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9.1 Parametric Curves 9.2 Calculus with Parametric Curves

9.1 Parametric Curves 9.2 Calculus with Parametric Curves. We can do this by writing equations for the x and y coordinates in terms of a third variable (usually t or ). There are times when we need to describe motion (or a curve) that is not a function. These are called

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9.1 Parametric Curves 9.2 Calculus with Parametric Curves

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  1. 9.1 Parametric Curves9.2 Calculus with Parametric Curves

  2. We can do this by writing equations for the x and y coordinates in terms of a third variable (usually t or ). There are times when we need to describe motion (or a curve) that is not a function. These are called parametric equations. “t” is the parameter. (It is also the independent variable)

  3. Circle: If we let t = the angle, then: We could identify the parametric equations as a circle.

  4. Ellipse: This is the equation of an ellipse.

  5. Tangents The formula for finding the slope of a parametrized curve is: We assume that the denominator is not zero. To find the second derivative of a parametrized curve, we find the derivative of the first derivative:

  6. Tangents Example:

  7. Example (cont.): • Find the first derivative (dy/dx).

  8. 2. Find the derivative of dy/dx with respect to t.

  9. 3. Divide by dx/dt.

  10. Areas under parametric curves • If a curve is given by parametric equations x=f(t), y=g(t) and is traversed once as t increases from α to β, then the area under the curve is • Examples on the board

  11. Lengths of parametric curves • If a curve C is described by the parametric equations x=f(t), y=g(t), α≤ t ≤β, where f’ and g’ are continuous on [α, β] and C is traversed exactly once as t increases from α to β, then the length of the curve is • Examples on the board

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