University:
North Central College
Course:
MATH 152 | Calculus II
Academic year:

2024
Views:

465

Pages:

2
Author:

nesikhiyawl0

Question 1 The loudness L(x), measured in decibels (dB), of a sound of intensity x, measured in watts per square meter, is defined as L(x) = 10log(x/Io), where Io = 10⁻¹² watt per square meter is the least intense sound that a human ear can detected. Determine the loudness, in decibels, of each of the following sounds. Heavy city traffic: intensity of x = 10⁻³ watt per square meter. Answer It is observed that the loudness is a logarithmic function. Hence, the different properties of logarithmic function can be used to find the find the intensity. The different properties of the logarithmic function are found from the properties of the exponential functions as exponential functions are inverse functions of logarithmic functions. Now, to find the intensity at x = 10⁻³ watt per sq meter, substitute this value in the loudness function given by L(x). The logarithmic properties log(xⁿ) = nlog(x) and log(10) = 1 can be used simplify the equation. L(x) = 10 log(x/I₀) L(10⁻³) = 10 log(10⁻³/10⁻¹²) = 10 log(10⁹) = 9 * 10 log(10) = 9 * 10 = 90 dB Question 2 When a < 0 and b>1,y=abˣ models negative exponential growth. a. Write an exponential function that models negative growth. b. Give an example of a situation that could be modeled by your function. c. Explain one difference between negative exponential growth and exponential decay. Answer 18000 = Pe^.06(4) A = Pe^rt 28000 = P(1.27124915) Solve for e.24 3P=6,293.02 Divide 8000 by 1.27124915 Question 3 Determine whether the function given by the table is linear, exponential, or neither. If the function is linear, find a linear function that models the data. If it is exponential, find an exponential function that models the data. x f(x) -1 8/7 0 8 1 56 2 392 Answer The function is linear if there is a common difference between the y-values for a constant increase in the z-values while the function is exponential if there is a common ratio between the y-values for a constant increase in the z-values. Since there is a common ratio of 7 between the y-values for << constant increase of 1 in the x-values, then the function is exponential. Use the exponential model y = abˣ where a is the initial value (where x= 0) and b is the common ratio. Substitute a = 8 and b= 7 so: y = 8(7)ˣ or f(x) = 8(7)ˣ Question 4 How do you tell which function-linear, exponential, or quadratic-best represents data? Answer A linear model will likely fit well if the first differences between consecutive y-values are constant or very close to constant. If the 1st differences are not constant then use those numbers to find the 2nd differences. If they are constsnt or nearly constant then a quadratic model will probably fit well/ If the ratios of consecutive y-values are constant or very nearly constant then an exponential model will likely work well. Question 5 How can you tell if an exponential function models growth or decay? Answer An exponential function of the form y=abˣ is an exponential growth function is b>1 and an exponential decay function is 0 < b < 1