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Approximating instantaneous rate of change with average rate of change

Sal approximates the instantaneous velocity of a motorcyclist. Created by Sal Khan.

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  • starky ultimate style avatar for user James Taylor
    When does creating a tangent line from the smooth curve become applicable in life?
    (0 votes)
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    • leaf blue style avatar for user Stefen
      Consider this . . . .
      When does skipping a rope become applicable to a boxer's life?
      They never jump rope in the ring.

      When does running through tires become applicable to a football player's life?
      There are no tires on a football field.

      When does a c major scale become applicable in a musician's life?
      They never give a concert featuring the c major scale.

      The answer to the purpose of what you are asking is the same answer for the others, it is training for what is to come, and a skill that will yield benefits later even though you may not be able to appreciate (or even in some cases, understand) it at the moment.

      Most of the math things you will do in "life" will not be as easy as graphing a tangent line, you will need much more sophisticated math, but to understand that math, some of it depends on building a foundation of what a tangent line actually means.

      In the end, it is possible you may have a career that, from your point of view as a student now, seems to be math free. But learning math is more than just numbers and tangent lines; it is an ordered system of logic that helps you be a better thinker, no matter what you choose to do. Being a better thinker will help you in all areas of your life, from managing your income and expenses, to being able to see logical holes in ideas so that people can't take advantage of you; both very important life skills.
      (111 votes)
  • leaf red style avatar for user John Donaldson
    At Sal says we can get the slope of the tangent line @ t=2 seconds by averaging the "slopes of the tangent lines between 1.5 and 2 and 2 and 2.5" ... Aren't they secant lines, i.e. the slope of a line between two points on a curve?
    (8 votes)
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    • blobby green style avatar for user Creeksider
      We don't actually have a curve in this case, just a list of data points, but conceptually they're points on a curve and of course there's no such thing as a tangent between two points on a curve, so Sal misspoke when he referred to them as tangents. He meant to say secant lines, or just lines.
      (4 votes)
  • piceratops ultimate style avatar for user Mike
    Why does simply taking away 18.7 from 46.1 give the same answer of 27.4?
    (4 votes)
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    • leaf green style avatar for user nick.embrey
      Its not a coincidence. Think about all the steps that went in to finding that 27.4. First was finding the average velocity between those two intervals. Then, you averaged those two to get 27.4. Plugging everything in ...

      27.4 = (((30.2 - 18.7) / 0.5) + ((46.1 - 30.2) / 0.5))/2.

      The two terms that are divided by 0.5 are the first part of calculating the average velocity. Then you add these together and divide by 2 to average them.
      The 2 cancels with the 0.5's, so you have 27.4 = 30.2 - 18.7 + 46.1 - 30.2,
      or, 27.4 = 46.1 - 18.7

      (Its helpful if you write it out on paper as its hard to format things here)
      (7 votes)
  • purple pi purple style avatar for user john carmack
    I s it possible to further simplify the final equation? While working the problem before I finished the video I ended at (s=27.4t-24.6) Since "s" is the f(t) isn't the equation now in slope-intercept form, or have I made some mathematical gaff?
    (3 votes)
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    • piceratops ultimate style avatar for user Just Keith
      Simplifying the final answer is not as important in calculus the way it is in algebra. We are more interested in having the answer in a useful form.

