slope Archives - Infinity is Really Big

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

January 8, 2019July 13, 2019

Deconstructing the Mean Value Theorem, Part 1

Visual interpretation of the Mean Value Theorem. — The Mean Value Theorem says there’s at least one point where the slope of the tangent line on a smooth curve equals the slope of the secant line between the endpoints.

Many mathematicians start chanting “MVT! MVT! MVT!” when they see the Mean Value Theorem in action. This is indeed because it is so valuable.

What does it mean … Read the rest