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.
$$x=e^{\prime}, \quad y=t e^{\prime}, \quad z=t e^{t^{2}} ; \quad(1,0,0)$$

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Key Concepts

Parametric Equations of Curves

Parametric equations express the coordinates of the points on a curve as functions of a parameter, typically denoted by t. This form of representation allows the description of complex curves and motions that are difficult or impossible to express with a single function of x or y. It is a fundamental concept in calculus and geometry for modeling the trajectory of moving objects and analyzing curves in space.

Differentiation of Vector-Valued Functions

Differentiating a vector-valued function with respect to its parameter yields the tangent (or velocity) vector to the curve at a given point. This process involves computing the derivative of each component of the function. The resulting tangent vector provides both the direction and rate of change of the curve at that point, which is essential for understanding the instantaneous behavior of the curve.

Tangent Line to a Curve in Space

The tangent line at a given point on a curve is the straight line that best approximates the curve near that point. Once the tangent vector at the point is found, the parametric equations of the tangent line can be constructed using the point and the direction provided by the tangent vector. This gives a linear representation of the curve’s instantaneous direction at the specified point.

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