Calculus Summary

Calculus has two main parts: differential calculus and integral calculus. Differential calculus studies the derivative and integral calculus studies (surprise!) the integral. The derivative and integral are linked in that they are both defined via the concept of the limit: they are inverse operations of each other (a fact sometimes known as the fundamental theorem of calculus): and they are both fundamental to much of modern science as we know it.


The limit of a function f(x) as x approaches a is equal to b if for every desired closeness to b, you can find a small interval around (but not including) a that acheives that closeness when mapped by f. Limits give us a firm mathematical basis on which to examine both the infinite and the infinitesmial. They are also easy to handle algebraically:

where in the last equation, c is a constant and in the first two equations, if both limits of f and g exist.

One important fact to keep in mind is that

doesn't depend at all on f(a) -- in fact, f(a) is frequently undefined. In the happy case where

we say that f is continuous at a. It is also sometimes useful to talk about one-sided (left or right) limits, where we only care about the values of x that are less than or greater than a.

The derivative of f(x) at x=a (or (a) ) is defined as

wherever the limit exists. The derivative has many interpretations and applications, including velocity (where f gives position as a function of time), instantaneous rate of change, or slope of a tangent line to the graph of f. Using the algebraic properties of limits, you can prove these extremely important algebraic properties of derivatives:

These rules, for example, allow you to calculate the derivative of any rational (= ratio of two polynomials) function. The chain rule in particular has many applications. For one thing, if you have two inverse functions f and g, that is if f(g(x)) = x, then the chain rule implies that (g) = 1/(x).

Also, if you have an implicitly defined function between x and y like x2 - 2 x y + y2 = 1, then you can perform implicit differentation (basically, just taking the derivative of everything with respect to both x and y are tacking on dxs and dys to indicate which) to get 2x dx - 2 x dy - 2 y dx + 2 y dy = 0. Then if you solve for dy/dx, this will be equal (by the chain rule) to y´ and if you solve for dx/dy, this will be equal to x´. Note that in this case, either derivative will be in terms of both x and y.

You may be wondering about the derivatives of your favorite trigonometric functions. Well,

These two facts, combined with the rules above, allow one to calculate easilythe derivatives of the rest of the trigonometric functions and their inverses. The derivatives of the hyperbolic functions are similar, except that

Many physical applications of derivatives reduce to finding solutions to differential equations: equations relating a function and its derivatives. For example, both sine and cosine satisfy the differential equation f´´(x) = -f(x), which models ideal pendulums, springs, and other examples of simple harmonic motion. The equation f´(x) = k f(x) comes up in modeling population growth and radioactive decay, and is solved by the function f(x) = ekx, where

is called Euler's constant and is defined to be the unique real number e such that (ex)´ = ex. The inverse of the exponential function ex is the natural logarithm function log(x), which has many useful and interesting properties, including: