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HARVARD MATH 19 - Lecture 2

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Math 19. Lecture 2A Calculus ToolkitT. JudsonFall 20051 The Coffee ProblemTwo identical cups of dark liquid are left in a 70◦F laboratory cool. At timet = 0, the first cup’s temperature was 190◦F, and was d ropping at a rateof 12◦F per minute. When did this cup’s temperature f all to 130◦F? Thesecond cup was at 130◦F after 10 minutes. Could this liquid be coffee?2 Differential EquationsThe most basic type of differential equation has the formdydx= f(x, y).In other words, if we know how a function changes, can we find the function?A solution to a differential equationdydx= f (x, y).is a function y = y(x) that satisfies the equation. For example, y = x4/4+Cis a solution to y′= x3, where C is an arbitrary constant. If we specify aninitial condition, y(0) = y0, then we can find a unique solution.In ge ne ral, differential equations are very difficult to solve. However, itis quite easy to check if a function is a actually a solution. For example,y(t) = 70 + 120e−0.1tis a solution to the differential equationdydt= −110(y − 70),y(0) = 190.13 Newton’s Law of CoolingAn object cools at a rate proportional to the temperature difference betweenthe objec t and its envi ronment. As a differential equation, Newton’s Law ofCooling can be stated asy′= k(y − Te).If we know the initial temperature at t = 0, then we have an initial valueproblem, which has a unique solution. We can state the initial condition asy(0) = T0.4 Continuity and Differentiability in Biology• If the true function under discussion jumps in value, then its replace-ment with a continuous function is reasonable when the experimentalerror is larger than any of the jumps.• Once the step to a continuous function is made, the step to d ifferen-tiability r arely adds trauma.5 What Do You Need from Calculus?• You need to und erstand the derivative and what it means.• To be able to compute derivatives of elementary functions.• To understand the definite integral and the relationship between an-tiderivatives an d the definite integral.• To be able to compute antiderivatives of elementary functions.• To understand and be able to apply Taylor’s Theorem.• To understand how a curve can be parameterized.Homework• None2Readings and References• C. Taubes. Modeling Differential Equations in Biology. Prentice Hall,Upper Saddle River, NJ, 2001, pp. 7, 81–85.• A. Ostebee and P. Zorn. Calculus from Graphical, Numerical, andSymbolic Points of View, second edition. Harcourt, Orlando, FL, 2002.• J. S tewart. Calculus and Concepts, second edition. Brooks/Cole, Bel-mont, CA,


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HARVARD MATH 19 - Lecture 2

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