61-66 - Linear Functions A linear function is described eitherverbally, numerically, or graphically. Express f in the formf(x) = ax + b.61. The function has rate of change -2 and initial value 3.62. The graph of the function has slope and y-intercept – 1.63.64.f(x)f(x)36.525.5245364.54118465.66.0.mn7 92.2.

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Algebra & Trigonometry with Analytic Geometry

13th Edition

ISBN:9781133382119

Author:Swokowski

Publisher:Swokowski

Chapter5: Inverse, Exponential, And Logarithmic Functions

Section5.2: Exponential Functions

Problem 70E

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Decay of Litter Litter such as leaves falls to the forest floor, where the action of insects and bacteria initiates the decay process. Let A be the amount of litter present, in grams per square meter, as a function of time t in years. If the litter falls at a constant rate of L grams per square meter per year, and if it decays at a constant proportional rate of k per year, then the limiting value of A is R=L/k. For this exercise and the next, we suppose that at time t=0, the forest floor is clear of litter. a. If D is the difference between the limiting value and A, so that D=RA, then D is an exponential function of time. Find the initial value of D in terms of R. b. The yearly decay factor for D is ek. Find a formula for D in term of R and k. Reminder:(ab)c=abc. c. Explain why A=RRekt.
Average Rate of change A fucntion f is given. (a) Find the average rate of change of f between x=0 and x=0 ,and the average rate of change of f between x=15 and x=50 . (b) Were the two average rates of change that you found in part (a) the same? (c) Is the function linear? If so, what is its rate of change? f(x)=12x6
Sales Growth In this exercise, we develop a model for the growth rate G, in thousands of dollars per year, in sales of the product as a function of the sales level s, in thousands of dollars. The model assumes that there is a limit to the total amount of sales that can be attained. In this situation, we use the term unattained sales for difference this limit and the current sales level. For example, if we expect sales grow to 3 thousand dollars in the long run, then 3-s is the unattained sales. The model states that the growth rate G is proportional to the product of the sales level s, and the unattained sales. Assume that the constant of proportionality is 0.3 and that the sales grow to 2 thousand dollars in the long run. a.Find the formula for unattained sales. b.Write an equation that shows the proportionality relation for G. c.On the basis of the equation from the part b, make a graph of G as a function of s. d.At what sales level is the growth rate as large as possible? e.What is the largest possible growth rate?
Spawner-Recruit Model In fish management it is important to know the relationship between the abundance of the spawners also called the parent stock and the abundance of the recruitsthat is, those hatchlings surviving to maturity. According to the Ricker model, the number of recruits R as a function of the number of spawners P has the form R=APeBp for some positive constants A and B. This model describes well a phenomenon observed in some fisheries: A large spawning group can actually lead to a small group of recruits. In a study of the sockeye salmon, it was determined that A=4 and B=0.7. Here we measure P and R in thousands of salmon. a. Make a graph of R versus P for the sockeye salmon. Assume there are at most 3000 spawners. b. Find the maximum number of salmon recruits possible. c. If the number of recruits R is greater than the number of spawners P, then the difference R-P of the recruits can be removed by fishing, and next season there will once again be P spawners surviving to renew the cycle. What value of P gives the maximum value of R-P, the number of fish available for removal by fishing?
Average Rate of change A fucntion f is given. (a) Find the average rate of change of f between x=0 and x=0 , and the average rate of change of f between x=15 and x=50 .(b) Were the two average rates of change that you found in part (a) the same? (c) Is the function linear? If so, what is its rate of change? f(x)=83x
Learning Curve A learning curve is a graph of a function P(t) that measures the performance of someone learning a skill as a function of the training time t. At first, the rate of learning is rapid. Then, as performance increases and approaches a maximal value M, the rate of learning decreases. It has been found that the function P(t)=MCekt where k and C are positive constants and CM is a reasonable model for learning. a Express the learning time t as a function of the performance level P. b For s pole-vaulter in training, the learning curve is given by P(t)=2014e0.024t where P(t) is the height he is able to pole-vault after t months. After how many months of training is he able to vault 12 ft? c Draw a graph of the learning curve in part b.
On June 1st, a company has $4,000,000 profit.If the company then loses 150,000 dollars perday thereafter in the month of June, what is thecompany’s profit nthday after June 1st?
Filling a PondA large koi pond is filled from a garden hoseat the rate of 10 gal/min.Initially,the pond contains 300 galof water. (a) Find a linear function V that models the volume of waterin the pond at any time t . (b) If the pond has a capacity of 1300 gal, how long does it take to completely fill the pond?
Minimizing a Distance When we seek a minimum or maximum value of a function, it is sometimes easier to work with a simpler function instead. Suppose g(x)=f(x) where f(x)0 for all x. Explain why the local minima and maxima of f and g occur at the same values of x. Let gx be the distance between the point 3,0 and the point (x,x2) on the graph of the parabola y=x2. Express g as a function of x. Find the minimum value of the function g that you found in part b. Use the principle described in part a to simplify your work.

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