Final answer:
The question focuses on calculating the work done by a force through integration and the approximation of a function using Fourier sine series. It involves evaluating an integral for work done, plotting the series for different numbers of terms, and discussing the boundary behavior.
Step-by-step explanation:
The student's question revolves around the calculation of work done by a variable force and the approximation of a function using a Fourier sine series. To determine the work done by the force F(x) = (10 N)sin[(0.1 m⁻¹)x] from x = 0 to x = 10 m, one needs to evaluate the integral of the force function over the given distance. The work can be represented as the area under the force-displacement graph, which is the integral:
W = ∫ F(x) dx = ∫ (10 N)sin[(0.1 m⁻¹)x] dx from x = 0 to x = 10
The Fourier sine series question is more complex as it involves graphing the function m(x) = x - x³ and its Fourier sine series approximation. Multiple terms (at least 10) of the series need to be calculated, graphs plotted, and then the error at different approximations as well as the boundary behavior needs to be discussed.
Plotting and interpreting the partial sums of Fourier series would demonstrate how varying numbers of terms (N) impact the accuracy of the series approximation to the original function. The error at the boundaries tends to be more significant due to the Gibbs phenomenon.