Questions 4 and 5 please!

4. Suppose a population of deer in a wildlife refuge has a carrying capacity of 1500 deer and its change in population follows the logistic differential equation model. (4 points) a. If the initial population is 200 deer and after 2 years the population is 300 deer, determine the exact value of k in the logistic model and write a formula for population of deer at time , in years. Show your work. b. Use your result from part(a) to find the approximate population after 10 years. c. At what time does the population reach 1200? Show your work. 5. A 500-gal tank initially contains 20 Ib of salt dissolved in 100 gal of water. Brine containing 1 Ib/gal of salt flows into the tank at the rate of 10 gal/min, and the well-stirred mixture flows out of the tank at the rate of 2 gal/min. How much salt does the tank contain when it is full? In order to earn full credit, you must set up a differential equation that models this situation, solve the differential equation by hand, and show all relevant steps. (9 points) 6. Consider the function f(x)=sin"'(x). (4 points) a. Use technology to complete the table of derivatives for the function centered at 0. Use your results to write the Maclaurin polynomial of degree 5 of f (x). Use exact values. Simplify the coefficients. n (%) ORI (%) (0) 0 3 1 4 b. Use ps (x) to approximate the value of sin\"'! (;) . (6 decimal places) 7. Use a Taylor series to approximate the definite integral to within 0.001 of the actual sum using the least possible number of terms. Show all relevant steps. Round your approximation to 5 decimal places. (5 points) j;xoos(Zx?') dx