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Tangent and Velocity Problem Notes

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1 2.1 – Tangent and Velocity Problems (Two problems with one theme) 1. Slope of a tangent line à 2. Instantaneous Velocity à geometric in nature very old – dates back to great Greek scientist Archimedes (287 -212 BC) mechanical in nature more recent – grew out of attempts by Kepler (1571 – 1630), Galileo (1564-1642), Newton (1642-1727) to describe the speed of a moving body. These two problems appear unrelated, but they are actually twins! 2 ∎ Tangent Line How to define the tangent line to a curve 𝒚 = 𝒇(𝒙) at point P? Tangent line to a circle at point P Tangent line to curve y=f(x) at P -line that intersects with the circle at only one point, P ??? *Note that Euclid’s definition for the tangent line to a circle is not sufficient The tangent line to a curve may intersect the curve at more than one point. Intuitive idea: Define the tangent line to a curve y=f(x) at point P to be the line that best approximates the curve near P – i.e. zoom in until the curve at P looks like a line (if possible) Better, but still too vague mathematically ( 𝐶𝑜𝑛𝑐𝑒𝑝𝑡 𝑜𝑓 𝑙𝑖𝑚𝑖𝑡𝑠 𝑝𝑟𝑜𝑣𝑖𝑑𝑒𝑠 𝑎 𝑤𝑎𝑦 𝑡𝑜 𝑔𝑒𝑡 𝑎 𝑚𝑜𝑟𝑒 𝑝𝑟𝑒𝑐𝑖𝑠𝑒 𝑑𝑒𝑠𝑐𝑖𝑟𝑝𝑡𝑖𝑜𝑛 𝑓𝑜𝑟 𝑡ℎ𝑒 𝑡𝑎𝑛𝑔𝑒𝑛𝑡 𝑙𝑖𝑛𝑒 3 Start with -Curve 𝑦 = 𝑓(𝑥) -point P on the curve -Point Q near point P (Q is movable) Dra ...
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Tags: Calculus Problems Slope of a tangent line instantaneous velocity tangent line Slope of the tangent line

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