91Ó°ÊÓ

An oscilloscope often displays a sawtooth curve. This curve can be approximated by sinusoidal curves of varying periods and amplitudes.

Part (a): Use a graphing utility to graph the following function, which can be used to approximate the sawtooth curve.

$f\left(x\right)=\frac{1}{2}\sin \left(2\mathrm{Ï€³\ae }\right)+\frac{1}{4}\sin \left(4\mathrm{Ï€³\ae }\right),0≤x≤4$

Part (b): A better approximation to the sawtooth curve is given by

$f\left(x\right)=\frac{1}{2}\sin \left(2\mathrm{Ï€³\ae }\right)+\frac{1}{4}\sin \left(4\mathrm{Ï€³\ae }\right)+\frac{1}{8}\sin \left(8\mathrm{Ï€³\ae }\right)$

Use a graphing utility to graph this function for$0≤x≤4$ and compare the result to the graph obtained in part (a).

Part (c): A third and even better approximation to the sawtooth curve is given by

$f\left(x\right)=\frac{1}{2}\sin \left(2\mathrm{Ï€³\ae }\right)+\frac{1}{4}\sin \left(4\mathrm{Ï€³\ae }\right)+\frac{1}{8}\sin \left(8\mathrm{Ï€³\ae }\right)+\frac{1}{16}\sin \left(16\mathrm{Ï€³\ae }\right)$

Use a graphing utility to graph this function for$0≤x≤4$and compare the result to the graphs obtained in parts (a) and (b).

Part (d): What do you think the next approximation to the sawtooth curve is?

Step by step solution

01

Part (a) Step 1. Given information.

Consider the given function,

$f\left(x\right)=\frac{1}{2}\sin \left(2\mathrm{Ï€³\ae }\right)+\frac{1}{4}\sin \left(4\mathrm{Ï€³\ae }\right),0≤x≤4$

02

Part (a) Step 2. Plot the function.

On plotting the function,

03

Part (b) Step 1. Given information.

Consider the given function,

$f\left(x\right)=\frac{1}{2}\sin \left(2\mathrm{Ï€³\ae }\right)+\frac{1}{4}\sin \left(4\mathrm{Ï€³\ae }\right)+\frac{1}{8}\sin \left(8\mathrm{Ï€³\ae }\right)$

04

Part (b) Step 2. Plot the function on the same graph.

On plotting the function on the same graph,

On comparing the two graphs, we can say that the second graph has more peak points, the greater number of peaks create a better presentation for a sawtooth curve.

05

Part (c) Step 1. Given information.

Consider the given function,

$f\left(x\right)=\frac{1}{2}\sin \left(2\mathrm{Ï€³\ae }\right)+\frac{1}{4}\sin \left(4\mathrm{Ï€³\ae }\right)+\frac{1}{8}\sin \left(8\mathrm{Ï€³\ae }\right)+\frac{1}{16}\sin \left(16\mathrm{Ï€³\ae }\right)$

06

Part (c) Step 2. Plot the function on the same graph.

On plotting the function on the same graph,

On comparing the three graphs, we can say that the second graph has more peak points, the greater number of peaks create a better presentation for a sawtooth curve.

07

Part (d) Step 1. Find the next approximation to the sawtooth curve.

Consider the given question,

The actual equation is given below,

$f\left(x\right)=\underset{i=1}{\overset{n}{∑}}\frac{1}{2n}\sin \left(2n\mathrm{Ï€³\ae }\right)$

Therefore, the next approximation is given below,

$f\left(x\right)=\frac{1}{2}\sin \left(2\mathrm{Ï€³\ae }\right)+\frac{1}{4}\sin \left(4\mathrm{Ï€³\ae }\right)+\frac{1}{8}\sin \left(8\mathrm{Ï€³\ae }\right)+\frac{1}{16}\sin \left(16\mathrm{Ï€³\ae }\right)+\frac{1}{32}\sin \left(32\mathrm{Ï€³\ae }\right)$

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