question 1

by Taylor series, write on paper don't type plz.

The rate at which an albatross flaps its wings is dependent on its altitude. From measuring many albatrosses, it has been determined that the rate which an albatross flaps its wings is given by 1000 ram) = W\" where h is the altitude in kilometres (km) and h) is in flaps per hour [flaps/h}. If one bird has a particular trajectory where the altitude is a function of time h : Mt). where t is in hours. then the total number of flaps in T hours is given by no = f fthttndt One day, researchers observe an albatross with the following trajectory I----Innmm I_-------II We wish to estimate the total number of flaps in 3 hours F(8). We will do this in two ways. [a] (5 point) Approximate f[h[t)) by a degree 2 polynomial centred at t = 0 using the chain rule. {b} {2 points] Use your answer in [a] to approximate F(8). {c} {2 points] Use a lefthand Riemann sum to approximate F(S). [ d} (1 point} Compare your two answers. Which one do you expect to be more accurate? Why? Consider the infinite series 1 2n + 2 n! In this problem, we are going to use Taylor series to find the exact value. (a) (4 points) Due to the factors of the form 1/(2n + 2) in the sum, we might suspect that this is actually the integral of some power series. That is, 1 2n + 2 7 T da, n=0 where c > 0 is a fixed number and pn is some sequence of numbers. Find suitable values of c and Pn so that the above is an equality. (b) (4 points) In (a), you obtained a power series n! This is, in fact, the Taylor series of some function f(@) centred at c = 0. Using the Taylor series of familiar functions like polynomials, the exponential, n! n=0 logarithms, etc., find this function f(x). (c) (2 points) Using your function f(x) from (b), find the exact answer to 1 1 2n+ 2 n!' by evaluating where c is the bound you found in part (a)