Algebra and Precalculus Archives - Infinity is Really Big

September 26, 2019September 26, 2019

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.*

*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

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.

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.

*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

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

February 26, 2019February 27, 2019

New Video: Dynamic Pascal’s Triangle

Made with Mathematica

The top 6 rows of Pascal’s triangle. Each row starts and ends with a “1”. Any number in-between the ends is the sum of the two numbers most close to it on the preceding line.

Pascal’s triangle is an infinite triangle of numbers with many internal patterns.

Start by creating a list of “1’s” in an upside-down … Read the rest

February 8, 2019December 1, 2019

Why Does Slicing a Cone Give an Ellipse?

The intersection of the pink plane with the blue cone is an ellipse (shaded orange when y > 0 and brown when y < 0). The length of either red line added to the length of either yellow line is constant.

Currently my favorite YouTube channel is “3Blue1Brown“, created by Grant Sanderson.

With excellent visuals and high production quality, Grant both shows and explains mathematics in a way that … Read the rest

January 16, 2019January 28, 2019

Financial Mathematics Terms: Effective Versus Nominal Rates of Interest

One of the most confusing topics for many people when starting to learn about financial mathematics is the various ways of measuring compound interest.

Terms like “effective interest rate”, “nominal interest rate”, “continuous interest rate”, and “real interest rate” show up in various articles and textbooks. Exploring the distinctions and relationships between effective and nominal interest rates is the … Read the rest

Category: Algebra and Precalculus

Transformations of Functions