For each of the following pairs of functions, either f(n) is O(g(n)), f(n) is (g(n)), or f(n) = Ⓒ(g(n)). Determine which relationship is correct. Show work. Note: You should use limits as shown in the slides from lecture. a) f(n) = log₂ (n³), g(n) = n³ log₂ n b) f(n) = 5¹, g(n) = 5n+3 c) f(n) = 4", g(n) = 9n d) f(n)= 27, g(n) = log₂ 2048

The half-life of plutonium-244 is 80,000,000 years. Find function gives the amount of carbon-14 remaining as a function of time, measured in years.

What is the composition of two functions, fg ?

Find the average rates of change of f(x)=x2+2x (a) from x1=3 to x2=2 and (b) from x1=2 to x2=0.

The number of bacteria in a refrigerated food product is given by N(T)=23T256T+1,3T33 ,where T is the temperature of the food. When the food is removed from the refrigerator, the temperature is given by T(t)=5t+1.5 , where t is the time in hours. a. Find the composite function N(T(t)) . b. Find the time (round to two decimal places) when the bacteria count reaches 6,752.

Table 6 shows the population, in thousands, of harbor seals in the Wadden Sea over the years 1997 to 2012. a. Let x represent time in years starting with x=0 for the year 1997. Let y represent the number of seals in thousands. Use logistic regression to fit a model to these data. b. Use the model to predict the seal population for the year 2020. c. To the nearest whole number, what is the limiting value of this model?

When examining the formula of a function that is the result of multiple transformations, how can you tell a reflection with respect to the x-axis from a reflection with respect to the y-axis?

Sketch the graph of fx=5e0.17x.

What is the y -intercept on the graph of the logistic model given in the previous exercise?

In this problem you are asked to find a function that models in real life situation and then use the model to answer questions about the guidelines on page 273 to help you. Minimizing Time A man stands at a point A on the bank of a straight river, 2 mi wide. To reach point B, 7 mi downstream on the opposite bank, he first rows his boat to point Pon the opposite bank and then walks the remaining distances x to B, as shown in the figure. He can row at a speed or 2 mi/h and walk as a speed of mi/h (a) Find a function that models the time needed for the trip. (b) Where should he land so that he reaches B as soon as possible?