Find formulas for the following animal populations. (a) In year t = 0, this population

Graph y = t + 5 sin t and y = t on 0 t 2. Where do the two graphs intersect if t is not restricted to 0 t 2?

Graph the function f(x) = ex(2 cos x + sin x) on the interval 0 x 12. Use a calculator or computer to find the maximum and minimum values of the function.

Find formulas for the following animal populations. (a) In year t = 0, this population of 5000 began growing at the rate of 300 per year. (b) In year t = 0, this population of 3200 began growing at an annual rate of 4% per year. (c) This population oscillates over a 5-year period between a low of 1200 in year t = 0 and a high of 3000.

For time t in months since January 1, a national park rabbit population is modeled by R = f(t) = 5000 cos(t/6) + 10,000. (a) Evaluate and interpret f(3) f(2). (b) Find and interpret all solutions to the equation f(t) = 12,000 0 t 12

From its past behavior, John knows that the value of a stock has a cyclical component that increases for the first three months of each year, falls for the next six, and rises again for the last three. In addition, inflation adds a linear component to the stocks price. John is seeking a model of the form f(t) = mt + b + A sin t 6 . with t in months since Jan 1. He has the following data: Date Jan 1 Apr 1 Jul 1 Oct 1 Jan 1 Price $20.00 $37.50 $35.00 $32.50 $50.00 (a) Find values of m, b, and A so that f fits the data. (b) During which month(s) does this stock appreciate the most? (c) During what period each year is this stock actually losing value?

A power company serves two different cities, City A and City B. The power requirements of both cities vary in a predictable fashion over the course of a typical day. (a) At midnight, the power requirement of City A is at a minimum of 40 megawatts. (A megawatt is a unit of power.) By noon the city has reached its maximum power consumption of 90 megawatts and by midnight it once again requires only 40 megawatts. This pattern repeats every day. Find a possible formula for f(t), the power, in megawatts, required by City A as a function of t, in hours since midnight. (b) The power requirements, g(t) megawatts, of City B differ from those of City A. For t, in hours since midnight, g(t) = 80 30 sin  12 t  . Give the amplitude and the period of g(t), and a physical interpretation of these quantities. (c) Graph and find all t such that f(t) = g(t), 0 t < 24. Interpret your solution(s) in terms of power usage. (d) Why should the power company be interested in the maximum value of the function h(t) = f(t) + g(t), 0 t < 24? What is the approximate maximum of this function, and approximately when is it attained? (e) Find a formula for h(t) as a single sine function. What is the exact maximum of this function?

Let f(t) = cos(et ), where t is measured in radians. (a) Note that f(t) has a horizontal asymptote as t . Find its equation, and explain why f has this asymptote. (b) Describe the behavior of f as t . Explain why f behaves this way. (c) Find the vertical intercept of f. (d) Let t1 be the least zero of f. Find t1 exactly. [Hint: What is the smallest positive zero of the cosine function?] (e) Find an expression for t2, the least zero of f greater than t1.

Let f(x) = sin  1 x  for x > 0, x in radians. (a) f(x) has a horizontal asymptote as x . Find the equation for the asymptote and explain carefully why f(x) has this asymptote. (b) Describe the behavior of f(x) as x 0. Explain why f(x) behaves in this way. (c) Is f(x) a periodic function? (d) Let z1 be the greatest zero of f(x). Find the exact value of z1. (e) How many zeros do you think the function f(x) has? (f) Suppose a is a zero of f. Find a formula for b, the largest zero of f less than a.

Derive the following identity used in an electrical engineering text3 to represent the received AM signal function. Note that A, c, d are constants and M(t) is a function of time, t. r(t)=(A + M(t)) cos ct + I cos(c + d)t = ((A + M(t)) + I cos dt) cos ct I sin dt sin ct

In the July 1993 issue of Standard and Poors Industry Surveys the editors stated:4 The strength (of sales) of video games, seven years after the current fad began, is amazing . . . . What will happen next year is anything but clear. While video sales ended on a strong note last year, the toy industry is nothing if not cyclical. (a) Why might sales of video games be cyclical? (b) Does s(t) = a sin(bt), where t is time, serve as a reasonable model for the sales graph in Figure 9.23? What about s(t) = a cos(bt)? (c) Modify your choice in part (b) to provide for the higher amplitude in the years 19851992 as compared to 19791982. (d) Graph the function created in part (c) and compare your results to Figure 9.23. Modify your function to improve your approximation. (e) Use your function to predict sales for 1993.

An amusement park has a giant double Ferris wheel as in Figure 9.24. The double Ferris wheel has a 30-meter rotating arm attached at its center to a 25-meter main support. At each end of the rotating arm is attached a Ferris wheel measuring 20 meters in diameter, rotating in the direction shown in Figure 9.24. The rotating arm takes 6 minutes to complete one full revolution, and each wheel takes 4 minutes to complete a revolution about that wheels hub. At time t = 0 the rotating arm is parallel to the ground and your seat is at the 3 oclock position of the rightmost wheel. (a) Find a formula for h = f(t), your height above the ground in meters, as a function of time in minutes. [Hint: Your height above ground equals the height of your wheel hub above ground plus your height above that hub.] (b) Graph f(t). Is f(t) periodic? If so, what is its period? (c) Approximate the least value of t such that h is at a maximum value. What is this maximum value?

A rope has one free end. If we give the free end a small upward shake, a wiggle travels down the length of the rope. If we repeatedly shake the free end, a periodic series of wiggles travels down the rope. This situation can be described by a wave function: y(x, t) = A sin(kx t). Here, x is the distance along the rope in meters; y is the displacement distance perpendicular to the rope; t is time in seconds; A is the amplitude; 2/k is the distance from peak to peak, called the wavelength, , measured in meters; and 2/ is the time in seconds for one wavelength to pass by. Suppose A = 0.06, k = 2, and = 4. (a) What is the wavelength of the motion? (b) How many peaks of the wave pass by a given point each second? (c) Construct the graph of this wave from x = 0 to x = 1.5 m when t is fixed at 0. (d) What other values of t would give the same graph as the one found in part (c)?