(a) Find the equation of the tangent line to the curve x=2 t+4, y=8 t^2-2 t+4 at t=1 without e

step 1 This question is looking for a tangent line to a parametric curve, both with and without removin

(a) Find the equation of the tangent line to the curve x=2...

Key Concepts

Parametric Equations

Parametric equations describe a curve by expressing the coordinates of the points on the curve as functions of an independent parameter. This representation is useful because it can simplify the description and analysis of curves, particularly when the curve cannot be represented as a single-valued function in the Cartesian plane.

Differentiation of Parametric Equations

When dealing with parametric equations, finding the slope of the tangent line requires the use of derivatives relative to the parameter. The slope is found by taking the derivative of y with respect to the parameter and dividing it by the derivative of x with respect to the parameter, using the formula dy/dx = (dy/dt) / (dx/dt). This method circumvents the need to eliminate the parameter.

Tangent Line Equation

The tangent line to a curve at a given point is defined as the line that just 'touches' the curve at that point and has the same instantaneous rate of change as the curve. Once the slope at the point (obtained using parametric differentiation) and the coordinates of the point (by substituting the parameter value into the parametric equations) are computed, the equation of the tangent line can be written using the point-slope form.

Eliminating the Parameter

Eliminating the parameter involves expressing one variable purely in terms of another to obtain a Cartesian equation of the curve. This method provides an alternative approach to verify results obtained from parametric differentiation by converting the curve into a familiar form and then applying standard differentiation techniques.

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