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Unformatted text preview: 1 Tangents and velocities On the first day, I talked very briefly about where calculus comes from. One branch comes from studying tangent lines to curves. Heres a more in-depth overview. A tangent line to a circle is basic. (draw picture) But a tangent line to a curve is far more difficult to see. A tangent to a circle intersects only in one place, but tangents to curves can intersect in more than one place: (draw picture) Example 1.1. Use y = x 2 and look at difference quotient. Use left and right limits with a chart. Introduce limit notation and show limit process. x m PQ 1 0.5 1.5 0.9 1.9 0.99 1.99 0.999 1.999 1.001 2.001 1.01 2.01 1.1 2.1 1.5 2.5 2 3 Velocity is a problem that might sound different, but turns out to be exactly the same as finding tangent lines. Example 1.2. Use falling ball problem. Galileo found distance travelled at time t is proportional to the square of the time (ignoring resistance). Use the equation s ( t ) = 4 . 9 t 2 Average velocity is easy, but velocity at a specific moment in time is different. Again, we do approximations. (difference quotient). Time Average Velocity 5-6 53.9 5-5.1 49.49 5-5.05 49.245 5-5.01 49.049 5-5.001 49.0049 Again, state limiting process The second example may seem similar to the first. The reason is because theyre the same. First, we draw the graph of s ( t ). Then we look at the graph of the line at ( a, 4 . 9 a...
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- Spring '08