The slope of the normal line to the curve x=3 cost, y=3 sint at t=π/ 4 is -1 . Justify your answ

step 1 The slope of the normal line to the curve x equals 3 cosine t, y equals 3 sine t, t equals pi ov

The slope of the normal line to the curve x=3 cost, y=3 sint...

Key Concepts

Tangent and Normal Lines

The tangent line to a curve at a given point touches the curve without crossing it and its slope is given by the derivative dy/dx. The normal line at that point is perpendicular to the tangent line; hence, its slope is the negative reciprocal of the slope of the tangent line. When the tangent line has a slope of 1, for example, the corresponding normal line will have a slope of -1.

Geometry of the Circle

For a circle centered at the origin, the radius that connects the center to a point on the circle is always normal (perpendicular) to the tangent at that point. At t = ?/4, the coordinates are equal, and the slope of the line joining the origin and the point is tan(?/4) = 1. Since this radius is the normal to the circle at that point, the slope of the normal is 1, confirming that the tangent line has the negative reciprocal slope, -1.

Parametric Equations

Parametric equations express the coordinates of the points on a curve as functions of a parameter, usually denoted by t. They provide a useful way to represent curves where the relationship between x and y might be difficult to describe explicitly. In this context, x = 3 cos t and y = 3 sin t represent a circle of radius 3 centered at the origin, with the parameter t corresponding to the angle in standard position.

Differentiation of Parametric Equations

To find the slope of a curve given in parametric form, we differentiate the coordinate functions with respect to the parameter t. The slope of the tangent line to the curve is calculated by taking the quotient of the derivatives, dy/dx = (dy/dt) / (dx/dt), which gives insight into the rate of change in y with respect to x at any point on the curve.

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