Thought a little background is required.
The place I teach students learn Calculus in school, not the theory but only the computational aspect.
I am trying to teach Calculus to first year Engineering students using Apostol but there is huge resistance from my own Department. The arguments range from Engineers don't need this, they won't understand theory and its useless for them, they should focus more on concepts and visualization (geometric intuition). I am feeling a bit defenseless and cornered as I never been part of discussions where people discussed pedagogy.
All I know, I spent 15 odd years in an Engineering Institute and 95% percent of my friends are Engineers and they mostly know/did their maths courses rather well; which was a proper course, as in they did proofs, knew about the real number system (lub axiom) , Sequences (Bolzano Weierstrass) , Continuity, Roll's Theorem and MVT, Taylor's Expansion all, Fundamental Theorem of Calculus and they did all these in their first semester along with proofs.
Here I am facing a scenario where Cauchy Sequence is not to be touched on as if it is going to make students head explode. Taylor's Expansion is not to be talked about, forget Taylor's Series I am not allowed to teach Series. The problem is the resistance comes from my own head/department and not the Engineering Department.
I am totally confused and do not know how can I argue my case. Apparently all these is part of Real Analysis and not Calculus and also Proofs are not required, how does one argue against this ?
I am not complaining at all, I am asking for advise regarding how to present my case in a compelling and convincing way.
Added in Response to comment by Douglas Zare
those with different preparation would probably go through a preparatory course I suppose. What I mean to what extent do the Engineering students need to know Calculus to do well in their Career. What is the basic minimum. Where they begin is a different issue, what's the least they must & should know ?