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By Marcel B. Finan

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Suppose that you deposit P dollars into a saving account that pays annual interest r and the bank agrees to pay the interest at the end of each time period( usually expressed as a fraction of a year). ). Thus, at the end of the first period the balance will be B=P+ r r P =P 1+ . n n At the end of the second period the balance is given by B =P 1+ r r r r + P 1+ =P 1+ n n n n 2 . e. after n periods, is B =P 1+ r n n . If the investment extends to another year than the balance would be given by r 2n P 1+ .

For example, ln 11 = ln 2 whereas ln11 = − ln12 . 3 Sketch the graphs of the functions y = ln x and y = ex on the same axes. 35 Solution. The graphs of y = ln x and y = ex are reflections of one another across the line y = x as shown in Figure 15. Figure 15 Continuous Growth Rate and the Number e ¯ When writing y = bet then we say that y is an exponential function with base e. Suppose that Q(t) = bat . Then a = ek where k = ln a. Thus, Q(t) = b(ek )t = bekt . Note that if k > 0 then ek > 1 so that Q(t) represents an exponential growth and if k < 0 then ek < 1 so that Q(t) is an exponential decay.

Solution. (x) f (x) = limh→0 f (x+h)−f h = limh→0 k−k = 0. h Thus, f (x) = 0. 4 (Derivative of a Linear Function) Find the derivative of the linear function f (x) = mx + b. Solution. (x) f (x) = limh→0 f (x+h)−f h = limh→0 m(x+h)+b−(mx+b) h = limh→0 mh = m. h Thus, f (x) = m. Recommended Problems (pp. 104 - 6): 5, 6, 7, 9, 11, 13, 15, 19, 21. 63 13 Leibniz Notation for The Derivative When dealing with mathematical models that involve derivatives it is convenient to denote the prime notation of the derivative of a function y = f (x) dy by dx .

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A Reform Approach to Business Calculus by Marcel B. Finan


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