<p>So I 5'ed AP Calculus BC and am enrolled in Calculus III for next semester. What should I do -- if anything -- to prepare for the next level of math?</p>
<p>Learn Partial Diff Eq and Multivariable calc.</p>
<p>Make sure you know what a limit is (no it isn’t just plugging a into the expression). Make sure you know what the derivative is (no it isn’t slope). Make sure you know what an integral is (no it isn’t area under the curve). If you know these things, you’ll be fine.</p>
<p>I bought Schaum’s Outline of Calculus. Should I read the chapters corresponding to calculus I & II to keep my calculus knowledge/skils sharp?</p>
<p>Calculus is about one thing: practice. Just keep practicing, and make the easy stuff second nature and you’ll be fine.</p>
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<p>i’m scared :(</p>
<p>i’m afraid of the theoretical stuff too. But really if you can’t get yourself to learn all that, make the easy stuff second nature like DrAhumada said. It’s tough to learn the theory on your own but that’s what they’ll teach you in you classes. What I’m doing is just reviewing over and over again until Calculus becomes as close to easy as Algebra. if that makes sense.</p>
<p>i think you’ll be absolutely fine… yes, college professors probably teach calc 1&2 a little differently than your AP calc teacher in HS did. But the fact that you took the AP tests and got 5’s tells me you are more than prepared to take calc 3. Even if you hadnt 5’d the AP tests i would proabably tell you the same thing. Lets face it, the only prerequisite for calc 3 is that you really know your derivatives and integrals.</p>
<p>You could get a calc III text and start working through problems.</p>
<p>If you want a little theory, see if you can get your hands on a copy of Spivak and work through the problems in the first couple of chapters.</p>
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<p>I got a 5 on AP calc AB and really struggled through calculus II in college. If you go to a good school, the classes there will be more challenging than AP courses.</p>
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<p>Don’t worry, I’m being super lame and pedantic about calculus. It’s funny because I’m actually a little bit of a math poseur–I’m only an engineering major.</p>
<p>My point is that the interpretations (derivative as slope, integral as area under the curve) that may have got you through AP calculus lose their usefulness when you generalize things a bit in calculus III. </p>
<p>If you never really understood the definitions of a limit, a derivative, or an integral or never learned their more powerful interpretations (the derivative tells you about how a function behaves locally–the integral is really just the continuous analogue of the sigma (a sum)) you may find calc III a little tricky.</p>
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<p>To OP: Spivak’s Calculus is pretty hard. The stuff you’ll learn in that book is way overkill for calculus III. But if you are interested and think you can handle it, try it out!</p>
<p>Hmm, so I had an idea. Instead of doing curriculum math, I’m thinking of just working on competition-level math problems to sharpen up my math abilities so picking stuff up becomes a lot easier.</p>
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<p>Silence kit, if you don’t mind, can you explain those interpretations? </p>
<p>I interpret the derivative as the instantaneous rate of change. And the integral as yes, the area under the curve, or the total amount of ____ in a function (eg. gallons of water)</p>
<p>Can you explain your interpreations?</p>
<p>silence_kit: if you don’t mind me asking, what school do you go to? I’m going to be an engineering major too and am now scared. I’m only planning on going into Calc II as well.</p>
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<p>If the math competitions are like “you have 45 minutes to solve 30 problems”, then probably it won’t be very helpful. </p>
<p>If the math competitions are like “here are 5 wicked hard problems. you have 4 hours to figure them out”, then working those problems will be very useful.</p>
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<p>Sure, but they aren’t really mine–I learned them from my calculus professors. I apologize in advance for being such a blowhard–but for some reason, I really like to lecture about this.</p>
<p>Instantaneous rate of change is a good way to think about the derivative. Evaluate the derivative of a function at a point, and the output will tell you how the original function changes near that point (aka gives you the instantaneous rate of change). This is pretty much a rephrase of what I said “the derivative tells you how a function behaves locally”. </p>
<p>The reason why the slope interpretation breaks down I will explain. In calculus III, you’ll talk about what it means to take the derivative of a function that takes in multiple numbers and outputs multiple numbers. It turns out that the derivative here, loosely, is just the the collection: the rate of change of output 1 wrt input 1, the rate of change of output 1 wrt to input 2, …, the rate of change of output 1 wrt input n, the rate of change of output 2 wrt input 1, …, the rate of change of output 2 writ input n, …, the rate of change of output m wrt input n. All these things together tell you about how the function behaves locally.</p>
