Welcome to Calculus One
Welcome to Calculus! Join me on this journey through one of the great triumphs of human thought.
Functions and Limits
Functions are the main star of our journey. Calculus isn't numbers: it's relationships between things, and how one thing changing affects something else.
The End of Limits
People have thought about infinity for thousands of years; limits provide one way to make such ponderings precise. Continuity makes precise the idea that small changes in the input don't affect the output much.
The Beginning of Derivatives
It is time to change topics, or rather, to study change itself! When we wiggle the input, the output value changes, and that ratio of output change to input change is the derivative.
Techniques of Differentiation
With the product rule and the quotient rule, we can differentiate products and quotients. And since the derivative is a function, we can differentiate the derivative to get the second derivative.
The chain rule lets us differentiate the composition of two functions. The chain rule can be used to compute the derivative of inverse functions, too.
Derivatives of Transcendental (Trigonometric) Functions
So far, we can differentiate polynomials, exponential functions, and logarithms. Let's learn how to differentiate trigonometric functions.
Derivatives in the Real World
Derivatives can be used to calculate limits via l'Hôpital's rule. Given a real-world equation involving two changing quantities, differentiating yields "related rates."
In the real world, we must makes choices, and wouldn't it be great if we could make the best choice? Such optimization is made possible with calculus.
Replacing the curved graph by a straight line approximation helps us to estimate values and roots.
Antidifferentiation is the process of untaking derivatives, of finding a function whose derivatives is a given function. Since it involves working backwards, antidifferentiation feels like "unbreaking a vase" and can be just as challenging.
By cutting up a curved region into thin rectangles and taking a limit of the sum of the areas of those rectangles, we compute (define!) the area of a curved region.
Fundamental Theorem of Calculus
Armed with the Fundamental Theorem of Calculus, evaluating a definite integral amounts to finding an antiderivative.
Substitution systematizes the process of using the chain rule in reverse. Considering how often we used the chain rule when differentiating, we will often want to use it in reverse to antidifferentiate.
Techniques of Integration
Integration by parts is the product rule in reverse. Integrals of powers of trigonometric functions can be evaluated.
Applications of Integration
We have already used integrals to compute area; integration can also be used to compute volumes.