Use the Newton forward difference formula to construct interpolating polynomials of degree three for the following data. Approximate f(–0.3) f(-0.75) = -0.0718, f(-0.5) = -0.0247, f(-0.25) = 0.3349, f(0) = 1.1010

The rate of change of an autocatalytic chemical reaction is kQxkx2 where Q is the amount of the original substance, x is the amount of substance formed, and k is a constant of proportionality. Factor the expression.

The population P of Buffalo, New York, from 1980 to 2018 can be approximated by the function P(t)= 356,656(0.991)t  where t represents the number of years since 1980. Interpret the meaning of the base of the function.

The table gives two functions of the form P (t) = a b' that represent the populations of a Central American country and a Northern European country, t years after a baseline census is taken. Part: 0/3 Part 1 of 3 (a) For each function given in the table, write an equivalent function using base e. That is, write a function of the form P (t)=Poet. Also, determine the population of each country in the baseline census. Round the value of k to five decimal places. Country P(t)=ab' Central American P(t)=4.3 (1.0135) Northern European P(t)=4.6 (1.0062) P(t)=Poekt P(t) = P(t) = Baseline Population (in millions) P(0) - P(0) =

Evaluate the function f(x) at the given numbers (correct to six decimal places). x² - 8x f(x) f(8.5) f(8.1) f(8.05) = f(8.01) = f(8.005) = f(8.001) = f(7.9) f(7.95) = f(7.99) = f(7.995) = f(7.999) = = = = = lim X-8 x²- x². - X- - x - 56 Guess the value of the limit of f(x) as x approaches 8, correct to six decimal places. (If an answer does not exist, enter DNE.) x28x 56 = x = 8.5, 8.1, 8.05, 8.01, 8.005, 8.001, 7.9, 7.95, 7.99, 7.995, 7.999 1

A stock analyst plots the price per share of a certain common stock as a function of time and finds that it can be approximated by the -0.071 function S(t)=26+28c where t is the time (in years) since the stock was purchased. Find the average price of the stock over the first six years. The average price of the stock is $. (Round to the nearest cent as needed.)

On a certain winter day, the temperature of a northern city from 6 a.m. to 6 p.m. can be modeled by the function T(x) = -0.3x² - 8.6x - 2.3 where T(x) is the temperature (in degrees Celsius) and x is the number of hours after 6 a.m. Which of the following expressions would give the average temperature of the city between 8 a.m. (x = 2) and 3 p.m. (x = 9)? 9 (-0.12³ (-0.1x³-4.3x² - 2.3x) dx 9 © ¦ / (-0.12³ 4.3x² - 2.3x) dx 9 (-0.3x² -8.6x - 2.3) dr (-0.3x² -8.6x - 2.3) de

A creative group of students produces a reality television show that features aspects of college and dorm live. Once the show begins airing on the university's public access cable channel, the function shown below represents the number of weeks that pass until x percent of the campus is watching the show. Compute w(50), to one decimal place, and interpret the results in the context of this problem. wx=(x+125)1/3 A.It will be about until 6 weeks until 50 percent of the campus is watching the show. B.It will be about 5.6 weeks until about 50 percent of the campus is watching the show. C.It will be about 5.7 weeks until about 60 percent  of the campus is watching the show. D.It will be about 5.5 weeks until about 50 percent of  the campus is watching the show.

a cylindrical tank with a diameter of 20 feet is filled with oil to a depth of 40 feet. The oil begins draining at a constant rate of 2 cubic feet per second. write the volume of the oil remaining in the tank t seconds later as a function of t

A scientist observed that the speed S at which certain ants ran was a function of T, the ambient temperature.† He discovered this could be determined by the formula below where S is measured in centimeters per second and T is in degrees Celsius. S = 0.2T − 2.7 (a) Using functional notation, express the speed of the ants when the ambient temperature is 14 degrees Celsius.S (  )Calculate that speed using the formula above. (Round your answers to one decimal place.) cm/sec(b) Solve for T in the formula above to obtain a formula expressing the ambient temperature T as a function of the speed S at which the ants run.T =        (c) If the ants are running at a speed of 3 centimeters per second, what is the ambient temperature? (Round your answer to one decimal place.) °C