Sunday, September 24, 2006

Review: 2.1 The Tangent and Velocity Problems

Definitions:

  • tangent line: a tangent to a curve is a line just touching one point on the graph of a function; finding the slope of the tangent line finds the slope at that particular point and the limit as that point is approached; it should go in the same direction as the curve at the point of contact
  • secant line: these go through the point of the tangent line and another nearby point so that we are able to find a slope and eventually a limit
  • limits: secant lines approaching tangent lines illustrate the idea of these; they arise when we attempt to find the tangent to a curve or the velocity of an object
  • instantaneous velocity: the limiting value of average velocities over shorter and shorter time periods

Formulas:

  • lim x--> a f(x) - f(a) / x - a
If f(x)=x2a , and we wanted the equation of the tangent line through P (1,1), we could solve for x increasingly closer to 1 until we find that we are clearly approaching 2. So, lim x--> 1 x2-1 / x-1 = 2 We can check with x=1.001, 1.0012-1 / 1.001-1 = 2 approximately
  • average velocity = distance traveled / time elapsed (by using a very small interval of time, we can guess the instantaneous velocity)

Sample problem: A ball is dropped from 450 m above ground. What is the velocity of the ball after 5 seconds?

We must know Galileo’s law that tells us s(t)=4.9t2

Because we are dealing with an instant, there is no time interval and an average for a short interval must be used.

First, we will solve for the velocity between 5 and 5.1 seconds

=s(5.1)-s(5) / 0.1 = 4.9(5.1)2 - 4.9(5)2 / 0.1 = 49.49 m/s

Now, we do this again and again with smaller intervals (making a table is recommended) until about 5.001, at which point we can tell the average velocity is getting very close to 49m/s.

Time interval (s)

Average velocity (m/s)

5 to 5.1

49.49

5 to 5.01

49.049

5 to 5.001

49.0049

Thus the instantaneous velocity at 5 is v=49m/s.

Links:

  • http://faculty.rmc.edu/bsutton/131-2006-fall/lecture3.1.pdf#search=%22tangent%20and%20velocity%22
  • http://math.usask.ca/maclean/101/Limits/Printables/BW/Limits.pdf#search=%22tangent%20and%20velocity%22 (this one goes into section 2.2, sorry Evan!)

Reminder: Evan, I do believe Monday’s notes on 2.2 are yours to post.

Personal Touches:

  • This stuff has a clear real-world application! We can find instantaneous velocity.
  • Why don’t we use point C to find a limit? Because c can’t line. (Yes, it is a pitiful joke)
  • a limerick I found on the web that does not have anything to do with the lesson:
        Integral z-squared dz
               from 1 to the square root of 3
               times the cosine
               of three pi over 9
               equals log of the cube root of 'e'.

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