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Tangents, x(t), v avg, and v Otherwise known as a really little bit of calculus

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What is a tangent? x(t) t tftf xfxf titi xixi Slope = v avg from t i to t f titi xixi titi xixi xixi titi Slope of tangent = v at t f Explanation #1: A chord is a line connecting two points on a graph A tangent is a chord with a time span of 0 = t i xi=xi=

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Comparison to geometry Explanation #1: A chord is a line connecting two points on a graph A tangent is a chord with a time span of 0 x(t) t tftf xfxf titi xixi Slope = v avg from t i to t f titi xixi Slope = v at t f xixi titi Slope = v avg from t i to t f titi xixi chord tangent

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What is a tangent? x(t) t tftf xfxf Slope = v at t f Explanation #2: If you zoom in close enough on any graph, the graph will look linear If this linear trend is extrapolated into a line, that line is a tangent x(t) t tftf xfxf t tftf xfxf t tftf xfxf t tftf xfxf

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Don’t lose sight of what we’ve done before t x(t) Position vs. Time Plot Position Only + -- 0

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Slope of a Tangent = v x(t) t t3t3 t1t1 t2t2 t5t5 t4t4 very steep line = very high slope = very high v at t 1 moderately steep line = moderate slope = moderate v at t 2 shallow line = low slope = low v at t 3 flat line: slope = v = 0 at t 4 “downhill” line: slope = v < 0 at t 5 very high v at t 1 moderate v at t 2 low v at t 3 v(t 1 ) > v(t 2 ) > v(t 3 ) > v(t 4 ) = 0 > v(t 5 ) Path + --

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Slope of a Tangent = v x(t) in m t (sec) t3t3 t1t1 t2t2 t5t5 t4t4 slope at t 4 = v at t 4 = v(t 4 ) = 0 m/s slope at t 5 = v at t 5 = v(t 5 ) = -0.35 m/s slope at t 1 = v at t 1 = v(t 1 ) = 2.71 m/s slope at t 2 = v at t 2 = v(t 2 ) = 0.69 m/s slope at t 3 = v at t 3 = v(t 3 ) = 0.36 m/s v(t 1 ) = 2.71 m/s v(t 2 ) = 0.69 m/s v(t 3 ) = 0.36 m/s v(t 4 ) = 0 m/s v(t 5 ) = -0.35 m/s v(t) (m/sec) t (sec) t3t3 t1t1 t2t2 t5t5 t4t4

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A special case: constant velocity Velocity does not change with time time in seconds x(t) in meters 4 4 8 8 In the special case of constant velocity graph of x(t) vs t is a straight line v avg = v = x/ t because v avg = v and v avg = x/ t Chord from t = 0.0 sec to t = 5.0 sec slope of chord = v avg v avg =(8.0 m – 0.0 m)/(5.0 sec – 0.0 sec) v avg = 1.6 m/s Tangent to a straight line is exactly the same as the line slope of tangent = v v= (8.0 m – 0.0 m)/(5.0 sec – 0.0 sec) v= 1.6 m/s

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A special case: constant velocity Velocity does not change with time time in seconds x(t) in meters 4 4 8 8 t3t3 t1t1 t2t2 t 1 : slope of tangent = v(t 1 ) = 1.6 m/s t 2 : slope of tangent = v(t 2 ) = 1.6 m/s t 3 : slope of tangent = v(t 3 ) = 1.6 m/s v(t) (m/sec) t (sec) 4 t3t3 t1t1 t2t2 8 If velocity is constant, then the plot of v(t) vs time is a horizontal line with y=slope of x(t) vs t graph

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Rate of change and tangent lines

Rate of change and tangent lines

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