Yes and no. You have to understand the fundamental concepts of integration, differentiation, differential equations etc. and what they mean and represent. All

Yes and no. You have to understand the fundamental concepts of integration, differentiation, differential equations etc. and what they mean and represent. All the time spent learning how to do symbolic integration by hand is however time that could be much better spent. I also think you could and should start by teaching probability and statistics and using that to introduce calculus as and when it shows up naturally, rather than teaching calculus in the abstract first and then showing the applications much later.