Euler's method

Use Euler's method with h = 0.05 to approximate the solution, and compare it with, actual values of y. See attached file for full problem description.

Runge-Kutta-Fehlberg using MATLAB

Runge-Kutta-Fehlberg using MATLAB. See attached file for full problem description.

Solve the given natural and clamped cubic splines problems.

What is the difference between natural and clamped Cubic Splines? Solve the following problems with a clear explanation. [1] A natural cubic spline S on [0,2] is defined by S(x) = { S0(x) = 1 + 2*x - x^3 , if 0 <= x <= 1 S(x) = { S1(x) = 2 + b*(x-1) + c*(x-1)^2 + d*(x-1)^3 , if 1 <= x <= 2 Find b, c and d. [2] A clamped cubic spline S for a function f is defined by S(x) = { S0(x) = 1 + B*x + 2*x^2 - 2*x^3 , if 0 <= x <= 1 S(x) = { S1(x) = 2 + b*(x-1) - 4*(x-1)^2 + 7*(x-1)^3 , if 1 <= x <= 2 Find f'(0) and f'(2).

Question about Banach Spaces

Show that (X, ||*||) is a Banach space if and only if {x in X: ||x||=1} is complete. Know that in the first direction, we must show that {x in X: ||x||=1} is closed subset of X. For the reverse direction, I know I have to take a cauchy sequence and translate it to the unit circle and then show that if it is convergent there, it is convergent outside of the unit circle.

Limit Superior

See attached for full problem description.