The concepts of limits, infinitesimal partitions, and continuously changing quantities paved the way to Calculus, the universal tool for modeling continuous systems from Physics to Economics.
What's the sum of the first 100 positive integers? How about the first 1000?
Infinitely many mathematicians walk into a bar. The first says "I'll have a beer". The next ones say "I'll have half of the previous guy". The bartender pours out 2 beers and says "Know your limits".
What happens when a function's output isn't calculable directly – e.g., at infinity – but we still need to understand its behavior? That's where limits come in.
If there are no holes in your function, it's continuous! Many powerful theorems in Calculus only apply to these special types of functions.
How does infinity really work? Is it the biggest number? Is it even a number at all?
A derivative is simply a rate of change. Whether you're modeling the movement of a particle or a supply/demand model, this is a key instrument of Calculus.
These are the rules that explain how to take derivatives of any functions: from polynomials to trigonometric functions to logarithms.
The first derivative is the slope of a curve, and the second derivative is the slope of the slope, like acceleration. So what's the third derivative? (Fun fact: it's actually called jerk.)
Not all equations can be written like y = f(x), so taking the derivative can be tricky. Save the mess and do it directly with implicit differentiation.
Those friendly functions that don't contain breaks, bends or cusps are "differentiable". Take their derivative, or just infer some facts about them from the Mean Value Theorem.
When you've got a limit that looks like 0/0 or ∞/∞, L'Hôpital's rule can often find its value -- and make it clear that not all infinities are equal!
How does pressure change in a combustion engine? How fast does the water level rise when filling a pool? Calculus quantifies the impact of change on areas, angles, distances, temperatures, and more.
How can you maximize your happiness under a budget? When does a function reach its minimum value? When does a curve change direction? The calculus of extrema explains these "extreme" situations.
You don't need a calculator or computer to draw your graphs! Derivatives and other Calculus techniques give direct insights into the geometric behavior of curves.
How fast are you moving? How fast is how fast you are moving changing? Displacement, velocity, and acceleration form the art of understanding movement, Calculus-style.
The Taylor series is a polynomial of infinite degree used to represent functions like sine, cube roots, and the exponential function. They're how some calculators (and Physicists) make approximations.
This is the opposite of the derivative - and it's an integral part of Calculus. It's uses range from basic integrals to differential equations, with applications in Physics, Chemistry, and Economics.
The definite integral of a function computes the area under the graph of its curve, allowing us to calculate areas and volumes that are not easily done using geometry alone.
Writing an integral down is only the first step. A toolkit of techniques can help find its value, from substitutions to trigonometry to partial fractions to differentiation.
In the language of Calculus, the area between two curves is essentially a difference of integrals. The applications of this calculation range from macroeconomic tax models to signal analysis.
The integrals of these special functions have relationships to probability distributions, factorials, quantum physics, and fluid dynamics.
Integration extends to the third dimension with the means to compute the volumes of shapes obtained by revolving regions around an axis, such as spheres, toroids, and paraboloids.
Finding the perimeter of arbitrary curves and the area of 3D shapes foils the traditional tools of Geometry, and calls for the help of integrals and derivatives to make these calculations.
Derivatives are rates of change, and in the physical world that means things like velocity and acceleration. In fact, studying these quantities played a major role in the invention of Calculus.
As a particle moves, its position can often be written in terms of time. Systems like these can be modeled with parametric equations, which cleverly write coordinates in terms of a parameter.
Whether you're charting the seas or just doing some plotting in the complex plane, polar equations are the go-to tool for describing points via a distance and angle with respect to the origin.
Parametric equations can be quite handy, and we don't want to unravel them just to do Calculus. Some tricks can bend traditional derivative and integral methods to apply to parametric equations.
Just because your equation is polar doesn't mean you can't do Calculus! Some tricks can extend the toolkit of Calculus to these special equations.
What are the zeroes of a function? When an exact answer isn't attainable, Calculus provides approximation techniques - the same ones that allow calculators to find your roots.
Whether you want to program a calculator or just approximation an integral on paper, break out some numerical approximation tools like Simpson's Rule and the Trapezoid Rule.
When differential equations can't be solved explicitly, approximate solutions can be found via clever numerical methods.