September 2019 - Infinity is Really Big

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