Calculus problem!! please help!!

This requires the use of RStudio.

Problem 1: Conservation of whooping cranes Whooping Cranes are large, heron-like birds that live in the west/mid-west of America (Alberta and Wisconsin) in the summer, and migrate south (Texas and Florida, respectively) in the winter. A combination of hunting and habitat loss nearly drove them to extinction in the

1930s. In 1967, the Whooping Crane was declared endangered, and it has been the subject of intense conservation efforts ever since. The conservation effort has had limited success due, in part, to the longevity of the birds (they can live to be over

25 years), and their long, difficult migration.

(a) Plot the raw data with the following commands:

t=seq(from = 0, to = 70, by = 10)

N=c(26,32,37,58,79,146,180,283)

plot(t,N)

(b) We begin by investigating whether the population exhibits exponential growth.

We will plot the data on a semilog graph and find the best fit line to log10(N) vs. t. Assuming you have already entered t and N into R (if not use the code above), the commands for making the semilog plot with the data and best fit line are as follows:

logN = log10(N)

F = lm( logN~t )

plot(t,logN)

lines(t,fitted(F))

(c) The equation of the best fit line defines a model; i.e. an equation which gives an approximation to the number of cranes as a function of time. To get the equation for the best-fit line, type in F (what we named the output from the fitting command) at the prompt in R. If the linear fit is of the form y = mx + b, then the coefficient called intercept is b and the other coefficient is the slope. Let N(t) represent the model equation for the number of whooping cranes as a function of time (where time is measured in years since 1940).

Use the equation of the best fit line to find the equation for N(t).

Plot the original data and a graph of N(t) on the same plot. The commands for plotting the data are given in part (a). To add the model, evaluate your expression for N(t) at the same time points, and use the lines command to add the curve to the plot.

(d) Discuss and answer the following (in complete sentences):

Could you predict the number of cranes 10 years in the future? 100 years?

Explain. Would you have confidence in these predictions?

Even though we dont have data for the years between our 10 year mea- surements, the model predicts the number of cranes at any year. In fact, the model predicts the number of cranes down to the minute. Do you think the model is reliable in its predictions of the number of cranes over very short interval