2311lectureoutline3.7 - These are just a few of the...

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MAC2311, Calculus I 3.7 Rates of Change in the Natural and Social Sciences We know that the derivative of y = f(x) can be interpreted as the instantaneous rate of change of y with respect to x . In Review, we know that the difference quotient = 2 1 2 1 ( ) ( ) f x f x y x x x - Δ = Δ - = the average rate of change of y with respect to x , but when we let 0 x Δ → , 2 1 0 0 2 1 ( ) ( ) lim lim ( ) x x f x f x y f x x x x Δ → Δ → - Δ = = Δ - , we got the derivative of f(x) which is the instantaneous rate of change of y with respect to x. We will now look at just a few of the applications of this idea in the Natural and Social Sciences. Physics: velocity, density, current, power, and temperature gradient Chemistry: rate of reaction and compressibility Biology: rate of growth and blood velocity gradient Economics: Marginal cost, marginal demand, marginal revenue and marginal profit Geology: rate of heat flow Psychology: rate of improvement of performance Sociology: rate of spread of a rumor
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Unformatted text preview: These are just a few of the applications of a valuable mathematical concept the derivative. In each case, we will be examining the rate at which something is changing. This is an illustration of the fact that part of the power of mathematics lies in its abstractness. A single abstract mathematical concept (such as the derivative) can have different interpretations in each of the sciences. When we develop the properties of the mathematical concept once and for all, we can then turn around and apply these results to all of the sciences. This is much more efficient than developing properties of special concepts in each separate science. The French mathematician Joseph Fourier (1768-1830) put it succinctly: Mathematics compares the most diverse phenomena and discovers the secret analogies that unite them. Try exercises 2, 8, 10, 14, 20a, 26, 28a on pages 230-232 in your text....
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This note was uploaded on 12/12/2011 for the course MAC 2311 taught by Professor Teague during the Spring '11 term at Santa Fe College.

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