- Calculus, Volume 1: One Variable Calculus, with an Introduction to Linear Algebra
- Tom M. Apostol
- Second Edition
- 1967
- 978-1-119-49673-1
14.13 Exercises
- A nonnegative function $f$ has the property that its ordinate set over an arbitrary interval has an area proportional to the arc length of the graph above the interval. Find $f.$
- The problem statement asks for a function $f(x)$ with speed $v(x),$ such that over an arbitrary interval $[a,b]$ and for some constant $k > 0:$ $$ \int_a^b f(x)\,dx = k\int_a^b v(x)\,dx $$
- We want to find a function $f,$ whose curve is described by the position vector $\mathbf{r}(x) = x\,\mathbf{i} + f(x)\,\mathbf{j},$ with velocity $\mathbf{v}(t) = \mathbf{i} + f'(x)\,\mathbf{j}$ and speed $v(x) = \sqrt{1 + [f'(x)]^2}$ such that: $$ \int_a^b f(x)\,dx = k\int_a^b v(x)\,dx $$ Extending the results of 14 and 15, we see that for an arbitrary constant $C,$ $f(x) = k\cosh\left(\frac{x}{k} + C\right)$ satisfies this condition. Additionally, noting that the derivative of a constant function is $0,$ we see that the function $f(x) = k$ also satisfies this conditon.