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MAT 1221 Survey of Calculus Section 2.3 Rates of Change http://myhome.spu.edu/lauw
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Expectations Use equal signs Show formula steps Show individual derivatives steps Double check the algebra
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HW WebAssign HW 2.3 There is a hint on problem 1 at the end of your HO. Additional HW listed at the end of the handout (need to get done, but no need to turn in)
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Fact: Slope of Tangent Line
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What is “Rate of Change”? We are going to look at how to understand and how to find the “ rate of change ” in terms of functions. (The connection between derivatives, slope of tangent lines and the rates of change.)
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Two Worlds and Two Problems ?
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The Velocity Problem y = distance dropped (ft) t = time (s) Displacement Function (Positive Downward) Find the velocity of the ball at t=2.
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The Velocity Problem Again, we are going to use a limiting process. Find the average velocity of the ball from t=2 to t=2+h by the formula
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The Velocity Problem thAverage Velocity (ft/s) 2 to 31 2 to 2.10.1 2 to 2.010.01 2 to 2.0010.001
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The Velocity Problem We “see” from the table that velocity of the ball at t=2 should be ____ft/s.
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The Velocity Problem We “see” from the table that velocity of the ball at t=2 should be ____ft/s. The instantaneous velocity at t=2 is _____ ft/s. (The ball is traveling at____ ft/s 2 seconds after it dropped.)
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Limit Notations When h is approaching 0, is approaching 64. We say as h 0, Or,
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Definition For the displacement function, the instantaneous velocity at time t is if it exists.
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Two Worlds and Two Problems
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Remarks In the context of moving objects, the independent variable is time t. We use the following notations Distance function Velocity function Acceleration function : rate of change of the velocity function
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Example 2 Given where s is in meters and t is in seconds, find (a) v(t) (b) a(t) (c) The velocity and acceleration at t=2s (d) The time when the velocity is 5m/s.
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Remarks When units are given, you answers in (c) and (d) should have units. The wonderful design of the notations helps you to get the units easily.
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Example 2 Suppose we model the amount of certain drug inside a patient’s body by mg after t hours of injection. (a) Find (b) Explain the meaning of the answer in (a)
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Definition For y=f(t), the (instantaneous) rate of change at t is
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Expectations Show the substitution step. Units are required for some of the answers. Use equal signs
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