Estimate the error in the the trapezoid rule approximation

Question

Problem 1:

a) Write an approximation of the integral from 0 to pi of "sin(x) dx" using the Trapezoid rule and Simpson's rule with n = 4. Evaluate sums

b) Estimate the error in the the Trapezoid rule approximation.

c) How large must n be so that Tn is accurate within 0.001?

Problem 2:

Sketch a direction field for the ordinary equation y' = k(y-a). Then use it to sketch a solution curve that passes through the origin.

Problem 3:

If y is the solution of the initial value problem

dy/dt = 2y(1-y/1), y(0) = 1

then limit as t approaches infinity is y = e⁷.

If false, what is limit as t approaches infinity of y? Justify answers.