Problems

Explain diagrammatically, or otherwise, why \[ \frac{\mathrm{d}}{\mathrm{d}x}\int_{a}^{x}\mathrm{f}(t)\,\mathrm{d}t=\mathrm{f}(x). \] Show that, if \[ \mathrm{f}(x)=\int_{0}^{x}\mathrm{f}(t)\,\mathrm{d}t+1, \] then \(\mathrm{f}(x)=\mathrm{e}^{x}.\) What is the solution of \[ \mathrm{f}(x)=\int_{0}^{x}\mathrm{f}(t)\,\mathrm{d}t? \] Given that \[ \int_{0}^{x}\mathrm{f}(t)\,\mathrm{d}t=\int_{x}^{1}t^{2}\mathrm{f}(t)\,\mathrm{d}t+x-\frac{x^{5}}{5}+C, \] find \(\mathrm{f}(x)\) and show that \(C=-2/15.\)