Find an equation for the line tangent to the curve at the...

Find an equation for the line tangent to the curve at the point defined by the given value of $t .$ Also, find the value of $d^{2} v / d x^{2}$ at this point.
$$x=t, \quad y=\sqrt{t}, \quad t=1 / 4$$

Key Concepts

Parametric Equations

Parametric equations define a curve by expressing the coordinates as functions of a parameter. This representation is especially useful for describing curves that may not be easily expressed as functions in a standard Cartesian form. In problems involving tangents and curvature, these equations allow the computation of derivatives with respect to the parameter, which can then be used to analyze the behavior of the curve.

First Derivative of a Parametric Curve

The first derivative in the context of parametric curves, given by dy/dx, is found by dividing the derivative of y with respect to the parameter by the derivative of x with respect to the parameter (i.e., dy/dt divided by dx/dt). This derivative represents the slope of the tangent line to the curve at a specific point, which is essential for determining the tangent line equation.

Tangent Line to a Curve

The tangent line to a curve at a given point is the line that best approximates the curve near that point. Its slope is determined by the first derivative at that point. Once the slope and the coordinates of the point are known, the point-slope form of the line can be used to write the equation of the tangent line.

Second Derivative in Parametric Form

The second derivative, d²y/dx², for a parametric curve provides information about the curvature or concavity of the curve. It is computed by differentiating the first derivative (dy/dx) with respect to the parameter and then dividing by dx/dt. This measure is important in analyzing how the slope of the tangent line changes along the curve, which in turn gives insight into the curve's behavior near the point of tangency.

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