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

Key Concepts

Second Derivative in Parametric Form

The second derivative, d²y/dx², provides information about the curvature of the curve. For a curve defined parametrically, it is found by differentiating the first derivative dy/dx with respect to the parameter and then dividing by dx/dt. This process reveals how the slope of the curve changes, giving insights into the concavity or convexity at a given point.

Tangent Line to a Curve

The tangent line to a curve at a given point is the straight line that best approximates the curve near that point. For curves defined parametrically, the slope of the tangent line is given by dy/dx computed at the specific parameter value. With the slope and the point on the curve, the equation of the tangent line can be formed using the point-slope formula.

Differentiation in Parametric Form

When working with parametric equations, derivatives are computed with respect to the parameter. The first derivative dy/dx is obtained by taking the derivative of y with respect to t and dividing by the derivative of x with respect to t, assuming that dx/dt is not zero. This technique applies the chain rule and is fundamental for analyzing the slope of the tangent to the curve.

Parametric Equations

Parametric equations express the coordinates of the points on a curve as functions of a parameter. This approach is especially useful for curves that are difficult or impossible to represent using a single function y = f(x). Each coordinate, such as x and y, is given separately as a function of an independent parameter, often denoted by t.

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