By Marcel B. Finan
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Additional resources for A Reform Approach to Business Calculus
022. 022t . Next,we want to find the time when P (t) = 10. 022t = 10. 37. 3. 022 Next, in order to convert from Q(t) = bekt to Q(t) = bat we let a = ek . 35. 35)t . 886)t . Solution. 4%. 1%. Recommended Problems (pp. 42 - 3): 3, 5, 9, 11, 15, 17, 19, 22, 23, 26, 27, 31. 37 7 Exponential Growth and Decay In this section, we consider some applications of exponential functions. Doubling Time In some exponential models one is interested in finding the time for an exponential growing quantity to double.
60 12 The Derivative Function Recall that a function f is differentiable at x if the following limit exists f (x + h) − f (x) . h→0 h f (x) = lim (2) Thus, we associate with the function f , a new function f whose domain is the set of points x at which the limit (2) exists. We call the function f the derivative function of f. The Derivative Function Graphically Since the derivative at a point represents the slope of the tangent line then one can obtain the graph of the derivative function from the graph of the original function.
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. We call the constant k the continuous growth rate. 072)t is rewritten as f (t) = 3ekt , find k. Solution. 072. 695. 2% per year. Find a formula for the population P (t) as a function of the year t. When does the population reach 10 million? 36 Solution. 022. 022t . Next,we want to find the time when P (t) = 10.