A tangent line to a circle intersects the circle at exactly one point on its circumference. The radius drawn from the center of the circle to the point of tangency is always perpendicular to the tangent line.
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To find the equation of the tangent line to a polar curve at a particular point, we’ll first use a formula to find the slope of the tangent line, then find the point of tangency (x,y) using the polar-coordinate conversion formulas, and finally we’ll plug the slope and the point of tangency into the point-slope formula for the equation of a tangent line.
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There’s a special relationship between two secants that intersect outside of a circle. The length outside the circle, multiplied by the length of the whole secant is equal to the outside length of the other secant multiplied by the whole length of the other secant.
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When a problem asks you to find the equation of the tangent line, you’ll always be asked to evaluate at the point where the tangent line intersects the graph. You’ll need to find the derivative, and evaluate at the given point.
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The definition of the derivative, also called the “difference quotient”, is a tool we use to find derivatives “the long way”, before we learn all the shortcuts later that let us find them “the fast way”.
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