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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 indepth 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 56 53.9 55.1 49.49 55.05 49.245 55.01 49.049 55.001 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
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 Calculus

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