Estimating π. Using random numbers can accomplish many tasks. For example, it is possible to estimate π using Monte Carlo methods. To get an estimate, you place random points in the square [0, 1] × [0, 1] and estimate π/4 as the ratio of the points that landed inside the unit circle to the total number of points (because the ratio of the area of the circle to the area of the square is π/4). Here is what a few test runs and the corresponding estimates for π might look like with the number of points equal to 100, 1000, and 10,000 respectively?

Estimations of π/4:

100 points: 0.75

1,000 points: 0.768

10,000 points: 0.7819

Step-by-step explanation:

To get an estimate of π/4 you can place random points in the square [0, 1] × [0, 1] and estimate it as the ratio of the points that landed inside the unit circle to the total number of points (because the ratio of the area of the circle to the area of the square is π/4).

We do it for 100 xy points and we get:

Point inside the circle area = 75

Estimation of π/4 = 0.75

We do it for 1,000 xy points and we get:

Point inside the circle area = 768

Estimation of π/4 = 0.768

If we do it fo 10,000 xy points, we get

Point inside the circle area = 768

Estimation of π/4 = 0.768

The value of π/4 (4 decimals) is 0.7854.

The simulation gets more precise with the increase in the number of points.

The spreadsheet and the graphs are attached.

answer: on graphs, the open and closed circles, or vertical asymptotes drawn as dashed lines us identify discontinuities.

as before, graphs and tables allow us to estimate at best.

when working with formulas, getting zero in the denominator indicates a point of discontinuity.

when working with piecewise-defined functions, check for discontinuities at the transition points where one piece ends and the next begins.

step-by-step explanation:

step-by-step explanation:

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