While it's indeed likely that many of your students will not extensively use the material they learn in your class, it may not be encouraging to them to hear that they won't use it. And indeed, they do learn important critical thinking skills, but they may be wondering what is special about the sort of critical thinking in calculus and linear algebra that they couldn't get out of another course.
In this vein, one solution may be to present real-world examples of how calculus and linear algebra are used today, get the students involved as you present, and relate the topic to the students' lives.
For instance, after you present a basic optimization example (maximize fence perimeter given the area it contains) you might present a slightly more complicated problem (building a window of a particular shape that maximizes the amount of light coming through with a fixed amount of building material, for instance), and get the class involved in trying to solve the problem. Ask them what's different about it, maybe have them briefly discuss a strategy with the person next to them. (This is not usually done in a large class, but at least one very popular professor at my school does this in his large classes.)
Continuing with the example of optimization, there's a great topic on matheducators.stackexchange about real-life optimization problems here. In particular, I've linked to a problem that video-game players might find to be really cool and related to their everyday lives. Of course, it may also help to branch out and briefly discuss some current, advanced applications of calculus and linear algebra. An excellent example where ideas from calculus and linear algebra are essential is the story of how UPS route optimization involves the minimization of left turns. This last example might inspire a lively discussion in your classes, because it shows how calculus and linear algebra can trump everyday intuition... and how pushing beyond a basic understanding can lead to new, interesting ideas! (I'm sure if you asked students in class, they could come up with reasons for why UPS's strategy might make sense.)
On the other hand, some students might not be moved by industry applications and might benefit from seeing mathematics as the kind of "safe place" you suggest where perfect formulations lead to perfect results. Strogatz has some interesting words in a Math Horizons article about the importance of enabling others to experience the beauty of math, and he provides a great example of why math is beautiful. I also just found a TED talk discussing the use of calculus in some parts of modern architecture. Though this last one is admittedly again an industry application, it at least sheds some light about how aesthetics and mathematics can interact in interesting ways.