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

step 1 Okay, this question gives you x equals e to the t, Y equals e to the negative t, and they want t

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

Key Concepts

Parametric Equations

Parametric equations describe a curve by expressing the x and y coordinates in terms of a separate variable, known as the parameter. This form is useful when the relationship between x and y is not easily expressed as a single function, allowing for a clearer description of complex or multi-valued curves.

Differentiation of Parametric Equations

To find the slope of a curve defined by parametric equations, you differentiate both x and y with respect to the parameter and then take the ratio dy/dx = (dy/dt) / (dx/dt). This method allows one to calculate the derivative at a particular value of the parameter without the need to eliminate the parameter from the equations.

Tangent Line to a Curve

The equation of the tangent line to a curve at a given point is derived using the point-slope form of a line. For a parametric curve, once the coordinates and the slope (from the derivative) at a specific parameter value are known, the tangent line equation can be constructed. This line approximates the curve near that point.

Elimination of Parameter

Eliminating the parameter involves expressing the relationship between x and y directly, without the intermediary parameter. This process can provide a check on the correctness of the tangent line equation derived from parametric differentiation, by enabling the use of standard differentiation with respect to x.

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