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

(a) Find the slope of the tangent line to the parametric curve $x=t / 2, y=t^{2}+1$ 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

Elimination of the Parameter

Eliminating the parameter involves rewriting the parametric equations to express y directly as a function of x. This Cartesian representation can then be differentiated using standard techniques, providing a way to verify results obtained via parametric differentiation.

Parametric Representation

Parametric representation involves expressing the coordinates of points on a curve as functions of a third variable (the parameter). This method is particularly useful for describing curves that cannot be represented by a single function y = f(x), and it allows for a more flexible description of curves in the plane.

Tangent Line and Slope

The tangent line to a curve at a point is the straight line that just touches the curve at that point and indicates the direction in which the curve is heading. The slope of this tangent line is determined by the derivative of the curve at that point, reflecting the instantaneous rate of change.

Derivative of Parametric Curves

When a curve is defined parametrically by functions x(t) and y(t), the slope of the tangent line at any point is given by dy/dx, which is calculated using the chain rule as (dy/dt) divided by (dx/dt). This approach is essential when differentiating curves whose x and y coordinates are expressed as functions of a parameter.

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