49-52= Find parametric equations for the tangent line to the...

$49-52=$ Find parametric equations for the tangent line to the
curve with the given parametric equations at the specified point.
$$x=\sqrt{t^{2}+3}, \quad y=\ln \left(t^{2}+3\right), \quad z=t ; \quad(2, \ln 4,1)$$

Key Concepts

Evaluating Derivatives at a Specific Point

To find the tangent line at a particular point on a parametric curve, one must compute the derivatives of the parametric functions at the corresponding parameter value. This evaluation yields the specific tangent vector at that point, which is then used to define the slope and direction of the tangent line.

Parametric Equations for Curves

This concept involves representing curves in space by expressing each coordinate as a function of an independent parameter. It allows for the description of complex curves that may not be easily expressed as simple functions of x or y alone, enabling the exploration of their properties in a multidimensional context.

Tangent Vector Calculation

The tangent vector to a parametric curve is found by differentiating the coordinate functions with respect to the parameter. This derivative provides the direction in which the curve is heading at any given point, and is crucial in determining the instantaneous rate of change of the curve’s position.

Parametric Equations for a Line

Once a point on the curve and a tangent vector at that point are determined, the tangent line can be expressed parametrically. This involves writing each coordinate of the line as the sum of the base point and the product of the tangent component with an independent scalar parameter, thereby providing a complete description of the line in space.

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