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Section 10.3 – Parametric Equations and Calculus

Section 10.3 – Parametric Equations and Calculus. Derivatives of Parametric Equations. To analyze a parametric curve analytically, it is useful to rewrite the equations in the form . But what if the parametric equations are difficult to convert to a single Cartesian equation? Consider: .

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Section 10.3 – Parametric Equations and Calculus

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  1. Section 10.3 – Parametric Equations and Calculus

  2. Derivatives of Parametric Equations To analyze a parametric curve analytically, it is useful to rewrite the equations in the form . But what if the parametric equations are difficult to convert to a single Cartesian equation? Consider: We must find a way to analyze the curves without having to convert them.

  3. Derivatives of Parametric Equations Let a differentiable parametric curve be defined as and . Consider : By the Chain Rule. OR... So...

  4. Derivatives of Parametric Equations If and are differentiable functions of and , then If is also a differentiable function of , then Nothing is new. All results about derivatives from earlier chapters still apply.

  5. Example 1 Consider the curve defined parametrically by and for . (a) Find the highest point on the curve. Justify your answer. Find dy/dx: Find the critical points. Test the critical points and the endpoints to find the maximum y.

  6. Example 1 (continued) Consider the curve defined parametrically by and for . (b) Find all points of inflection on the curve. Justify your answer. Find d2y/dx2: Check to see if there is a sign change in the second derivative. Find the critical points of the first derivative. 0 2.798 Find the x and y value: is also undefined at the endpoint

  7. White Board Challenge Let and . Find the equation of the tangent line at .

  8. Example 2 Let and . Find: Find dy/dx: (a) The coordinate(s) where the tangent line is vertical. Although t=2 makes the denominator 0, t=0 is the only value that satisfies both conditions. This occurs when: (b) The coordinate(s) where the tangent line is horizontal. Although t=2 makes the numerator 0, t=-2 is the only value that satisfies both conditions. This occurs when:

  9. Example 2 Let and . (c) Prove the relation is differentiable at . Prove that it is Continuous The point (x,y) for t=2 exists The limit exists Since the limits equal the values of the coordinate, the relation is continuous at t=2. Prove the Right Hand Derivative is the same as the Left Hand Derivative (and non-infinite) The one-sided derivatives are equal and non-infinite. Thus the derivative exists, at t=2.

  10. Arc Length of Parametric Curves Let and be continuous functions of . Consider: Regular Arc Length Formula.

  11. Arc Length of Parametric Curves Let be the length of a parametric curve that is traversed exactly once as increases from to . If and are continuous functions of , then:

  12. Example 1 Calculate the perimeter of the ellipse generated by and . We must find the limits for the integral. We already graphed this curve. If the curve starts at , it will complete a cycle at . For most arc length problems, the calculator needs to evaluate the definite integral.

  13. Example 2 A particle travels along the path and . Find the following: (a) The distance traveled during the interval . Use arc length. (b) The displacement during the interval . Coordinate at t=0: Coordinate at t=4: Use the Distance Formula

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