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The Tangent and Velocity Problems The Tangent Problem Can a tangent to a curve be a line that touches the curve at one point? animation

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Example Guess an equation of the tangent line to the exponential function y = 2 x at the point P(0,1). animation

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Example A cardiac monitor is used to measure the heart rate of a patient after surgery. It compiles the number of heartbeats after t minutes. When the data in the table is graphed, the slope of the tangent line represents the heart rate in beats per minute. t (min)3638404244 Heartbeats2530266128062948 3080

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Use the data to estimate the patient’heart rate after 42 minutes using the secant line between a )t = 36 and t = 42 b) t = 38 and t = 42 c) t = 40 and t = 42 d) t = 40 and t = 44 t (min)3638404244 Heartbeats2530266128062948 3080

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The Velocity Problem When you watch the speedometer of a car as you travel in the city traffic, you see that the needle doesn’t stay still for very long which means the velocity of the car isn’t constant.

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Example If an arrow is shot upward on the moon with a velocity of 58 m/s, its height in meters after t seconds is given by h = 58t – 0.83t 2. a)Find the average velocity over the given time intervals:[1,2] [1,1.1] [1,1.5] b) Find the instantaneous velocity after 1 second.

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Example The position of a car is given by the values in the table: a)Find the average velocity for the time period beginning when t = 2 and lasting 3s, 2s, 1s. t (sec)012345 s (feet)0103270119178 b)Use the graph of s(t) to estimate the instantaneous velocity (the limit of average velocities) when t = 2.

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Definition (Limit) We write lim f(x) = L (or f(x) - > L as x - > a) if we can make the values of f(x), arbitrarily close to L by taking x to be sufficiently close to a but not equal to a. x - > a

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Definition (Right-hand limit) We write lim f(x) = L (or f(x) - > L as x - > a + ) if we can make the values of f(x), arbitrarily close to L by taking x to be sufficiently close to a and larger than a. x - > a +

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Definition (Left-hand limit) We write lim f(x) = L (or f(x) - > L as x - > a - ) if we can make the values of f(x), arbitrarily close to L by taking x to be sufficiently close to a and less than a. x - > a -

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Observation If lim f(x) = L and lim f(x) = L, then lim f(x) = L. x - > a - x - > a + x - > a

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