Let $(M^n,g)$ be a complete Riemannian manifold, let $p\in M$, and let $d_g$ denote the Riemannian distance. Define $r_p:M\to[0,\infty)$ by $r_p(x)=d_g(p,x)$. For $k\in\mathbb{R}$, define the model cotangent function $\operatorname{ct}_k:I_k\to\mathbb{R}$ by