sec10_2

sec10_2 - y = g ( t ), with x ( ) = a and x ( ) = b , then...

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10.2 Calculus with Parametric Curves 1. Slopes of Tangent Lines Recall from 10.1 the graphs that may be drawn from the parametric equations x = cos t , y = sin t . (1) The graph with t as the horizontal axis and x as the vertical axis. (2) The graph with t as the horizontal axis and y as the vertical axis. (3) The graph with x as the horizontal axis and y as the vertical axis. The slope of the line tangent to the graph in 1 is found using . The slope of the line tangent to the graph in 2 is found using . To Fnd the slope of the line tangent to the graph in 3 we need to Fnd . dy dx = d 2 y dx 2 = 1
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Section 10.2 Calculus with Parametric Curves 2 Example 1.1. (problem 4) Find an equation of the tangent to the curve x = t - t - 1 , y =1+ t 2 at t =1 . Example 1.2. (problem 15) Find dy/dx and d 2 y/dx 2 of x = 2 sin t , y = 3 cos t for 0 <t< 2 π . For which values of t is the curve concave upward? Example 1.3. (problem 18) Find the points on the curve x =2 t 3 +3 t 2 - 12 t , y = 2 t 3 +3 t 2 +1 , where the tangent is horizontal or vertical.
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Section 10.2 Calculus with Parametric Curves 3 2. Area If x and y are parameterized by t as x = f ( t ) and
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Unformatted text preview: y = g ( t ), with x ( ) = a and x ( ) = b , then b a y dx = Example 2.1. (problem 31) Use the parametric equations of an ellipse, x = a cos , y = b sin , 2 , to Fnd the area that it encloses. 3. Lengths of Curves Recall we Fnd the length of a curve by b a 1 + dy dx 2 dx or d c 1 + dx dy 2 dy If x and y are parameterized by t as x = f ( t ) and y = g ( t ), with t , and the curve is traversed exactly once as t increases from to , then the length is given by Example 3.1. (problem 47) ind the length of the curve. x = e t-t , y = 4 e t/ 2 , and-8 t 3 . Section 10.2 Calculus with Parametric Curves 4 4. Surface Area Remark 4.1. ds = 1 + ( dy dx ) 2 dx = 1 + dx dy 2 dy = Example 4.1. (problem 60) Find the area of the surface obtained by rotating about the x-axis. x = 3 t-t 3 , y = 3 t 2 , and t 1 ....
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sec10_2 - y = g ( t ), with x ( ) = a and x ( ) = b , then...

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