lecture6 - = Introduction to Limits Def. We write lim x ! a...

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Lecture 6: Tangent Lines and Slope Sec. 2.1 Tangent line to a circle Tangent line to a curve Slope of the Tangent Line Approximate slope
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ex. Let f ( x ) = x 2 +1. How can we ±nd the equation of the tangent line at (1 ; 2)? 6 - ? ± 1) Find the slope of the secant line through the points (1 ; 2) and (1 : 1 ; 2 : 21): m = 2) Find the slope through (1 ; 2) and (1 : 01 ; 2 : 0201): m =
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3) Now let P be (1 ; 2) and Q be ( x;x 2 + 1) slope of secant line through P and Q : m sec = slope of tangent line at (1 ; 2): m = 4) Equation of the tangent line to f ( x ) = x 2 + 1 at (1 ; 2)
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Velocity ex. Suppose an object is s ( t ) feet from its start at t seconds. Find the average velocity on the time interval from t = 2 to t = 2 + h seconds where s ( t ) = 3 t 2 . Average Velocity
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Find the velocity at the instant when t = 2 seconds. Instantaneous velocity
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Unformatted text preview: = Introduction to Limits Def. We write lim x ! a f ( x ) = L if we can make the values of f ( x ) arbitrarily close to L by taking x suf-ciently close to a (on either side of a ) but not equal to a . We say that f ( x ) as x ex. Let f ( x ) = x 2 1 x 1 : To nd lim x ! 1 f ( x ) : 1) use a table of values x f ( x ) x f ( x ) 2) Sketch the graph of f ( x ) = x 2 1 x 1 . 6-? lim x ! 1 f ( x ) = Now consider g ( x ) = 8 < : x 2 1 x 1 x 6 = 1 3 x = 1 Sketch the graph of g ( x ) : 6-? lim x ! 1 g ( x ) = NOTE: lim x ! 1 f ( x ) = lim x ! 1 g ( x ) but f (1) = and g (1) =...
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lecture6 - = Introduction to Limits Def. We write lim x ! a...

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