Category: Calculus

Calculus is the study of the relationships between quantities that change in a continuous way. The idea of a limit forms the foundation for the main topics of derivatives, integrals, and infinite series.

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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.

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

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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

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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