<p>This is a lot of information to keep track of! You can’t easily see it graphically . . . thinking about slope won’t help you much here.</p>
<p>For similar reasons, thinking about integration as “area under the curve” breaks down in calculus III. Here is a pretty concrete example. Let’s say that I want to figure out the mass of a 3-D object. I know how it is shaped, and I also know the density of the object. That density is the function p(x, y, z). It takes in a coordinate in space, and returns the density of the object at that point or 0 if the object isn’t at that position.</p>
<p>You can integrate the density function over the shape of the object (this is the higher dimensional version of integrating a function over an interval) to find the mass. To plot the function so you can “find the area under the curve” you’d need 3 dimensions for x, y, z and then 1 more to plot the density. Finding the area under the curve becomes “find the 4-D hypervolume under the volume”. That’s pretty hard to think about!</p>
<p>Here is a better interpretation:</p>
<p>Chop up the shape into a bunch of tiny cubes. Multiply the volume of the tiny cube by the density where the cube is located. The cube is so small that the density is constant in the cube. Now you have the mass of the tiny cube. Add up all the cubes. You have the mass of the object. This interpretation (integral as sum) is extremely useful in physics and engineering. </p>
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<p>I go to the University of Illinois. </p>
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<p>Don’t worry, they’ll teach you what you need to know. I’m exaggerating a bit with the macho math posturing. You actually can get through these classes by going through the motions with the formulas, but taking the time to learn the math a little better will pay off in later classes.</p>
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<p>While I don’t recommend wrote review, working on your integrating skills will definitely be valuable come Differential Equations. Just like learning calc really solidified my algebra skills, differential equations really improved my integration.</p>
<p>and silence_kit is talking about topics like partial derivatives and gradients. with these as well as curvature, divergence, curl, etc., derivatives and integrals become a tool to understand theoretical and physical phenomena. it’s appealing to us engineers, because there are more physical examples and we’re in three or four dimensions. some of the math majors hate it, but that’s why they’re math majors :)</p>
<p>Read ahead. There’s nothing like having previewed the material to do well in a class. You’ll be surprised by how much you can understand by yourself by simply reading a Cal III textbook and doing an exercise or two. You’ll also identify trouble areas so that you can pay extra attention to those topics in class.</p>
<p>To get you started, read through the following:</p>
<p>[Multivariable</a> Calculus](<a href=“http://people.math.gatech.edu/~cain/notes/calculus.html]Multivariable”>http://people.math.gatech.edu/~cain/notes/calculus.html)</p>
<p>Here are a few questions you can try to answer…</p>
<p>(1) What is the dot product or scalar product of the vectors <3, 7, 2> and <1, 1, 0>? What is their cross product? What is their sum and what is their difference? What is the angle between them? What are the magnitudes of each? What are they in cylindrical and spherical coordinates?</p>
<p>(2) Find f ’ if f(t) = <2t + sin(t), 3 - exp(t), tan(t)>. Also find F, an antiderivative of f, for the same f. Find the curvature, torsion, and second derivative of f.</p>
<p>(3) Find the gradient of f where f(x, y, z) = 2x^2 + 3y^2 - exp(xyz). Find the directional derivative in the direction of the vector <1, 1, 0>. Say if this function has any minima, maxima, or saddle points, and if so, where they are. Find the maximum, minimum, or saddle-point of the function subject to the constraint that x+y+z=0.</p>
<p>(4) Integrate the function in (3) with respect to each of the variables (x from 0 to 1, y from 1 to 2, and z from 2 to 3). Do this in each of the 6 possible orders (x, y, z or x, z, y or y, x, z or y, z, x or z, x, y or z, y, x).</p>
<p>Once you’ve polished these off, or determined that you can’t figure out the text well enough to do them, you’ll be in a great position to ace that class.</p>
<p>yep…this class was easier than calc 1&2 (for me)…most of my friends say the same thing)…i really dont think there’s any reason to be scared…maybe the last couple of sections are a little difficult but theres nothing like so hard about the class…Plus this kid is going to UVM, not MIT. He’s going to be one of the smarter students.</p>
<p>a little off-topic here, but I’m currently in Cal 1 and am really amazed at how easy it is so far.</p>
<p>yeah calc 1 is easy up until you have to do integrals. even then they are pretty easy once you get the hang of them since you only learn a few of the techniques of integration in calc 1.</p>
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<p>Heh, if I could solve USAMO problems, I wouldn’t be going to UVM :P</p>