Let y=f(x) be the particular solution to the differential equation dxdy=(3−x)y2 with initial condition f(1)=−1.
5(A)Separable and Linear First-Order EquationsMedium3 marks
Find f′′(1), the value of dx2d2y at the point (1,−1).
5(B)Power Series, Radius of Convergence and Taylor SeriesEasy2 marks
Write the second-degree Taylor polynomial for f about x=1.
5(C)Power Series, Radius of Convergence and Taylor SeriesEasy2 marks
The second-degree Taylor polynomial for f about x=1 is used to approximate f(1.1). Given that ∣f′′′(x)∣≤60 for all x in the interval 1≤x≤1.1, use the Lagrange error bound to show that this approximation differs from f(1.1) by at most 0.01.
5(D)Numerical Solution of Differential EquationsEasy2 marks
Use Euler's method, starting at x=1 with two steps of equal size, to approximate f(1.4). Show the work that leads to your answer.