(a) Find the slope of the tangent line to the parametric...

(a) Find the slope of the tangent line to the parametric curve $x=t^{2}+1, y=t / 2$ at $t=-1$ and at $t=1$ without eliminating the parameter.
(b) Check your answers in part (a) by eliminating the parameter and differentiating an appropriate function of $x$.

Key Concepts

Eliminating the Parameter

Eliminating the parameter involves rewriting the parametric equations as a single equation that relates y directly to x. This method can simplify the analysis of the curve and serves as an alternative approach to verify results, such as the slope of the tangent line, by applying traditional differentiation methods to the resulting function.

Differentiation with Respect to the Parameter

Differentiating functions defined parametrically involves finding the derivatives of the individual functions x(t) and y(t) with respect to the parameter t. This process often employs standard differentiation techniques and is crucial in calculating the slope of the tangent line and other related geometric properties.

Parametric Equations

Parametric equations describe a curve by defining both x and y as functions of a third variable, usually denoted as t. This method allows the description of more complex curves that cannot be represented as a single function y = f(x), and sets the stage for analyzing properties such as tangent lines and curvature using the parameter t.

Slope of the Tangent Line (dy/dx) in Parametric Context

When working with parametric equations, the slope of the tangent line at any point on the curve is found by computing the derivative dy/dx, which is obtained by dividing dy/dt by dx/dt. This concept is essential when exploring the geometric behavior of curves, as it provides a means to understand instantaneous rates of change along the parametric path.

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