      But, yes, I think that is the correct way to put that tangent line in slope-intercept form.
      (5 votes)
  • piceratops ultimate style avatar for user Anirudh Roy
    What did the question mean by saying t is an independent varible? How di it affect the answer?
    (1 vote)
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  • blobby green style avatar for user Bennett Engelman
    The general question is why the 'secant' line rather than the 'tangent' line?
    (1 vote)
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    • leaf green style avatar for user jude4A
      A secant line is just the line between two points on a graph. A tangent line is a line that touches exactly one point on the graph. We can express tangent lines in calculus by saying that they are secant lines that have two points that get infinitely close together, such that they look like exactly one point.
      (4 votes)
  • hopper cool style avatar for user Arnab Chowdhury
    So what form is y = mx+c called?
    (1 vote)
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  • piceratops seedling style avatar for user Joshua Decosta
    at Sal says to take the Tagent we use the average of [1.5, 2] and [2, 2.5]. why don't we take the average of the [1.5, 2] and [1, 1.5]?
    (1 vote)
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  • blobby green style avatar for user Jimmy Clary
    find slope tangent line to graphof function f(x)=-2x+6 at (-1,8)
    (0 votes)
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  • starky sapling style avatar for user Abhinav Shrestha
    can u do the same calculation if, lets say, the slopes were from 1 to 2 and 2 to 2.5 as they're not equal distance away from 2
    (2 votes)
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    • blobby green style avatar for user barbaramcboy
      Seems to me that no, you couldn't. You'd have to use a weighted average since they're not equidistant from 2. In your example, the first interval is twice as large as the latter. Thus, the slope from 1 to 2 would have a weight of (2-1)/(2.5-1)=2/3, and the slope from 2 to 2.5 would have a weight of (2.5-2)/(2.5-1)=1/3. So, in your example, the approximation would be equal to (2/3)(slope from 1 to 2)+(1/3)(slope from 2 to 2.5). If I'm incorrect, I hope someone can please correct me.
      (1 vote)

Video transcript

The table gives a position s of a motorcyclist for t between 0 and 3, including 0 and 3. This is just saying that t is part of the interval, or t in the interval between 0 and 3. And we see that right over here, where the distance traveled s is measured in meters, and t is time in seconds. Assume the motorcyclist is accelerating during a three-second period. And they give us the information. This is time between 0 seconds and 3 seconds. And here we have the corresponding distance that you could view as a function of time. The average velocity for t between 1.5 and 2, so t between 1.5 and 2 is 23 meters per second. So what they did over here is they figured out, well, what is delta s over delta t in this interval? And they figured out that it was 23 meters per second. And you can verify that. Your change in s looks like it's 12.5. Your change in time is point-- or actually, this looks like it's 11.5. Yeah, 11.5. Your change in time is 0.5. 11.5 divided by 0.5 is 23. So that makes sense. And then they tell us the average velocity for t between 2 and 2.5. So change in our distance over change in time, they say is 31.8 meters per second. And then they say, estimate the instantaneous velocity at t equals 2 seconds and use this value to write the equation of a line tangent to s of t at the time t equals 2. So we can try to approximate. We can approximate the slope of the tangent line right over here, right when t equals 2 seconds, by taking the average of the slopes of the tangent lines between 1.5 and 2, and 2 and 2.5. So essentially, to approximate the slope of the tangent line, we're going to take the average of these two rates of change right over here, the average of these two slopes. So let's do that. So the average is going to be 23 plus 31.8 over 2. And let's see, what is that equal to? That is equal to 54.8/2. And what is that equal to? Let's see. 54 divided by 2 is 27. So it's 27.4. So we can use that as our approximation for the instantaneous rate of change for the slope of the tangent line. And now we have to actually figure out what that equation actually is. They don't just want the slope. So this is the slope right over here. And they say that they want it in point-slope form. And they remind us that t is the independent variable. So when you're putting something in point-slope form, it really just comes out of the definition of a line. A line always has a constant slope. So let's just imagine taking a random point on that line, t-- let me write it this way-- t comma capital S-- a random point on the line, on the tangent line here. Well, the slope between that and this point is always going to be constant. So what's the slope between this point and this point? Well, your change in S is going to be S minus 2 over your change in t, which is t minus-- oh sorry, your S is S. This is confusing sometimes. S minus 30.2 over your change in t, t minus 2, is equal to your slope. Anywhere along that tangent line, you're going to have that slope, 27.4. And then you multiply both sides by t minus 2, and you've put it in point-slope form. So this is the same thing as S minus 30.2 is equal to 27.4 times t minus 2.