Calculus students usually see the theorem that a continuous function on a closed interval that takes on different signs at the intervals endpoints must have a zero in the interval. This can be proved rigorously via the bisection algorithm. In addition to being clearer than a more abstract proof, this can be used as a first introduction to numerical methods.
The geometric interpretation of Newton's method requires an understanding of the relation between the derivative of a funciton and the line tangent to the graph of the function. Consequently, Newton's method fits well in the early discussion of derivatives. On the other hand, it gives an application of the derivative (to root finding) that might help motivate interest in derivatives.
And so forth. Nearly every introductory topic can be related with numerical methods. That this is not customary everywhere reflects the inertial resistance to change in elementary curricula.
(However, coding these algorithms should be left for a subsequent class devoted to that task. Many students struggle with coding loops and conditionals and this requires its own pedagogy. But it is easier if the students already understand the algorithms they are asked to code.)
Calculus students usually acquire the mistaken notions that most integrals can be evaluated exactly and that exact expressions/formulas are useful. In fact most computation is numerical and most evaluation of exact expressions/formulas is also numerical. Particularly with students who will mostly use calculus on computers it is important to introduce them early to the issues related to practical computation of integrals and derivatives.