History of Calculus
This is the mathematics of (instantaneous rates of) change.
What is calculus?
This mathematician is known for his mathematical description of the forces acting on a lever.
Who is Archimedes?
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 derivative defined in terms of limits.
What is the limit as the change in x approaches 0 of the change in y over the change in x.
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.
The mathematization of motion.
What is the Platonic-Pythagorean project?
It is the independent variable in the function d(t)=16t^2.
What is time?
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?
How one derives the velocity function from Galileo’s law of free fall.
What is the first derivative of Galileo's law of free fall function?
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.
Morris Kline unpacked the Pythagorean motto "All is number" this way.
What is...
1. The universe is ordered according to perfect mathematical laws.
2. Divine reason is the orderer.
3. Human reason can discern the divine mathematical pattern.
The difference between average speed and instantaneous speed?
What is average speed is an interval of distance divided by an interval of time whereas instantaneous speed is distance divided by an exact moment in time.
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?
The general form of the power function and the Power Rule in Calculus.
What is
f(x)=kx^n
and
f '(x)= nkx^n-1?
This man reasoned that motion does not exists.
Who is Zeno?
The three central concepts of calculus.
What are limits, derivatives, and integrals of functions?
What we say when we represent physical properties with numbers.
What is quantifying that property?
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?
How one derives the acceleration function from Galileo’s law of free fall.
What is the second derivative of Galileo's law of free fall function?
Thales inaugurated Western philosophy circa 500 B.C. by asking this question and provided this answer.
What is "Why do things change?" What is water?
This mathematician said, "Philosophy is written in this all-encompassing book that is constantly open before our eyes, that is the universe; but it cannot be understood unless one first learns to understand the language and knows the characters in which it is written. It is written in mathematical language, and its characters are triangles, circles, and other geometrical figures; without these it is humanly impossible to understand a word of it, and one wanders around pointlessly in a dark labyrinth."
Who is Galileo?
What the constant represents in this function that shows the relationship between height and time:
h(t)=-16t^2 +12.
What is the initial height (12 feet)?
The definition of acceleration in terms of limits.
What is the limit as the change in t approaches 0 of the change in velocity over the change in time?
Taking the derivative of this function results in the formula for the circumference of a circle.
What is A(r)=𝝅r2?