Consider a polar curve such as this one. The distance from the pole to the curve is shown as a ray which is defined by the angle q. Therefore, this distance is function of q.

Recall that we can describe the curve in terms of x and y using parametric equations.

To determine the slope of a tangent to this polar curve, we can use the parametric form of the derivative.

When q equals p/3, the tangent is equal to zero. Therefore, the tangent line is horizontal.

Let's look at the general form of the parametric equations. If we differentiate x and y with respect to q, we can derive an equation for finding the slope of the tangent in terms of only r and q.

Consider this polar curve. We can determine the tangent slope directly without converting to rectangular coordinates.

We can substitute for r and dr/dq in the slope equation.

Let's look at a graph of this polar curve. When q is equal to zero, the slope is equal to infinity. Therefore, the tangent is vertical.

This polar curve is known as the spiral of Archimedes. Use the slider to vary the value of q. Notice how the length of the ray representing r changes as the point on the curve approaches the pole at the origin.

At the pole, the slope is just tan q. For this spiral curve, the slope is equal to zero when r is equal to zero. The tangent lies along the x-axis.

There can be multiple tangents at the pole. This curve has two tangents at the pole.

There are two values of q for which r equals zero. These produce the two tangent lines at the pole.

What is the slope of the tangent line whenq equals p/3? Click the "Submit" button after entering your answer.

What is the derivative dr/dq ? Click the "Submit" button after selecting your answer.

What is the slope of the tangent when q is equal to zero? Click the "Submit" button after selecting your answer.

If the derivative dr/dq is nonzero, what is the equation of the slope of the tangent at the pole? Click the "Submit" button after selecting your answer.

What is the value of the positive slope? Click the "Submit" button after entering your answer.

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