Section 2.1
        The Tangent and Velocity Problems

                           Math 1a


                       Februar...
Outline




   Bingo



   Velocity



   Tangents
Outline




   Bingo



   Velocity



   Tangents
Hatsumon



  Problem
  My speedometer is broken, but I have an odometer and a clock.
  How can I determine my speed?



 ...
Outline




   Bingo



   Velocity



   Tangents
A famous solvable problem

   Problem
   Given a curve and a point on the curve, find the line tangent to
   the curve at t...
A famous solvable problem

   Problem
   Given a curve and a point on the curve, find the line tangent to
   the curve at t...
A famous solvable problem

   Problem
   Given a curve and a point on the curve, find the line tangent to
   the curve at t...
Towards a definition of tangent

   This doesn’t work so well for general curves, though:
Towards a definition of tangent

   This doesn’t work so well for general curves, though:




                             ...
Towards a definition of tangent

   This doesn’t work so well for general curves, though:


                               ...
Towards a definition of tangent

   This doesn’t work so well for general curves, though:


                               ...
Tangent


  A line L is tangent to a curve C at a point P if
      L and C both go through P, and
      L and C have the s...
Tangent as a limiting process




       To find the tangent line through a curve at a point, we draw
       secant lines t...
Tangent as a limiting process




       To find the tangent line through a curve at a point, we draw
       secant lines t...
Same thing!




   The infinitesimal rate of change calculation is the same in both
   cases: finding velocities or finding s...
General rates of change




      The rate of change of f (t) at time t1 = the slope of y = f (t)
      at the point (t1 ,...
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Lesson 1: The Tangent and Velocity Problems

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Finding the speed of a moving object (without a speedometer) and finding the slope of a line tangent to a curve are two interesting problems. It turns out there are models of the same process.

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Lesson 1: The Tangent and Velocity Problems

  1. 1. Section 2.1 The Tangent and Velocity Problems Math 1a February 1, 2008 Announcements Grab a bingo card and start playing! Syllabus available on course website Homework for Monday 2/4: Practice 2.1: 1, 3, 5, 7, 9 Turn-in 2.1: 2, 4, 6, 8 Complete the ALEKS initial assessment (course code QAQRC-EQJA6)
  2. 2. Outline Bingo Velocity Tangents
  3. 3. Outline Bingo Velocity Tangents
  4. 4. Hatsumon Problem My speedometer is broken, but I have an odometer and a clock. How can I determine my speed? | | | | | | | | | −4 −3 −2 −1 0 1 2 3 4
  5. 5. Outline Bingo Velocity Tangents
  6. 6. A famous solvable problem Problem Given a curve and a point on the curve, find the line tangent to the curve at that point.
  7. 7. A famous solvable problem Problem Given a curve and a point on the curve, find the line tangent to the curve at that point. But what do we mean by tangent?
  8. 8. A famous solvable problem Problem Given a curve and a point on the curve, find the line tangent to the curve at that point. But what do we mean by tangent? In geometry, a line is tangent to a circle if it intersects the circle in only one place. •
  9. 9. Towards a definition of tangent This doesn’t work so well for general curves, though:
  10. 10. Towards a definition of tangent This doesn’t work so well for general curves, though: Is this a tangent line?
  11. 11. Towards a definition of tangent This doesn’t work so well for general curves, though: Is this a tangent line? • Is this a tangent line?
  12. 12. Towards a definition of tangent This doesn’t work so well for general curves, though: Is this a tangent line? • Is this a tangent line? We need to think of tangency as a “local” phenomenon.
  13. 13. Tangent A line L is tangent to a curve C at a point P if L and C both go through P, and L and C have the same “slope” at P. Slope of L = “m” in y = mx + b rise = run f (x) − f (a) Slope of C at a ≈ where x ≈ a x −a
  14. 14. Tangent as a limiting process To find the tangent line through a curve at a point, we draw secant lines through the curve at that point and find the line they approach as the second point of the secant nears the first.
  15. 15. Tangent as a limiting process To find the tangent line through a curve at a point, we draw secant lines through the curve at that point and find the line they approach as the second point of the secant nears the first. √ For instance, it appears the tangent line to y = x through (4, 2) has slope 0.25.
  16. 16. Same thing! The infinitesimal rate of change calculation is the same in both cases: finding velocities or finding slopes of tangent lines.
  17. 17. General rates of change The rate of change of f (t) at time t1 = the slope of y = f (t) at the point (t1 , f (t1 )). units of f (t) The units are . units of t

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