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Find dy/dx and d²y/dx² as a function of t if x=2sint and y=3cost. For which values of is the curve concave upward?

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Final answer:

To find the first and second derivatives of y with respect to x given the parametric equations x and y as functions of t, differentiate each with respect to t, and then use the chain rule for derivatives. The curve is concave upward where the second derivative is positive.

Step-by-step explanation:

To find the derivatives dy/dx and d²y/dx² as functions of t given the parametric equations x=2sin(t) and y=3cos(t), we first differentiate both x and y with respect to t to get dx/dt and dy/dt, respectively. Then, to find dy/dx, we simply divide dy/dt by dx/dt. To find d²y/dx², we differentiate dy/dx with respect to t and divide by dx/dt.

The curve is concave upward when d²y/dx² is positive. We evaluate the sign of d²y/dx² by finding the second derivative and determining the interval(s) for t where it is positive.

User Seongkuk Han
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