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Let me share with you something I found particularly weird when I was a student first learning calculus. Lets say you have a circle with radius 5 centered at the origin of the xy-coordinate plane, which is defined using the equation x^2 + y^2 = 5^2. That is, all points on this circle are a distance 5 from the origin, as encapsulated by the pythagorean theorem with the sum of the squares of the legs of this triangle equalling the square of the hypotenuse, 52. And suppose you want to find the slope of a tangent line to this circle, maybe at the point (x, y) = (3, 4). Now, if youre savvy with geometry, you might already know that this tangent line is perpendicular to the radius line touching that point. But lets say you dont already know that, or that you want a technique that generalizes to curves other than circles. As with other problems about slope of tangent lines, they key thought here is to zoom in close enough that the curve basically looks just like its own tangent line, the
