Circle Equation: Center (0,0)

For the questions below, be sure to zoom out if you need to!
1.
Suppose [i]P(x,y)[/i] = any point that lies on a circle with center (0,0) and radius 5. [br]Use what you've observed to write an equation that expresses the relationship among [i]x[/i], [i]y[/i], and [i]r[/i].
2.
What is the equation of a circle with center (0,0) and radius [i]r[/i] = 9?
3.
Suppose another circle has center (0,0). Suppose this circle also passes through the point (12, -5).[br]Write the equation of this circle. [br]
4. FINAL QUESTION:
Suppose [i]P(x,y)[/i] = any point that lies on a circle with center (0,0) and radius [i]r[/i], where [i]r[/i] > 0. [br]Use what you've observed to write an equation that expresses the relationship among [i]x[/i], [i]y[/i], and [i]r[/i].
Quick (Silent) Demo

Angle Relationships

Angle Relationships

Polynomials

Polynomials
https://teacher.desmos.com/activitybuilder/custom/561582ecbd554ea00760f933?collections=651ca31cf69ee59aa9e3818a,5e73b275913f047206662889

The Unit Circle

Move the theta slider to investigate unit circle values.
Unit Circle - The "Unit Circle" is a circle with Radius 1.
Circle with Radius r, move the radius value to see how our values change.

Function and Their Derivatives

Function and Their Derivatives

Properties of Power Series

Power Series
A power series has the general form[br][math]\sum_{k=0}^{\infty}c_k\left(x-a\right)^k[/math][br]where a and [math]c_k[/math] are real numbers and x is a variable. The [math]c_k[/math]'s are the [b]coefficients [/b]of the power series and a is the [b]center[/b] of the power series. The set of values of x for which the series converges is its [b]interval of convergence[/b]. The[b] radius of convergence[/b] of the power series, denoted [i]R[/i] is the distance from the center of the series to the boundary of the interval of convergence.
Convergence of Power Series
A power series [math]\sum_{k=0}^{\infty}c_k\left(x-a\right)^k[/math] centered at a converges in one of three ways.[br][b]1) [/b]The series converges for all x, in which the interval of converges is [math]\left(-\infty,\infty\right)[/math] and the radius of convergence is [math]R=\infty[/math][br][br][b]2) [/b]There is a real number R>0 such that the series converges for |x-a|<R and diverges for |x-a|>R, in which case the radius of convergence is R.[br][br][b]3)[/b] The series converges only at a, in which case the radius of convergence is R=0
Combining Power Series
Suppose the power series [math]\sum c_kx^k[/math] and [math]\sum d_kx^k[/math] converges to f(x) and g(x) respectively, on an interval I.[br][br][b]1. Sum and Difference[/b]: The power series [math]\sum\left(c_k\pm d_k\right)x^k[/math] converges to [math]f\left(x\right)\pm g\left(x\right)[/math] on I[br][br][b]2. Multiplication by a Power: [/b]Suppose m is an integer such that [math]k+m\ge0[/math] for all terms of the power series [math]x^m\sum c_kx^k=\sum c_kx^{k+m}[/math]. This series converges to [math]x^mf\left(x\right)[/math] for all [math]x\ne0[/math] in I. When x=0, the series converges to [math]lim_{x\longrightarrow0}x^mf\left(x\right)[/math][br][br][b]3. Composition: [/b]If[math]h\left(x\right)=bx^m[/math], where m is a positive integer and b is a nonzero real number, the power series [math]\sum c_k\left(h\left(x\right)\right)^k[/math] converges to the composite function [math]f\left(h\left(x\right)\right)[/math], for all x such that h(x) is in I.
Differentiating and Integrating Power Series
Suppose the power series [math]\sum c_k\left(x-a\right)^k[/math] converges for |x-a|<R and defines a function f on that interval[br][br][b]1.  [/b]Then f is differentiable (which implies continuous) fro |x-a|<R and f' is found by differentiating the power series for the f term; that is [br][math]f'\left(x\right)=\sum kc_k\left(x-a\right)^{k-1}[/math][br]for |x-a|<R[br][br][b]2.[/b] The indefinite integral of f is found by integrating the power series for f term by term; that is,[br][math]\int f\left(x\right)dx=\sum c_k\frac{\left(x-a\right)^{k+1}}{k+1}+C[/math][br]for |x-a|<R, where C is an arbitrary constant.

Comparing Dot Plots

Observe how variations in characteristics of data sets affect the shape of dot plots.
Putting It All Together
[i]Answer these open ended questions on your own or with others to form deeper math connections.[/i]
Modify the second data set and consider whether one center of distribution (one of the means) is much higher than the other, or if the centers (the means) are close together. How do the centers of distribution compare across the data sets of the dot plots in the following cases?[br]a) The difference in the means is less than 2 MADs[br]b) The difference in the means is more than 2 MADs
Modify the second data set so that the ranges of the dot plots are very different. Describe in your own words how the ranges compare with one another.
How do the variabilities of the two dot plots compare in the following cases?[br]a) The MAD of the original data set is higher[br]b) The MAD of the modified data set is higher[br]c) The MADs are close together
a) Modify the maximum of the second data set without changing the mean. How would you describe the difference between the two data sets?[br]b) Modify the maximum without changing the MAD. Compare the data sets in your own words again.
Imagine that the dot plots represent the heights of flowers in a garden bed.[br]a) How would you label the two different number lines?[br]b) What would a higher MAD mean for a population of flowers?
What other populations of people or things could the dot plots represent? How could you describe a comparison of the two populations?

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