Beyond Calculus - CALCULUS - MATHEMATICS IN HISTORY - Mathematics for the liberal arts

Mathematics for the liberal arts (2013)

Part I. MATHEMATICS IN HISTORY

Chapter 4. CALCULUS

4.20 Beyond Calculus

In this chapter we have seen that calculus is the mathematics we use to understand change. When quantities change, derivatives tell us the rate. If we know a rate, integrals allow us to transform that knowledge into a total change. Our study has allowed us to maximize and minimize functions, to analyze the history of fossil fuel use, and even to compute (roughly) the height of a rocket.

Yet as much as we have learned, our view of calculus has been very one-dimensional, literally. We worked solely with functions of a single variable. The world is filled with beautiful complicated things that can’t be expressed as functions of one variable. For example, when a rocket launches into orbit, it doesn’t fly straight up as we assumed in Section 4.17. It follows some arc into space that results in an elliptical shaped orbit. Truly modeling the height of a rocket requires a three-dimensional view and functions with variables x, y, z, and t, along with corresponding multidimensional calculus principles. There are whole courses of study that take the ideas of calculus and put them in a multi-variable context.

Many of the physical models we use for the universe around us are based on relationships defined in terms of derivatives. For example, Newton’s Law of Cooling says that the rate of change in the temperature of an object is proportional to the difference in temperature between the object and its surroundings. Informally, it says that very hot objects cool at a faster rate than slightly hot objects (but it still takes hot objects longer to completely cool since they have further to go). Formally, Newton’s Law of Cooling is a differential equation, an equation containing derivatives of a function,

images

To solve a differential equation is to find a function y that makes the equation true. For example, in the equation above, y = y(t) = A + Bekt works. Take a derivative for yourself and check. After a first course in calculus, many people go on to one or more courses in differential equations.

One can also study calculus where the functions are defined on the complex numbers (i.e., numbers that have both real and imaginary parts). The complex numbers are an especially beautiful domain in which to do calculus, and many elegant results exist in this context. One elegant fact is that any function that has a (single) derivative defined at every complex number is guaranteed to have a power series, which is a way of saying that it can be written as an infinite polynomial. For example, the exponential function has a power series,

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Calculus has helped us gain understanding in nearly every domain of mathematics, including number theory, geometry, game theory, topology, numerical analysis, and probability. It is an essential tool for physics, chemistry, biology, engineering, economics, and statistics. It is one of humankind’s most powerful and awe-inspiring creations.

EXERCISES

4.100 Let y(t) = 70 + 20e–2t, which might describe the temperature of an object that cools from 90°F to room temperature at 70°F.

a) Find 2(70 – y(t)).

b) Find images and verify that you get the same answer as in part (a).

4.101 Let y(t) = A + Bekt, where A, B, and k are constants.

a) Find k(Ay(t)).

b) Find images and verify that you get the same answer as in part (a).

4.102 Let images terms in the power series for ex.

a) Compute T(0), and verify that you get e0 = 1.

b) Compute T(1), and verify that the answer you get is close to e = e1 ≈ 2.718281828459045.

c) Compute T(2), and compare to the value of e2 from a calculator or computer.


1W. M. Priestley, “Sherlock Holmes Meets Pierre de Fermat,” Calculus: A Liberal Art, second edition, Springer-Verlag, New York, 1998.