This section is meant to provide the core curriculum for Calculus I, II, III, and IV. We’ll start with a short funny video going over some of the basics of calculus.

[vc_video link=”” align=”center”]

Why is Calculus Important for modeling things?

Calculus is the study of change, in the same way that geometry is the study of shape and algebra is the study of operations and their application to solving equations. Any problem involving rates of change or variables which don’t remain constant require calculus in order to solve. Lets begin with some of the history leading up to the development of calculus.

Xeno’s Paradox

The great Greek philosopher Zeno of Elea (born sometime between 495 and 480 B.C.) proposed four paradoxes in an effort to challenge the accepted notions of space and time that he encountered in various philosophical circles. His paradoxes confounded mathematicians for centuries, and it wasn’t until Cantor’s development (in the 1860’s and 1870’s) of the theory of infinite sets that the paradoxes could be fully resolved. Zeno’s paradoxes focus on the relation of the discrete to the continuous, an issue that is at the very heart of mathematics.

The Arrow Paradox


?AristotlePhysics VI:9, 239b5


The continuity of space or time, considered by Zeno and others, is represented in mathematics by the continuity of points on a line. As late as the seventeenth century, mathematicians continued to believe, as the ancient Greeks had, that this continuity of points was a simple result of density, meaning that between any two points, no matter how close together, there is always another. This is true, for example, of the rational numbers. However, the rational numbers do not form a continuum, since irrational numbers like ? 2 are missing, leaving holes or discontinuities. The irrational numbers are required to complete the continuum. Together, the rational and irrational numbers do form a continuous set, the set of real numbers. Thus, the continuity of points on a line is ultimately linked to the continuity of the set of real numbers, by establishing a one-to-one correspondence between the two. This approach to continuity was first established in the 1820s, by Augustin-Louis Cauchy, who finally began to solve the problem of handling continuity logically. In Cauchy’s view, any line corresponding to the graph of a function is continuous at a point, if the value of the function at x, denoted by f(x), gets arbitrarily close to f(p), when x gets close to a real number p. If f(x) is continuous for all real numbers x contained in a finite interval, then the function is continuous in that interval. If f(x) is continuous for every real number x, then the function is continuous everywhere.

Cauchy’s definition of continuity is essentially the one we use today, though somewhat more refined versions were developed in the 1850s, and later in the nineteenth century. For example, the concept of continuity is often described in relation to limits.


Whenever a point x is within ? units of cf(x) is within ? units of L.
For all x > Sf(x) is within ? of L.

Suppose f(x) is a real-valued function and c is a real number. The expression


means that f(x) can be made to be as close to L as desired by making x sufficiently close to c. In that case, it can be stated that “the limit of f of x, as x approaches c, is L“. Note that this statement can be true even if f(c) ? L. Indeed, the function f(x) need not even be defined at c.

For example, if


then f(1) is not defined (see division by zero), yet as x moved arbitrarily close to 1, f(x) correspondingly approaches 2:

f(0.9) f(0.99) f(0.999) f(1.0) f(1.001) f(1.01) f(1.1)
1.900 1.990 1.999 ? undef ? 2.001 2.010 2.100

Thus, f(x) can be made arbitrarily close to the limit of 2 just by making x sufficiently close to 1.

Augustin-Louis Cauchy in 1821,[3] followed by Karl Weierstrass, formalized the definition of the limit of a function into what became known as the (?, ?)-definition of limit in the 19th century. The definition uses ? (the lowercase Greek letter epsilon) to represent a small positive number, so that “f(x) becomes arbitrarily close to L” means that f(x) lies in the interval (L – ?, L + ?), which can also be written using absolute value as |f(x) – L| < ?.[3] The statement “x approaches c” then indicates that a positive number ? (the lowercase greek letter delta) exists within either (c – ?, c) or (c, c + ?), which can be expressed with 0 < |x – c| < ?. The first inequality means that the distance between x and c is more than 0 and that x ? c, while the second indicates that x is within distance ? of c.[3]

In addition to limits at finite values, functions can also have limits at infinity. For example, consider

f(x) = {2x-1 \over x}
  • f(100) = 1.9900
  • f(1000) = 1.9990
  • f(10000) = 1.99990

As x becomes extremely large, the value of f(x) approaches 2, and the value of f(x) can be made as close to 2 as one could wish just by picking x sufficiently large. In this case, the limit of f(x) as x approaches infinity is 2. In mathematical notation,

 \lim_{x \to \infty} f(x) = 2.



Limits and infinitesimals

Calculus is usually developed by manipulating very small quantities. Historically, the first method of doing so was by infinitesimals. These are objects which can be treated like numbers but which are, in some sense, “infinitely small”. An infinitesimal number dx could be greater than 0, but less than any number in the sequence 1, 1/2, 1/3, … and less than any positive real number. Any integer multiple of an infinitesimal is still infinitely small, i.e., infinitesimals do not satisfy the Archimedean property. From this point of view, calculus is a collection of techniques for manipulating infinitesimals. This approach fell out of favor in the 19th century because it was difficult to make the notion of an infinitesimal precise. However, the concept was revived in the 20th century with the introduction of non-standard analysis and smooth infinitesimal analysis, which provided solid foundations for the manipulation of infinitesimals.


