The Fundamental Theorems of Calculus (FTC)There are 2 parts of the Fundamental Theorem of Calculus.Part 1 of the FTC tells us that we can figure out the exact value of an indefinite integral (area under the curve) when we know the interval over which to evaluate.Part 2 (Evaluation theorem) states that if we can find an anti-derivative for the integrand, then we can evaluate the definite integral by evaluating the anti-derivative at the endpoints of the interval and subtracting.
DerivativesDerivatives in calculus, is the rate of change of a function with respect to a variable. They are a fundamental tool of calculus.Rate of Change refers to the slope of the tangent line to the graph of the function at a specific point.
What are Derivatives?
Yes! The are called "Anti-Derivatives" or "Integrals"Integrals is the reverse of a derivative and is the opposite of differential calculus. Integration can be used to find areas, volumes, central points and many useful things. But in class, we use it to find the area under the curve in most cases.In particle motion, Velocity & Acceleration have anti-derivatives. The integral of Velocity is Position. v(t) -> p(t)The integral of Acceleration is Velocity. a(t) -> v(t)
Absolutely! Some Rules includes:The function f(x) must be continuous during the the interval in question. The interval must be closed, which means that both limits must be constants (real numbers only, no infinity allowed).Examples:
But wait! Isn't there an opposite of a derivative?
100%, through... "Particle Motion"!!!Through Particle motion which introduces us to conceptions of physics such as Position, Velocity, & Acceleration.Using those concepts, we can theoretically use calculus calculate how long it would take for something or someone to get from Point A to B. The particle may be a “particle,” a person, a car, or some other moving object. The position, velocity, or acceleration may be given as an equation, a graph or a table and sometimes you will be given an initial condition to work with.
ConclusionDerivatives & Integrals are a fundamental tools of calculus. As seen in these comic strip panels, It is used in many different ways to tell us the rate of change, particle motion, area of a curve, and etc. It is useful for many real world situation which is why people study it.
So you can tell how fast I'm speeding down the supermarket aisle through calculus?
Are there any theorems related to Derivatives or Anti-Derivatives?