Problem 1 This problem focuses on the function f defined by Equation (1):

f(z) = 2
z
2
"
1
(1 − z
2
)
−1/2 − 1
+ 0#−1
. (1)
(1.1) Plot f(z) only for values of z in the interval where 0 < z < 10−7
.
The interval where 0 < z < 10−7 must fill the horizontal axis of the plot.
(1.2) Plot f(z) only for values of z in the interval where 0 < z < 10−15
.
The interval where 0 < z < 10−15 must fill the horizontal axis of the plot.

Here are the plots for the function \( f(z) = 2z^2 \left(1 - z^2\right)^{-\frac{1}{2}} - 1 \):

1. **Plot for \( 0 < z < 10^{-7} \)** (left plot): The function is shown over this small interval, filling the horizontal axis.
2. **Plot for \( 0 < z < 10^{-15} \)** (right plot): This shows the function over an even smaller interval, as requested.

Let me know if you'd like a detailed explanation of the behavior of the function in these intervals or if you have other questions.

Here are five related questions to consider:
1. What happens to the function \( f(z) \) as \( z \) approaches zero?
2. How does the behavior of \( f(z) \) change if we change the power in the denominator \( \left(1 - z^2\right)^{-1/2} \)?
3. What would the function look like for negative values of \( z \)?
4. How does this function compare to simpler functions like \( z^2 \)?
5. How does the term \( \left(1 - z^2\right)^{-\frac{1}{2}} \) affect the overall behavior of \( f(z) \)?

**Tip**: Be cautious with very small intervals like \( 10^{-15} \), as the function can be numerically unstable near zero!

This problem focuses on the function f defined by Equation (1):
f(z) = 2z^2 (1 − z^2)^{-1/2} − 1.
(1.1) Plot f(z) only for values of z in the interval where 0 < z < 10^{-7}.
(1.2) Plot f(z) only for values of z in the interval where 0 < z < 10^{-15}.

Explore the behavior of the function f(z) = 2z^2 (1 − z^2)^{-1/2} − 1 by plotting it over intervals 0 < z < 10^{-7} and 0 < z < 10^{-15}. Learn about limits, asymptotic behavior near z = 0, and numerical stability in function plotting. Suitable for undergraduate-level calculus or analysis students.

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