Derivatives Consider the following parametric curves. a....

Derivatives Consider the following parametric curves.
a. Determine dy/dx in terms of t and evaluate it at the given value of $t.$
b. Make a sketch of the curve showing the tangent line at the point corresponding to the given value of $t.$
$$x=3 \sin t, y=3 \cos t ; t=\pi / 2$$

Key Concepts

Parametric Equations

Parametric equations represent curves by expressing the coordinates as functions of an independent parameter. This approach is especially useful when the curve does not represent a function in the classical sense, or when a more flexible description of the path in the plane is required. It also helps in analyzing motion along the curve by associating positions with values of the parameter.

Parametric Differentiation

Parametric differentiation is a method used to compute the derivative dy/dx when both x and y are defined in terms of a third variable, usually denoted as t. This is done by differentiating x and y with respect to t, and then forming the ratio (dy/dt) / (dx/dt), provided that dx/dt is nonzero. This technique is fundamental for determining slopes and analyzing changes along curves represented parametrically.

Tangent Line to a Curve

The tangent line to a curve at a given point represents the line that just touches the curve at that point and has the same instantaneous direction (slope) as the curve. For parametric curves, once the derivative dy/dx is calculated at a particular parameter value, it provides the slope of the tangent, which is essential for sketching the behavior of the curve and understanding its local linear approximation.

Chain Rule

The chain rule is a key concept in calculus for differentiating composite functions. It is employed in the process of parametric differentiation when linking the derivatives of x and y with respect to the parameter. The chain rule ensures that the dependencies of functions on the parameter are correctly managed, allowing for the proper computation of the derivative dy/dx.

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