Newton and Calculus
This is the mathematics of (instantaneous rates of) change.
What is calculus?
A mathematical relationship such that for every input, there is exactly one output.
What is a function?
The definition of a limit in words
What is the value you approach but never reach?
The single law that express the three central concepts of calculus
What is the Fundamental Theorem of Calculus?
The two steps utilized in the method of increments
What is...
1. Find a formula for the average velocity
2. Take the limit of the average velocity as delta t approaches 0?
This man overthrew Aristotle's reign during the Scientific Revolution.
Who is Newton?
It is the independent variable in the function d(t)=16t^2.
What is time?
The scenario where instantaneous velocity describes no distance traversed and no time has lapsed
What is the paradox of speed?
This method involves multiple numerical calculations in which there are shrinking intervals showing the "path" to reach the instantaneous rate of change
What is the method of approximation?
In the Method of Increments, delta t may be rearranged in these two ways
What is
t-initial=t-final minus delta t
or
t-final=delta t plus t-initial?
Newton imagined a mathematical curve this way related to the problem of change.
What are the paths of moving points?
The Platonic-Pythagorean Project
What is to describe the physical world mathematically?
How limits solve the paradox of speed
What is to make the interval of time infinitesimally small.
The definition of average velocity using final and initial times.
What is (d-final minus d-initial)/(t-final minus t-initial)?
The reason why delta t is rearranged in the Method of Increments
What is to find an equation for average velocity that does not have delta t in the denominator?
These are the four main problems that calculus was invented to solve.
What is...
1. Describing continually changing velocity and acceleration.
2. Finding the slope of a curve
3. Finding the tangent line to a curve
4. Finding the area under the curve.
The three central concepts of calculus
What are limits, derivatives, and integrals of functions?
The limit as x approaches 0 of x^2 +2
What is 2?
The definition of instantaneous speed
What is the limit when delta t approaches 0 of the average velocity function?
Use this acronym to expand a squared binomial
What is F.O.I.L?
Three main characteristics of the Scientific Revolution
What is the...
1. Overthrow of Aristotle
2. Reintroduction of the Platonic-Pythagorean Project.
3. The mathematization of motion.
What we say when we represent physical properties with numbers
What is quantifying that property?
The limit when x approaches 4 of (x-4)/(x^2-x-12)
What is 1/7?
The instantaneous velocity of a falling object after one seconds and the distance traveled after one second
What is 32ft/s and 16ft?
The Method of Increments and this generic variable derives the instantaneous velocity function
What is the generic t for t-final?