Math 19. Lecture 2A Calculus ToolkitT. JudsonFall 20041 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 dropping at a rateof 12◦F per minute. When did this cup’s temperature fall to 130◦F? Thesecond cup was at 130◦F after 10 minutes. Cou ld this liquid be coffee?2 Differential EquationsThe most basic type of differential equation has the formdydx= f(x, y).In otherwords, 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 general, 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 object and its environment. 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 B iology• If the true function under discussion j umps in value, then its replace-ment with a continuous function is reasonable wh en the experimentalerror is larger th an any of the jumps.• Once the s tep to a continuous function is made, the step to differen-tiability rarely adds trauma.5 What Do You Need from Calculus?• You need to understand the derivative and what it means.• To be able to compute derivatives of elementary fu nctions.• To understand the definite integral and the relationship between an-tiderivatives and 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 Di fferential Equ ations 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,
View Full Document
Unlocking...