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Why did the calculus student who didn't know the fundamental theorem of calculus fail the entire year of calculus?
Wiki User
∙ 10y ago
I don't know the details about this particular student, but I would hazard a guess that he didn't know quite a few other things about calculus, either. In any case, if you don't know the fundamental theorem - at least, if you don't know how to apply it in practice - you'll have serious problems with many different problems - specifically when you need to do definite integrals.
Wiki User
∙ 10y ago
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What is calculus 1?
Traditionally, and in my learning experiences, calculus is taught in three stages, often referred to as Calculus I, Calculus II, and Calculus III (often shortened to Calc I, Calc II, Calc III). You are asking about Calculus I only, but it is easy to explain all three. Calc I usually covers only derivative calculus, Calc II covers integral calculus and infinite series, and Calc III covers both derivative and integral calculus, but in multiple variables instead of only one independent variable ( xyz = x+y+z as opposed to y = x). This is a traditional collegiate leveling of calculus. This is often changed around in secondary education (in the United States at least). Programs such as AP Calculus often change around this order. AP Calculus AB covers Calc I and introduces Calc II, while AP Calculus BC covers the remainder of Calc II. Now that you know the subject matter, what does it mean? Derivative calculus is a generalized category meant to encompass the computation and application of only derivatives, which are basically rates of change of a mathematical function. A basic mathematical function such as y = x + 2 describes a mathematical relationship: for every additional independent variable "x", a dependent variable "y" will have a value of (x + 2). But, how do you describe how quickly the value of "y" changes for each additional "x"? This is where derivatives come from. The derivative of the function y = x + 2, as you would learn in Calc I, is y' = 1. This means that y changes at a constant rate (called y') of "1" for each additional x. In more familiar terms, this is the slope of this function's graph. However, not all functions have constant slopes. What about a parabola, or any other "curvy" graph? The "slopes" of these graphs would be different for any given value of a dependent variable "x". A function such as y = x2 + 2 would have a derivative, as you would learn in Calc I, of y' = 2x, meaning that the original value of "y" will change at a rate of two times the value of "x" (2x), for each additional increment of "x". You can continue into further derivatives, called second, third, fourth (and so on) derivatives, which are derivatives of derivatives. This is essentially asking "At what rate does a derivative change?". The beginning of Calc I is concerned with introducing what a derivative is, ways to describe the behavior of mathematical functions, and how to compute derivatives. After this introduction is complete, you will begin to apply derivatives to mathematical problems. The description of how derivatives are used to solve these problems is not worth going into, because it would be better for you to connect derivatives to their applications on your own, but you can use derivatives to answer such questions as: What is the maximum/minimum value of a mathematical function on a given interval or on its entire domain? This kind of knowledge can be applied like so: Suppose a mathematical function is found that describes the volume of a box. Knowing that you can use the derivative of this function to find its maximum value, you can then find what value of a certain variable will yield the maximum volume of the box. Another type of application is called a "related rates" problem, in which a known mathematical relationship is used with some given information to describe another property. A question of this type could be: Suppose you have a cylindrical tank of water with a small hole in the bottom, and you measure that the water is flowing out at 2 gallons per minute. At what rate is the height of the water in the tank changing? (This is a simple related rates problem). A full description of integral calculus (Calc II and a basis of Calc III), would take far too long to explain, and it would be easier to explain once you have taken Calc I. Calc III takes the same idea as Calc I and Calc II, but instead of one independent variable "x" changing one dependent variable "y", there are several variables, although in most applications you will only see three, "x", "y", and "z", although the ideas you will learn in the class will apply to potentially infinite variables. The basic ideas of derivatives and integrals will hold here, but the mathematical methods needed and applications possible with multiple variables require additional learning.
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