Infinity is Really Big - Page 2 of 7 - Bill Kinney's Blog on Mathematics, Applications, Life, and Christian Faith

January 14, 2020January 14, 2020

Applied vs Pure Mathematics

Calculus 2, Lecture 1 (Videotaped Fall 2016)

Population modeling is an important part of applied mathematics in calculus. It is a good example to start with in looking at the nature of applied vs pure mathematics. — Population modeling is an important part of applied mathematics in calculus.

Calculus is at the heart of much of modern mathematics and its applications. Its discovery by Newton and Leibniz sparked the scientific revolution and enlightenment of the 1600’s and 1700’s and brought about our modern world.

Throughout mathematical history, … Read the rest

January 8, 2020June 29, 2021

Introduction to Multivariable Calculus

Calculus 1, Lectures 18B through 20B

In an introduction to multivariable calculus, the graph of a function of more than one variable should be sliced to help you understand it and its partial derivatives. — The graph of a multivariable function can be sliced to help you understand it and its partial derivatives.

In some ways, multivariable calculus seems like a minor extension of single-variable calculus ideas and techniques. In other ways, it’s definitely a major step up in difficulty.

This post will mainly be an introduction to multivariable … Read the rest

November 30, 2019November 30, 2019

Infinitesimal Calculus and Calculus Rules

Calculus 1, Lectures 16A through 18A

Infinitesimal calculus can be used to derive the derivative of the sine function.

Did you know that Newton and Leibniz did not know the precise definition of a limit?

Instead, they approached calculus in an intuitive way. Today, this intuitive method is called infinitesimal calculus. It is based on the concept of … Read the rest

October 29, 2019October 29, 2019

Differentiable Functions and Local Linearity

Calculus 1, Lectures 12 through 15B

An infinitely oscillating continuous and differentiable function. — This function is both continuous and smooth at $x=0$ , in spite of oscillating infinitely often in any neighborhood of $x=0$ .

x=0 — This function is both continuous and smooth at $x=0$ , in spite of oscillating infinitely often in any neighborhood of $x=0$ .

In Steven Strogatz’s excellent book, Infinite Powers, there is a big emphasis on an idea he calls The Infinity Principle.

The concepts of a continuous function and of a differentiable function are … Read the rest

October 15, 2019October 15, 2019

Limit Definition, Continuity, and Derivatives

Calculus 1, Lectures 8A through 11

The limit definition of the derivative leads to the graph of a function with a hole (a removable discontinuity). The limit still exists, based on the precise epsilon delta definition of a limit. — The graph of the difference quotient $DQ(h)=\frac{(2+h)^{2}-8}{h}$ for computing the derivative $f'(2)$ when $f(x)=x^{3}$ has a removable discontinuity (a hole) at $h=0$ . The limit still exists so that $f'(2)=12$ .

DQ(h)=\frac{(2+h)^{2}-8}{h} — The graph of the difference quotient $DQ(h)=\frac{(2+h)^{2}-8}{h}$ for computing the derivative $f'(2)$ when $f(x)=x^{3}$ has a removable discontinuity (a hole) at $h=0$ . The limit still exists so that $f'(2)=12$ .

Graphs with holes (missing points) might seem like anomalies — like they are mere curiosities hardly worth studying. However, they arise at the heart of differential calculus.

… Read the rest

September 26, 2019September 26, 2019

Transformations of Functions

Calculus 1, Lectures 6A through 7B

Visual for the fact that $\log_{2}(2x)=\log_{2}(x)+\log_{2}(2)=\log_{2}(x)+1$ in terms of transformations of functions.

\log_{2}(2x)=\log_{2}(x)+\log_{2}(2)=\log_{2}(x)+1 — Visual for the fact that $\log_{2}(2x)=\log_{2}(x)+\log_{2}(2)=\log_{2}(x)+1$ in terms of transformations of functions.

In the 1600’s, algebra and geometry got “married” to create a new subject called analytic geometry. Insights in either branch led to insights in the other. This has been very fruitful over the years.

The symbiosis between these subjects can be especially … Read the rest

September 20, 2019September 26, 2019

The Derivative of y = x^2

Calculus 1, Lectures 4A through 5B

The derivative of x^2 at an arbitrary value of x is 2x. This is the slope of the tangent line at that point. — The derivative of $y=f(x)=x^{2}$ at an arbitrary value of $x$ is equal by $2x$ . This is the slope of the tangent line at that point and is approximated by $2x+\Delta x$ , which are slopes of secant lines.

y=f(x)=x^{2} — The derivative of $y=f(x)=x^{2}$ at an arbitrary value of $x$ is equal by $2x$ . This is the slope of the tangent line at that point and is approximated by $2x+\Delta x$ , which are slopes of secant lines.

Lectures 4A through 5B of my series of Calculus 1 lectures have a lot of content. I will summarize the

… Read the rest

September 17, 2019September 26, 2019

Linear versus Exponential Growth and Decay

Calculus 1, Lectures 3A and 3B

For $f(x)=2\cdot 1.5^{x}$ , $f(x+2)=1.5^{2}f(x)$ , no matter what $x$ is. The exponential function is converting addition of inputs to multiplication of outputs.

f(x)=2\cdot 1.5^{x} — For $f(x)=2\cdot 1.5^{x}$ , $f(x+2)=1.5^{2}f(x)$ , no matter what $x$ is. The exponential function is converting addition of inputs to multiplication of outputs.

Linear functions are constructed from the arithmetic building blocks of repeated addition and subtraction. In a similar manner, exponential functions are constructed from repeated multiplication and division.

These sentences describe both the most basic similarity … Read the rest

September 12, 2019September 26, 2019

Proportionality and Linear Functions

Calculus 1, Lectures 2A and 2B

Photo by Thomas Ciszewski on Unsplash

Quantities that are directly proportional to each other arise everywhere in science. Examples include distance traveled with speed, volume with temperature for an ideal gas, and force with acceleration.

If the two quantities are thought of as variables, then the direct proportionality between them leads to an equation … Read the rest

September 9, 2019February 19, 2021

The Big Ideas of Calculus

Calculus 1, Lecture 1

The area of a circle can be found with the Infinity Principle. This is described by Steven Strogatz in his book Infinite Powers.

Calculus is often described as the mathematics of change. In terms of a short description, this is apt.

You might even say this description gives us a view of the big … Read the rest