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Unformatted text preview: 1 Tangents and Velocities Recall finding tangents numerically. 1.1 Tangents Remember formula for slope of secant line. Definition 1. The tangent line to the curve y = f ( x ) at the point ( a, f ( a )) is the line through the point with slope m = lim x → a f ( x ) f ( a ) x a Example 1.1. y = x 2 , P (1 , 1) Slope of tangent line = slope of curve. m = f ( a + h ) f ( a ) h Example 1.2. y = 3 x , P (3 , 1) Example 1.3. y = √ x , (1,1),(4,2),(9,3) 1.2 Velocity Same deal. ave. velocity = distance/time = secant line. Example 1.4. s ( t ) = 4 . 9 t 2 , t = 5, h = 450 m 1.3 Generic rates of change This formula works for any change in y wrt x . Example 1.5. Temp problem from book. 2 Derivatives Recall: formulas for slope of tangent lines and velocities. The limit lim h → f ( x + h ) f ( x ) h is actually used whenever we’re computing a rate of change. Definition 2. The derivative of a function f at a number a , denoted by f ( a ) is f ( a ) = lim h → f ( a + h ) f ( a ) h if the limit exists...
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This note was uploaded on 02/29/2008 for the course MAT 141 taught by Professor Varies during the Spring '08 term at Lehigh University .
 Spring '08
 varies
 Slope

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