How fast is the car going?

If the speed of a car is a linear function of time in seconds and 10 is less than or equal to t for time is less than or equal to 30. if after each second the speed of the car increases by 5 feet per second and after 20 seconds the speed feet per second is 80 feet per second how fast is the car going after 30 seconds?

Complex hilbert space Bra-ket

In a complex hilbert space, prove the following: < f|g > = ¼ [ ||f + g||^2 - ||f - g||^2 + i||f + ig||^2 - i||f - ig||^2 ]

Evaluate integrals using Rayleigh's energy theorem

See attachment for problem description. I have added another document with a definition of Rayleigh's theorem and how to use it in a problem.

Riemann Stieltjes Integration

Suppose that f^2 is Riemann Stieltjes Integrable. Is f Riemann Stieltjes Integrable as well? Explain.

Integration

Define the following functions on the closed interval [-1, 1]: β(x) = { 0, for x<0 ½, for x=0 1, for x>0 Let f:[-1,1] R such that f is bounded. Show that f is Riemann Stieltjes integrable with respect to β if and only if f is continuous at x=0.