2.5: Summary and Further Problems

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Geometric means, impedances, low-pass filters—these ideas are all abstractions. An abstraction connects seemingly random details into a higher-level structure that allows us to transfer knowledge and insights. By building abstractions, we amplify our intelligence.

Indeed, each of our reasoning tools is an abstraction or reusable idea. In Chapter 1, for example, we learned how to split hard problems into tractable ones, and we named this process divide-and-conquer reasoning. Don’t stop with this one process. Whenever you reuse an idea, identify the transferable process, and name it: Make an abstraction. With a name, you will recognize and reuse it

Exercise \(\PageIndex{1}\): From circles to spheres

In this problem, you first find the area of a circle from its circumference and then use analogous reasoning to find the volume of a sphere.

a. Divide a circle of radius r into pie wedges. Then snip and unroll the circle:

Use the unrolled picture and the knowledge that the circle’s circumference is \(2 \pi r\) to show that its area is \(\pi r^{2}\).

b. Now extend the argument to a sphere of radius r : Find its volume given that its surface area is \(4 \pi r^{2}\). (This method was invented by the ancient Greeks.)

Exercise \(\PageIndex{2}\): Gain of an LRC circuit

Use the impedance of an inductor (Problem 2.19) to find the gain of the classic LRC circuit. In this configuration, in which the output voltage measured across the resistor, is the circuit a low-pass filter, a high-pass filter, or a band-pass filter?

Exercise \(\PageIndex{3}\): Continued fraction

Evaluate the continued fraction

\[1 + \frac{1}{1 + \frac{1}{1+...}}\]

Compare this problem with Problem 2.8.

Exercise \(\PageIndex{4}\): Exponent tower

Evaluate

\[\sqrt{2}^{\sqrt{2}^{\sqrt{2}}}\]

Here, \(a^{b^{c}}\) means a^{(b^{(c)})}\).

Exercise \(\PageIndex{5}\): Coaxial cable termination

In physics and electronics laboratories around the world, the favorite way to connect equipment and transmit signals is with coaxial cable. The standard coaxial cable, RG-58/U, has a capacitance per length of 100 picofarads per meter and an inductance per length of 0.29 microhenries per meter. It can be modeled as a long inductor–capacitor ladder:

What resistance R placed at the end (in parallel with the last capacitor) makes the cable look like an infinitely long LC cable?

Exercise \(\PageIndex{6}\): UNIX and Linux

Using Mike Gancarz’s The UNIX Philosophy [17] and Linux and the Unix Philosophy [18], find examples of abstraction in the design and philosophy of the UNIX and Linux operating systems.