If you are asked to find the slope using the definition or using the difference

If you are asked to find the slope using the

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If you are asked to find the slope using the definition or using the difference quotient, this is the technique you will use.
In the previous example, the tangent line could be found using . ( 29 1 1 y y m x x - = - The slope of a curve at a point is the same as the slope of the tangent line at that point. If you want the normal line , use the negative reciprocal of the slope. (in this case, ) 1 2 - (The normal line is perpendicular.)
Example 4: a Find the slope at . x a = ( 29 ( 29 0 lim h f a h f a m h m + - = 0 1 1 lim h a h a h m - + = ( 29 ( 29 0 1 lim h h a a h a a h m - + = + ( 29 0 lim h a a h h a a h m - - = + 2 1 a = - Let ( 29 1 f x x = On the TI-89: limit ((1/( a + h ) – 1/ a ) / h , h , 0) F3 Calc Note: If it says “Find the limit” on a test, you must show your work ! ( 29 a a h + ( 29 a a h + ( 29 a a h + 0
Example 4: b Where is the slope ? 1 4 - Let ( 29 1 f x x = 2 1 1 4 a - = - 2 4 a = 2 a = On the TI-89: Y= y = 1 / x WINDOW 6 6 3 3 scl 1 scl 1 x y x y - < < - < < = = GRAPH
Example 4: b Where is the slope ? 1 4 - Let ( 29 1 f x x = On the TI-89: Y= y = 1 / x WINDOW 6 6 3 3 scl 1 scl 1 x y x y - < < - < < = = GRAPH We can let the calculator plot the tangent: F5 Math A: Tangent ENTER 2 ENTER Repeat for x = -2 tangent equation
Review: average slope: y m x = slope at a point: ( 29 ( 29 0 lim h f a h f a m h m + - = average velocity: ave total distance total time V = instantaneous velocity: ( 29 ( 29 0 lim h f t h f t V h m + - = If is the position function: ( 29 f t These are often mixed up by Calculus students! So are these! velocity = slope π

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