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.
$$x=t+e^{t}, \quad y=1-e^{t}, \quad t=0$$

Key Concepts

Parametric Equations

Parametric equations express the coordinates of the points on a curve as functions of an independent parameter. This representation allows one to describe curves that may not be easily represented by a single function y = f(x) and is especially useful in modeling scenarios where the parameter (often time) plays a key role.

First Derivative in Parametric Form

In the parametric framework, the first derivative (dy/dx) is obtained by dividing the derivative of the y-component with respect to the parameter by the derivative of the x-component with respect to the same parameter (dy/dt divided by dx/dt). This derivative reveals the slope of the tangent line to the curve at any given parameter value.

Tangent Line to a Curve

The tangent line at a specific point on a curve is the straight line that best approximates the curve near that point. By using the computed first derivative to obtain the slope and evaluating the original parametric equations at the given parameter value to get the point of tangency, one can construct the equation of the tangent line using the point-slope form.

Second Derivative in Parametric Form

The second derivative with respect to x (d²y/dx²) measures the curvature of a parametric curve, indicating how the curve is bending at a given point. It is computed by differentiating the first derivative (dy/dx) with respect to the parameter and then dividing by dx/dt, i.e., d²y/dx² = (d/dt (dy/dx))/(dx/dt).

Evaluation at a Specific Parameter Value

Evaluating expressions such as the function itself, its first derivative, or its second derivative at a particular value of the parameter is crucial for obtaining specific numerical information like the coordinates of a point, the slope of the tangent, or the curvature at that point. This step ensures that the derived expressions reflect the behavior of the curve at the desired location.

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