# HW4 - This is the 21st century. Find the point in the...

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MATH 102, ASSIGNMENT 4 DUE JUNE 9 Problem 1. Find y 0 from the implicit equation x 2 / 3 + y 2 / 3 = 1. Problem 2. Curves described by implicit equations of the form y 2 = x 3 + ax + b , where a and b are ﬁxed numbers, are called “elliptic curves”. They are important objects of study in one of the oldest branches of mathematics, number theory; they feature, for instance, in the proof of Fermat’s Last Theorem. More recently, elliptic curves have been used in cryptography. For each of the following elliptic curves, ﬁnd the number of points on the curve where the tangent line is horizontal: y 2 = x 3 + x , y 2 = x 3 - x , y 2 = x 3 - x + 1 Problem 3. The curve described by the implicit equation x 3 + y 3 = 3 axy , where a is a ﬁxed positive number, is called the “Folium of Descartes”. Descartes - to whom we owe the carte- sian coordinates - was the ﬁrst to discuss this curve. At that moment in time, around 1640, he was unable to ﬁnd the tangent line to the curve at an arbitrary point.
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Unformatted text preview: This is the 21st century. Find the point in the ﬁrst quadrant where the tangent to the curve is horizontal. Problem 4. Air is being pumped in a balloon at a rate of 2 cubic centimeters per minute. How fast is the surface of the balloon changing when the radius is 2 centimeters? Problem 5. A reservoir in the shape of an inverted cone has a radius of 2 meters at the top and a depth of 6 meters. Wine is poured into the reservoir at a rate of 1 m 3 / sec. At what rate is the depth of the wine increasing when the depth is 4 meters? J-RULE: Just answers - “yes”, “5” - won’t get you full marks, but (correctly) justiﬁed answers - “yes, because. ..”, “(some algebra). .. (sweat sweat) . .. 5” - will. 1...
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## This note was uploaded on 01/27/2012 for the course ECON 204 taught by Professor Beryl li during the Spring '11 term at University of Victoria.

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