Clicker Question 1

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Clicker Question 1

• Solve for x 5 3(x2) 12
• A. x ln(12)/ln(8) 2
• B. x ln(7/3) 2
• C. x ln(7)/ln(3) 2
• D. x ln(7) ln(3) 2
• E. x (ln(7) 2)/ln(3)

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Clicker Question 2

• What is the value of sin(arctan(-1))?
• A. 0
• B. -1
• C. - ?/4
• D. ?2 / 2
• E. - ?2 / 2

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Clicker Question 3

• Solve for t cos(3t -2) .4
• A. t (arccos(.4) 2) / 3
• B. t (cos(.4) 2) / 3
• C. t (arccos(.4) 3) / 2
• D. t arccos(.8)
• E. t (arccos(.8) 2) / 3

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Limits (2/4/09)

• Question How can we compute the slope of the
  tangent line to the curve y x 2 at the point
  (1, 1)?
• Possible approach Compute the slope of the
  secant line which the connects the points (1, 1)
  and (1 h, (1 h )2) for small values of h.
• Now try to see the limit as h goes toward 0.

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Instantaneous Velocity

• That example was the Tangent Problem. Now comes
  the Velocity Problem.
• Question Given the position of a moving car as a
  function of time, how can we compute the
  instantaneous velocity of the car at a
  specific moment?
• Possible approach Compute the average velocity
  over a short period of time, and find the limit
  as that period approaches zero.

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Limit of a Function at a Point

• In both problems above, we seek the limit of some
  function (often, but not always, the function is
  in the form of a ratio) as we approach some
  point.
• Definition We say the limit as x approaches a of
  f (x) is a number L, writing limx?a f (x) L, if
  f s values get closer and closer to L as x gets
  closer and closer to a.

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Some Examples of Limits

• Some limits are obvious
• limx ? 3 x 2
• limt ? ? cos(t)
• But some limits arent
• limt ? 0 sin(t) / t
• limh ? 0 ((3 h)2 - 9) / h
• What was problematic about these two?

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Clicker Question 4

• What is limx ? 8 (ex 8 log2(x ))?
• A. e 2
• B. 3
• C. e 3
• D. 4
• E. Does not exist

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Clicker Question 5

• What is limx ? 3 (x 2 9) / (x 3) ?
• A. 3
• B. 6
• C. 0
• D. 1
• E. Does not exist

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Other Not Obvious Limits

• What is limx ?4 (x 2 3x - 4)/(x 4) ?
• What is limx ?2 3/(x - 2)2 ?
• What is limx ?? 1/x ?
• Note that in the last two examples, we are
  allowing the idea of infinity to be involved in
  limits, either as the answer (meaning the output
  keeps getting bigger and bigger) or as what x
  approaches (meaning x gets bigger and bigger).

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Assignment for Friday

• Hand-in 1 is due next Tuesday.
• Here we go with calculus!
• Read Sections 2.1 and 2.2.
• In Section 2.1, do Exercises 3 and 5.
• In Section 2.2, do Exercises 1, 3, 5, 9, 15, 21,
  25, and 27.