For an embedded Riemannian manifold $M \subseteq \mathbb{R}^m$ and a point $x \in M$, there is a series expansion (page 8 of Monera's paper): $$\exp_x(t v) = x + t J_x(v) + \frac{t^2}{2!} Q_x(v) + \cdots$$ where $v \in T_x M$ satisfies $\|v\|=1$, $t \ge 0$, $u \mapsto x + J_x(u)$ is the (linear) embedding of the tangent space $T_x M$, and $Q_x(v) = (\nabla_v v)^\perp$, the component of $\nabla_v v$ that is normal to $T_x M$.

Question. Is the following or a variant of it true? For some constant $C$, we have that whenever $0 < t < C$, there is a vector $w > \in T_x M$ with $\|w \|\le 1$ such that: $$\exp_x(tv) = x + t J_x(v) + > \frac{t^2}{2!} Q_x(w)$$ A higher order (more than quadratic) analogue is also sought, if it exists.

My end goal of this analysis is to obtain a bound on the local deviation of an embedded manifold from being linear. Namely, upon accepting the claimed result, we have that $$\| \exp_x(tv) - (x + t J_x(v)) \| \le \frac{A}{2}\cdot t^2 \text{ where } A := \sup_{w \in TM, \|w \| \le 1 } (\nabla_w w)^\perp $$

And honestly, all this feels like they are likely to be some exercise problems to a Riemannian geometry course, but I can't find a reference for this type of result. Pointing towards a classical reference for this would be highly welcome as well.