Differential calculus

Tangent line at (xf(x)). The derivative f’(x) of a curve at a point is the slope (rise over run) of the line tangent to that curve at that point.


The most common symbol for a derivative is an apostrophe-like mark called prime. Thus, the derivative of the function of f is f’, pronounced “f prime.” For instance, if f(x) = x2 is the squaring function, then f’(x) = 2x is its derivative, the doubling function.


If a function is linear (that is, if the graph of the function is a straight line), then the function can be written as y = mx + b, where x is the independent variable, y is the dependent variable, b is the y-intercept, and:

m= \frac{\text{rise}}{\text{run}}= \frac{\text{change in } y}{\text{change in } x} = \frac{\Delta y}{\Delta x}.

This gives an exact value for the slope of a straight line. If the graph of the function is not a straight line, however, then the change in y divided by the change in x varies. Derivatives give an exact meaning to the notion of change in output with respect to change in input. To be concrete, let f be a function, and fix a point a in the domain of f. (af(a)) is a point on the graph of the function. If h is a number close to zero, then a + h is a number close to a. Therefore (a + hf(a + h)) is close to (af(a)). The slope between these two points is






The derivative f’(x) of a curve at a point is the slope of the line tangent to that curve at that point. This slope is determined by considering the limiting value of the slopes of secant lines. Here the function involved (drawn in red) is f(x) = x3 – x. The tangent line (in green) which passes through the point (-3/2, -15/8) has a slope of 23/4. Note that the vertical and horizontal scales in this image are different.



Leibniz notation





In this usage, the dx in the denominator is read as “with respect to x”. Even when calculus is developed using limits rather than infinitesimals, it is common to manipulate symbols like dx and dy as if they were real numbers; although it is possible to avoid such manipulations, they are sometimes notationally convenient in expressing operations such as the total derivative.

Integral calculus




A motivating example is the distances traveled in a given time.





To approximate that area, an intuitive method would be to divide up the distance between a and b into a number of equal segments, the length of each segment represented by the symbol ?x. For each small segment, we can choose one value of the function f(x). Call that value h. Then the area of the rectangle with base ?x and height h gives the distance (time ?x multiplied by speed h) traveled in that segment. Associated with each segment is the average value of the function above it, f(x)=h. The sum of all such rectangles gives an approximation of the area between the axis and the curve, which is an approximation of the total distance traveled. A smaller value for ?x will give more rectangles and in most cases a better approximation, but for an exact answer we need to take a limit as ?x approaches zero.

The symbol of integration is \int \,, an elongated S (the S stands for “sum”). The definite integral is written as:

\int_a^b f(x)\, dx.

and is read “the integral from a to b of f-of-x with respect to x.” The Leibniz notation dx is intended to suggest dividing the area under the curve into an infinite number of rectangles, so that their width ?x becomes the infinitesimally small dx. In a formulation of the calculus based on limits, the notation

\int_a^b \ldots\, dx


The indefinite integral, or antiderivative, is written:

\int f(x)\, dx.

Functions differing by only a constant have the same derivative, and therefore the antiderivative of a given function is actually a family of functions differing only by a constant. Since the derivative of the function y = x² + C, where C is any constant, is y’ = 2x, the antiderivative of the latter is given by:



Fundamental theorem


The Fundamental Theorem of Calculus states: If a function f is continuous on the interval [ab] and if F is a function whose derivative is f on the interval (ab), then


Furthermore, for every x in the interval (ab),


This realization, made by both Newton and Leibniz, who based their results on earlier work by Isaac Barrow, was key to the massive proliferation of analytic results after their work became known. The fundamental theorem provides an algebraic method of computing many definite integrals?without performing limit processes?by finding formulas for antiderivatives. It is also a prototype solution of a differential equation. Differential equations relate an unknown function to its derivatives, and are ubiquitous in the sciences.


Product Rule



or in the Leibniz notation thus:


The derivative of the product of three functions is:


Quotient Rule

In calculus, the quotient rule is a method of finding the derivative of a function that is the quotient of two other functions for which derivatives exist.[1][2][3]

If the function one wishes to differentiate, f(x), can be written as

f(x) = \frac{g(x)}{h(x)}

and h(x)\not=0, then the rule states that the derivative of g(x) / h(x) is

f'(x) = \frac{h(x)g'(x) - h'(x)g(x)}{[h(x)]^2}.

More precisely, if all x in some open set containing the number a satisfy h(x)\not=0, and g‘(a) and h‘(a) both exist, then f‘(a) exists as well and

f'(a)=\frac{h(a)g'(a) - h'(a)g(a)}{[h(a)]^2}.

And this can be extended to calculate the second derivative as well (you can prove this by taking the derivative of f(x) = g(x)(h(x)) ? 1 twice). The result of this is:


The quotient rule formula can be derived from the product rule and chain rule.


—Under Construction—-


Riemann Sums

Integral Sums

Chain Rule

Intergration by Parts

Substitution Method

Series and Expansions

Taylor Series

Fourier Series

Back to top