332
Chapter 3 | Derivatives
358. The formula I ( t ) = sin t e t
360. [T] An isotope of the element erbium has a half-life of approximately 12 hours. Initially there are 9 grams of the isotope present. a. Write the exponential function that relates the amount of substance remaining as a function of t , measured in hours. b. Use a. to determine the rate at which the substance is decaying in t hours. c. Use b. to determine the rate of decay at t =4 hours. 361. [T] The number of cases of influenza in New York City from the beginning of 1960 to the beginning of 1961 is modeled by the function N ( t ) =5.3 e 0.093 t 2 −0.87 t , (0≤ t ≤4), where N ( t ) gives the number of cases (in thousands) and t is measured in years, with t =0 corresponding to the beginning of 1960. a. Show work that evaluates N (0) and N (4). Briefly describe what these values indicate about the disease in New York City. b. Show work that evaluates N ′(0) and N ′(3). Briefly describe what these values indicate about the disease in New York City. 362. [T] The relative rate of change of a differentiable function y = f ( x ) is given by 100· f ′( x ) f ( x ) %. One model for population growth is a Gompertz growth function, given by P ( x ) = ae − b · e − cx where a , b , and c are constants. a. Find the relative rate of change formula for the generic Gompertz function. b. Use a. to find the relative rate of change of a population in x =20 months when a =204, b =0.0198, and c =0.15. c. Briefly interpret what the result of b. means. For the following exercises, use the population of New York City from 1790 to 1860, given in the following table.
is the formula for a
decaying alternating current. a. Complete the following table with the appropriate values. t sin t e t
0
(i)
π 2
(ii)
π
(iii)
3 π 2
(iv)
2 π
(v)
5 π 2
(vi)
3 π
(vii)
7 π 2
(viii)
4 π
(ix)
b. Using only the values in the table, determine where the tangent line to the graph of I ( t ) is horizontal. 359. [T] The population of Toledo, Ohio, in 2000 was approximately 500,000. Assume the population is increasing at a rate of 5% per year. a. Write the exponential function that relates the total population as a function of t . b. Use a. to determine the rate at which the population is increasing in t years. c. Use b. to determine the rate at which the population is increasing in 10 years.
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