(a) Find the equation of the tangent line to the curve x=e^t, y=e^-t at t=1 without eliminatin

step 1 In this question, we are given the x equal to e to the t, the y equal to e to the minus t. So fr

(a) Find the equation of the tangent line to the curve...

(a) Find the equation of the tangent line to the curve
$$
x=e^{t}, \quad y=e^{-t}
$$
at $t=1$ without eliminating the parameter.
(b) Find the equation of the tangent line in part (a) by eliminating the parameter.

Key Concepts

Parametric Equations

Parametric equations define a curve by expressing the coordinates of points on the curve as functions of an independent parameter. This method of describing curves is particularly useful because it allows for the representation of a wide variety of curve shapes that might be difficult or impossible to describe using a single function in Cartesian coordinates.

Parametric Differentiation

Parametric differentiation is a technique used to compute the derivative dy/dx for a curve represented by parametric equations. It involves differentiating both the x and y components with respect to the parameter and then forming the ratio (dy/dt)/(dx/dt). This method provides the slope at any point on the curve, which is necessary for finding tangent lines.

Tangent Line

The tangent line to a curve at a given point is the straight line that best approximates the curve at that point. Its slope is determined by the derivative at that point, and its equation can be written using the point-slope form. In the context of parametric equations, once the coordinates of the point and the derivative are known, the tangent line can be directly formulated.

Elimination of the Parameter

Eliminating the parameter involves rewriting the system of parametric equations as a single equation in Cartesian coordinates by expressing one variable in terms of the other. This approach can simplify the process of finding derivatives and the tangent line by reducing the problem to one involving standard function differentiation.

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