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 $d^{2} y / d x^{2}$
at this point.
$$
x=2 t^{2}+3, \quad y=t^{4}, \quad t=-1
$$

Key Concepts

Parametric Equations

Parametric equations describe a set of related quantities as functions of an independent parameter. Instead of writing y explicitly in terms of x, both x and y are expressed in terms of a third variable, usually denoted by t, which allows for a more flexible representation of curves, particularly when they cannot be represented as functions in the traditional sense.

Tangent Line to a Curve

The tangent line at a point on a curve is the straight line that just touches the curve at that point and has the same instantaneous direction as the curve. For parametric curves, the slope of the tangent is found by taking the derivative dy/dx, which requires calculating the derivatives of y and x with respect to the parameter t and then forming their ratio.

Differentiation of Parametric Equations

Differentiation of parametric equations involves computing the derivatives of the parametric expressions for x and y with respect to the parameter t. The first derivative, dy/dx, is obtained by dividing dy/dt by dx/dt. This process employs the chain rule, linking the rates of change of each variable with respect to the parameter.

Second Derivative in Parametric Form

The second derivative d²y/dx² for a parametric curve measures the curvature of the curve and is found by differentiating the first derivative dy/dx with respect to the parameter t, and then dividing the result by dx/dt. This formula, d²y/dx² = (d/dt(dy/dx))/(dx/dt), accounts for the variation of the slope along the curve.

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