  For smooth manifolds $M$ and $N$, let $\Ebar(M, N)$ be the homotopy fiber of
the map $\Emb(M, N)\longrightarrow \Imm(M, N)$. Consider the functor from the
category of Euclidean spaces to the category of spectra, defined by the formula
$V\mapsto \Sigma^\infty\Ebar(M, N\times V)$. In this paper, we describe the
Taylor polynomials of this functor, in the sense of M. Weiss' orthogonal
calculus, in the case when $N$ is a nice open submanifold of a Euclidean space.
This leads to a description of the derivatives of this functor when $N$ is a
tame stably parallelizable manifold (we believe that the parallelizability
assumption is not essential). Our construction involves a certain space of
rooted forests (or, equivalently, a space of partitions) with leaves marked by
points in $M$, and a certain ``homotopy bundle of spectra'' over this space of
trees. The $n$-th derivative is then described as the ``spectrum of restricted
sections'' of this bundle. This is the first in a series of two papers. In the
second part, we will give an analogous description of the derivatives of the
functor $\Ebar(M, N\times V)$, involving a similar construction with certain
spaces of connected graphs (instead of forests) with points marked in $M$.
