First of all, why study calculus? Calculus makes the connection between discrete and continuous numbers. Consider the graph of the sine function: a wiggly line that repeats every 2*pi along the horizontal axis. Pick any horizontal location, and the sine function will tell you the exact vertical location of the graph at that point.
Now just consider the sine function from 0 to pi along the horizontal axis, which is half a cycle of its oscillation (you remember your trigonometery, right). That segment of the graph starts at zero, rises to a maximum of one, then falls back to zero. Again, you can calculate the exact value of the sine function at any point you choose. You know everything about it. Or do you? What is the area under the curve? (I.e., between the function and the horizontal axis.)
Calculus lets you consider the area under a curve by breaking the area into an infinite number of vertical slices, and adding them up. By a beautiful twist of mathematics, the area under sin(X) (from 0 to X) turns out to be -cos(X). The process of adding up the little slice is called integration.
So who cares? Well, most interesting things that happen in the real world occur over either time or space, and can be modelled as areas under curves. Thus, an understanding of calculus allows you to create mathematical models of pretty much anything you want. Electrical engineers integrate voltages and currents over time. Mechanical engineers integrate forces across steel beams. Chemical engineers integrate the completion of a reacion over time. Physicists integrate all sorts of weird crap. Without calculus, you are like Copernicus, knowing only that the planets go around the sun. With calculus, you are like Netwon, and can calculate orbits. Incidentally, Netwon invented modern calculus in his efforts to understand the universe.
A knowledge of calculus, even at the C- level, gives you a powerful tool for dealing with the real universe of atoms and forces. Without calculus, you will be limited to text processing, database lookups, graph traversals and the like. You can make a perfectly good career out of that (e-commerce software doesn't have much call for integration), but knowing calculus will open doors into other jobs: signal processing, factory automation, network traffic analysis, physics modelling, etc.
In your calculus class, you will study many types of equations, but mostly you will concentrate on (i.e., beat your head against) two types: trigonometric and exponential. Why trigonometric functions? When you put feedback into a physical system (i.e., a microphone close to a speaker), the system will produce either a sine wave or something resembling one. It doesn't matter whether it's biological, electrical, psychological, or whatever kind of feedback: feedback loves sine waves. When you study differential equations a few courses down the road, it'll be obvious why sine waves occur everywhere. And of course, circular motion is naturally described by trigonometric equations (think motors and engines). Just take it for granted and learn to love the trigonometric functions. If you can't learn to love them, then learn to hate them with skill.
The other class of functions you'll learn to hate is the exponentials, and their brothers the logarithms. Again, it is due to practical interest: whenever something changes by a fractional rate over a period of time, you get an exponential function. You'll learn how to derive exact equations for radioactive decay and compound interest.
Unfortunately for you, your instructors and textbooks are likely to suck. Regarding your instructors, other posters have given advice about study groups and tutoring: this can help. Regarding textbooks, I wholeheartedly recommend Calculus, by Larson, Hostetler, and Edwards. It's a massive book (> 1000 pages), but it is comprehensive and has plenty of helpful diagrams. Inside the front and back covers, it has wonderful tables of common formulas and identities. As a practicing engineer, I still use this book occassionally. Don't be shortchanged by the many poor texts out there. (And don't assume the math department knows best. Math depts pick textbooks the way businessmen pick software, and you could end up with the Microsoft Outlook of textbooks.)
One final piece of advice: the great physicist Richard Feynman said that if you cannot explain an idea to another person, then you do not understand it. When studying complicated material, I pretend to explain it to another person. When you come to something unexplainable, you have found your area of ignorance, and can read the book or ask the professor again. Repeat this until you can teach everything you're supposed to be learning.
I don't want the world, I just want your half.
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