why do we have to learn calculus
Perhaps I'm being too sentimental here, but I do believe that everyone should (and can! ) learn some calculus. For me, calculus was when mathematics became beautiful. All my math before came together to form a whole. I can't tell you that calculus will save your life or you'll integrate functions on a daily basis (why would you want to? ), but I think the purpose of calculus goes beyond it's "practical" uses. It's an end in of itself. If you want some more "down-to-earth" reasons to study calculus, then here's my best shot. Discipline: Calculus is hard. It demands your attention if you want to succeed. You have to take notes, memorize trig identities, common derivatives and friends, and be able to recognize what to do when faced with a problem. If you didn't know how to study before learning calculus, you probably did afterwards. Mental Exercise: As litteacher8 noted, calculus is like cardio for your mind.
If you usually study history or literature, studying mathematics can be the change of pace your brain needs; after all, the brain thrives of diversity! Also learning a new skill is a great way of keeping your mind in shape. Self-Assessment: Challenging yourself is one of the best ways to get to know yourself. Taking math up to calculus is a great way to know whether or not you'll be happy handling the hard stuff. From a college courses viewpoint, calculus opens up avenues of study that wouldn't be available otherwise. It gives you choices. Hope this helps!
Somebody has to play devil's advocate here, so I guess it's up to me. The second semester of freshman calc is universally devoted to force-feeding students with tricks for integration which they will neither need nor retain after their final exam. This wouldn't be so bad if there were some other educational purpose being served. For comparison, a student taking an English class might neither need nor retain any specific skills or knowledge related to the novel White Teeth by Zadie Smith, but the student is arguably developing skills such as critical thinking and written expression.
There is unfortunately no such argument to be made for techniques of integration. Students learn this bag of tricks as a set of cookbook procedures -- the opposite of critical thinking. The analog of the skill in written expression taught in an Engligh course would be facility in performing integration. This analogy fails, however, because written expression is a skill that can only be performed by the human mind, whereas computers can now carry out integration so well that the human who can outperform Wolfram Alpha in integration is as rare as the human who can (on a good day) win a game of chess against Deep Blue. Since computers don't seem to have attained creativity or self-awareness, it should be deeply demoralizing to the practitioners of any art when their craft is computerized.
It shows that what they were doing could have been done better by a mindless automaton. For this reason, it's unlikely that humans a hundred years from now will be much interested in chess or integration techniques. In the nineteenth century, men who wanted to become officers in the British military were required to demonstrate proficiency in ancient Greek. The intention was to exclude the working class. Today, people who want to be doctors are required to demonstrate proficiency in techniques of integration. The intention is to weed out students from the "impacted major" of biology at the University of California. In defense of the nineteenth-century British ruling class, we can observe that translating English into ancient Greek is still a skill that can't be carried out by a computer.
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