calculus
For extra credit for my class we are supposed to explain or describe to my teacher "The Fundamental Theorem of Calculus".
Now I understand Calculus has a lot to do with integrals, differentiating, finding curves and the area between curves by using integrals.
But my teacher wants us to show us him an example using mathematical examples and such.
I was wondering if someone could maybe help guide me or give me some examples I could use to work off of for this?
I will not explain calculus. There are many websites that do it very well, probably much better than I can, so I'll leave it to them. I will address prerequisites for calculus with an emphasis on procedure rather than deep understanding. At some point, you would do well to revisit these concepts in a slow, measured pace, being very careful and filling in all the details. However, your position seems more of an "in over my head" thing so I'll try to address it from that perspective.
There are only two big concepts in calculus, the derivitive (verb: to differentiate) and the integral (verb: to integrate). Both of these are special uses of the extremely broad concept of a limit, but many introductory calculus classes only touch on limits in a very superficial way. Certainly to do a first physics course with calculus you will not need a deep understanding of limits, only an appreciation for why they let us get the results we want.
If you do not have a good grasp on trigonometry, you can still do/understand calculus but it will seem rather artificial. You should know the sine and cosine functions and it will help to know the other four that usually accompany them. You should know the unit circle and the special values on it. Some basic identities will make your life easier.
For limits, you can get by with an intuitive understanding of functions, and of real numbers. It is more helpful to have a good intuition for rational numbers, for example you should know that there are infinitely many rationals in between any two real numbers. You must understand the notion of the domain of a function. Experience with rational functions is extremely valuable. To do calculations with limits, you should be very comfortable with simplifying rational expressions, including domain issues and extraneous solutions. Again, you can do without rational functions, but you are much better equipped to understand the significance of limits if you can manipulate them without much trouble.
For derivatives, you will need to be familiar with operations of functions: addition, subtraction, multiplication, division. A special emphasis on composition of functions: for you this will probably be the most important prerequisite for solving physics problems. You should be familiar with but do not need to be extremely comfortable with implicitly defined curves: for example, the circle is not given in $y=f(x)$ form, but it is still well-defined. You should understand the domain issues that can arise when converting implicitly defined curves into function form.
For integrals, there are a lot of skills you could need, but I will try and keep it to a bare minimum. I would not try to understand the real definition of an integral (your resources may call it a Riemann integral), but it is absolutely essential that you understand the intuition behind it, and its link to limits.
Of course you must be familiar with finding area of basic shapes. You must be extremely comfortable with reading $\Sigma$ notation; if not in reading it directly, then at least to translating it into $+$ notation (you do not need to be able to write $\Sigma$ notation well). However, the most useful skills for cracking integrals are pattern-recognition and persistence; they can sometimes require quite a bit of creativity to solve.
There is another (shorter) list which I think is equally important for your situation: things which you are not expected to know, but are expected to pick up during a calculus class. These include deeply understanding inverse functions, familiarity with theorems of the form "If … then there exists …", high comfort with implicit curves, high comfort with recognizing compositions of functions [you will need to pick this one up], the significance of variables as distinct from numbers, skill at visualizing 3D space, distinction between functions and their values at points.
At some point while learning derivatives, you will come across the notion of related rates. Please learn this very carefully. Many students struggle a lot with this section — including me — but it is a very important use of calculus for physics. Perhaps it will not come up directly in your class, but if you know it well you will see it hiding just behind the things you discuss.
The higher you go in math, the less the class will focus on real life applications. The math professors leave that to the science and engineering professors to teach. This might be fore some of the following reasons:
Calculus and higher mathematics topics are very dense, and there just isn't time to teach both concepts, methods, and applications.
Calculus and higher mathematics are not "everyday math" for most people who use it. By this, I mean you won't use calculus to figure out how much change you should have received from the cashier, or calculating the amount of wallpaper you need to buy. Calculus is applied quite often, but in highly specialized settings. If calculus teachers also taught applications, such applications would be useless to most students. By leaving these topics to the science professors, these topics are taught just to students who will use them. In other words, a physicist, an engineer, and a chemist all use calculus, but they do it in different ways.
Mathematicians sometimes pride themselves in how disconnected from application they can be. I've even heard a mathematician say "If there's an application to this, I don't want to know it." (though he did not say that about calculus).
So now you know why you haven't learned applications, here are some:
Calculus is the study of change. Situations in science or engineering where nothing is changing are pretty boring, so we use calculus to study questions that do change. For example: Newton's second law of motion is $$F=ma$$ where $F$ is force, $m$ is mass, and $a$ is acceleration. Though it is not immediately apparent how this equation can be "solved," solutions to this equation describe the motion of an object when the force $F$ is applied it. Acceleration is the change in velocity per change in time: $$a=\frac{dv}{dt}.$$ Velocity, in turn, is the change in position per change in time, so $$a=\frac{d^2x}{dt^2}.$$ Thus, we are left with $$F=m\frac{d^2x}{dt^2}.$$ Often, force itself is a function of $x$, and possible time $t$ , so we have to solve: $$F(x,t)=m\frac{d^2x}{dt^2}.$$ So position in terms of time, $x(t)$, is a function whose second derivative times $m$ is equal to $F(x,t)$. This is called a differential equation, and is somewhat more difficult to solve than an algebraic equation. But by solving such, we can predict how an object will move in time. I have heard differential equations called the "language of physics," since almost every physical situation can be described in terms of a differential equation, and solving these differential equation is key to understanding how the physical situation evolves in time and space.
If we already know how an object moves through time, that is we know $x(t)$, we can find $v(t)$ and $a(t)$ by taking derivatives.
A very important problem in all of science, engineering, and even economics and finance is that of optimization. Figuring out what choices to make to get the best results is very applicable. A company can decide what price to sell their product at to make the most profit. An urban power can decide the best way to set up utilities. A chemist can calculate the optimal amount of reagent to use to complete an experiment. Optimization almost always means calculus because differentiable functions may reach minima or maxima when their derivatives are zero. So the challenge is to take the derivative, set it equal to zero, and solve for the value of $x$ that makes this true.
Calculus also involves an operation which is, in some sense, the opposite of differentiation called integration. You will probably discuss this in your class. Integration is useful when we must find a "total effect" of a constantly changing cause. For instance, a dam can only hold back a certain force of water behind it. The force from the water depends on water pressure. But the water pressure towards the bottom of the dam is more than at the top. So how can we find the total force? Integration is the key.
It can almost be said that multivaribale calculus was developed with the express purpose of studying fluid mechanics and electricity and magnetism.
Calculus is also heavily used in biology. This isn't really my area, so I don't know all the specifics, but I do know there are differential equations used to predict how a population will grow or shrink in time. This can be used to find the optimal amount of animals to hunt without risking extinction.
Best Answer
The Fundamental Theorem of Calculus states:
Dividing both sides by $b-a$ gives us $$\frac{1}{b-a}\int_a^b f'(x)dx=\frac{f(b)-f(a)}{b-a}.$$ You can think of $f'(x)$ as the slope of the tangent line at $x$, and note that $\frac{f(b)-f(a)}{b-a}$ is the slope of the line connecting $(a,f(a))$ and $(b,f(b))$. This way, the FToC can be rewritten as: