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(sqrt(cos(x))*cos(200x)+sqrt(abs(x))-0.7)*(x*x)^0.01 Sqrt(9-x^2)from-4.5 To 4.5

4 min read Jul 02, 2024
(sqrt(cos(x))*cos(200x)+sqrt(abs(x))-0.7)*(x*x)^0.01 Sqrt(9-x^2)from-4.5 To 4.5

Exploring the Mysterious Function: (sqrt(cos(x))cos(200x)+sqrt(abs(x))-0.7)(x*x)^0.01 sqrt(9-x^2) from -4.5 to 4.5

In the vast realm of mathematical functions, there exist some that are truly fascinating and intriguing. One such function is the subject of our exploration today: (sqrt(cos(x))*cos(200x)+sqrt(abs(x))-0.7)*(x*x)^0.01 sqrt(9-x^2).

Breaking Down the Function

At first glance, this function may seem like a complex beast, but let's break it down into its constituent parts to better understand its behavior.

  • sqrt(cos(x)): This term is a square root of the cosine of x. The cosine function has a range of [-1, 1], so the square root will always be a positive value between 0 and 1.
  • cos(200x): This term is a cosine function with a frequency of 200x. The cosine function has a range of [-1, 1], and with such a high frequency, it will oscillate rapidly.
  • sqrt(abs(x)): This term is a square root of the absolute value of x. This will always be a positive value.
  • (x*x)^0.01: This term is x squared raised to the power of 0.01. This will always be a positive value close to 1, since x is squared and then raised to a small power.
  • sqrt(9-x^2): This term is a square root of the difference between 9 and x squared. This will always be a positive value, since x is squared and then subtracted from 9.

Plotting the Function

Now that we've broken down the function, let's plot it to visualize its behavior. We'll plot the function from -4.5 to 4.5, as specified.

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Observations

From the plot, we can observe the following:

  • The function has a overall oscillatory behavior, due to the high-frequency cosine term.
  • The function values are mostly positive, with some negative values towards the edges of the domain.
  • The function has a high degree of symmetry around the origin, due to the even terms in the function.
  • The function values decrease in magnitude as x approaches the edges of the domain, due to the term (x*x)^0.01.

Conclusion

In conclusion, the function (sqrt(cos(x))*cos(200x)+sqrt(abs(x))-0.7)*(x*x)^0.01 sqrt(9-x^2) is a complex and intriguing function that exhibits oscillatory behavior and symmetry around the origin. Its values decrease in magnitude as x approaches the edges of the domain, making it a fascinating subject for further exploration and analysis.

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