Book Review: The Humongous Book of Calculus Problems

I bought this book a year ago as a refresher of Calculus and as of now, I am almost finished reading it. I think what separates this book from the rest are the numerous worked examples (well, 1000 of them) with detailed solutions and explanations. Additional pointers and explanations in layman’s words are provided as notes.

This book has 565 pages containing 28 chapters. The first 8 chapters contain a review about equations, polynomials, functions, and trigonometry, while the remaining chapters discussed topics in Calculus I and II: Limits, Differentiation, Integration, Parametric and Polar Equations, Sequences and Series. As a bonus, a chapter on Differential Equations is also included.  » Read more

Understanding Hilbert’s Grand Hotel Paradox

Long ago, in a land far away, there was a grand hotel where there were infinitely many rooms. This hotel was attended by a brilliant manager.

One night, a guest arrived, but  the hotel was full — each room was occupied by one guest. The newly arrived guest asked if a spare room was available. “Of course we have, we are the Infinite Grand Hotel. There is always a room for everyone,” the manager said proudly.

Now since each room was occupied by a guest, the manager requested the guest in Room 1 to move to Room 2, the guest in Room 2 to move to Room 3, the guest in Room 3 to move to Room 4, and so on. Basically, he told every guest in Room n to move to Room n + 1. Since the hotel had infinitely many rooms, there was no problem in moving, there was always a room to move to. This left Room 1 vacant, and therefore, the guest was accommodated. The guest was happy. The manager was happy.  » Read more

The 0.0001 Time Rate Approach, an Extended Application

In the previous post, I introduced to you the 0.0001 Time Rate approach This post is an extended application of the said technique on curvatures and radius of curvatures.

Curvature is defined (by The Facts on File Dictionary of Mathematics) as the rate of change of the slope of the tangent to a curve. For each point on a smooth curve there is a circle that has the same tangent and the same curvature at that point. The radius of this circle called the “radius of curvature”, is the reciprocal of the curvature, and its center is known as the center of curvature. If the graph of a function y=f(x) is a continuous curve, the slope of the tangent at any point is given by the derivative dy/dx and the curvature is given by

$C = \displaystyle\frac{\displaystyle\frac{d^2y}{dx^2}}{(1+(\displaystyle\frac{dy}{dx})^2)^{3/2}}$

and the radius of curvature is given by

$RC = \frac{1}{C} = \displaystyle \frac{(1+(\displaystyle \frac{dy}{dx})^2)^{3/2}}{\displaystyle \frac{d^2y}{dx^2}}$

where: $\frac{dy}{dx}$ is the derivative of $y$ with respect to $x$ and $\frac{d^2x}{dx^2}$  is the second derivative of y with respect to x or in simpler terms, it is the derivative of $\frac{dy}{dx}$. » Read more

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