used your current velocity - your instantaneous rate of change - to estimate where you're going to be in the future.

Understanding Linearization

Without knowing it, you used a concept called linearization. Linearization, formally, looks like this: f(x + delta x) is approximately equal to f(x) + delta xf`(x). So what does this mean for visiting Grandma's house? Well, the function that you're looking at is your position, and you said, 'What is my position at time, t, plus some delta t?' So t is 30 minutes, and delta t is going to be +1. You said, 'My position is going to be approximately equal to where I'm at now, x(t), plus the change in time, times the speed that I'm going, this f`(x), so that's x`(t) - this is your velocity.

Using Linearization to Estimate Travel

If you drive 8 hours at a velocity of 30 mph, you will have traveled a total of 240 miles
Linearization Travel Estimation Graph

Let's look at this slightly differently. Let's say you want to go on a road trip now - not just to Grandma's house. And you want to know how far away from home you're going to be at the end of the day. Well, as you start your road trip, you are perhaps going 30 mph. So you start your road trip going about 30 mph, and at this point you haven't gone anywhere yet, but you know that you're going to travel 8 hours today. Assuming that you're going to go 30 mph for 8 hours, you're going to end up 240 miles from where you started. That's assuming that you're going to continue at your current rate of change.

Let's say that after 2 hours, you again take a look at how far you've gone to estimate how far you're going to make it in that day. After 2 hours, you've gone 100 miles. You're now going 60 mph, and you know that because you've traveled 2 hours and you've planned for 8 hours total, you've got 6 more hours of travel in front of you. So if I'm going to travel 6 more hours at my current speed of 60 mph, that's going to be 360 more miles, and I'm going to add that to the 100 miles I've traveled already. That means that by the end of the day, I will have gone 460 miles, rather than 240. After 4 hours, you revisit your calculation again. Now you've traveled 200 miles, but you're at a dead stop on the freeway. Using your current velocity of 0 mph, it doesn't matter that you're going to travel another 4 hours, because you're not going to go anywhere. So your estimated distance at the end of the day is going to be 200 miles, which is exactly what you've traveled thus far.

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