Sec 10.1.pdf - Section 10.1 Parametric equations Tangent lines and arc length for parametric curves Section 10.1 Parametric equations Tangent lines and

Sec 10.1.pdf - Section 10.1 Parametric equations Tangent...

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Section 10.1: Parametric equations; Tangent lines and arc length for parametric curves Section 10.1: Parametric equations; Tangent lines and arc length
Parametric curves Now that we have finished chapter 9, we have completed what is generally considered “calculus”. For the remainder of the semester, we will discuss some topics that will prepare you for calculus III, as well as other advanced courses in mathematics. First we study: Parametric Curves. Section 10.1: Parametric equations; Tangent lines and arc length
Parametric curves We begin with the following definition: Definition 1 A parametric curve is a curve in the xy -plane whose coordinates ( x , y ) can be specified by functions f ( t ) and g ( t ) such that each point ( x , y ) on the curve satisfies x = f ( t ) and y = g ( t ) for some t . The variable t is called a parameter . By selecting a few values of t , we can plot points and sketch graphs of the curves in much the same way as we do for functions. Section 10.1: Parametric equations; Tangent lines and arc length
Parametric curves Example 1 Sketch a graph of the curve defined by x = t + 1, and y = t + 2. Solution. First we make a table of values: t -1 0 1 2 x 0 1 2 3 y 1 2 3 4 Section 10.1: Parametric equations; Tangent lines and arc length
Parametric curves If we plot ( x , y ) for each t , we get the following scatter plot: Section 10.1: Parametric equations; Tangent lines and arc length
Parametric curves If we draw a smooth curve through these points, we obtain the following graph: Remarks: Here, it is easy to see what the final curve will look like. For more complicated curves, the resulting scatter plot may not be so easy to sketch. Section 10.1: Parametric equations; Tangent lines and arc length
Parametric curves Example 2 Sketch a graph of the parametric curve defined by x = t - 3 sin t , y = 4 - 3 cos t . Solution. As before, we first make a table of values: t 0 1 2 3 4 5 6 7 8 9 10 x 0 -1.5 -0.7 2.6 6.3 7.9 6.8 5.0 5.0 7.8 11.6 y 1.0 2.4 5.2 7.0 6.0 3.1 1.1 1.7 4.4 6.7 6.5 Section 10.1: Parametric equations; Tangent lines and arc length
Parametric curves If we plot these points, we obtain the following scatter plot: Looking at these points, it is not so clear that what the curve should look like. If we add (many) more points to our scatter plot, you should (hopefully) be able to see that it should look like this: Section 10.1: Parametric equations; Tangent lines and arc length
Parametric curves Thus, drawing sketches of parametric curves can be quite difficult; more so than for functions y = f ( x ).

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