[table][br][br][tr][br][td]Maclaurin series[/td][br][td]マクローリン級数[/td][br][td]매클로린 급수[/td][br][td]麦克劳林级数[/td][br][/tr][br][tr][br][td]Taylor series[/td][br][td]テイラー級数[/td][br][td]테일러 급수[/td][br][td]泰勒级数[/td][br][/tr][br][tr][br][td]Function approximation[/td][br][td]関数近似[/td][br][td]함수 근사[/td][br][td]函数近似[/td][br][/tr][br][tr][br][td]Series expansions[/td][br][td]級数展開[/td][br][td]급수 전개[/td][br][td]级数展开[/td][br][/tr][br][tr][br][td]Polynomial[/td][br][td]多項式[/td][br][td]다항식[/td][br][td]多项式[/td][br][/tr][br][tr][br][td]Factorial[/td][br][td]階乗[/td][br][td]팩토리얼[/td][br][td]阶乘[/td][br][/tr][br][tr][br][td]Infinite series[/td][br][td]無限級数[/td][br][td]무한 급수[/td][br][td]无限级数[/td][br][/tr][br][/table][br]

[table][br][tr][br][td][b]Factual Inquiry Questions[/b][/td][br][td][b]Conceptual Inquiry Questions[/b][/td][br][td][b]Debatable Inquiry Questions[/b][/td][br][/tr][br][tr][br][td]What is the Maclaurin series, and how is it defined for a function?[/td][br][td]Why is the Maclaurin series considered a special case of the Taylor series?[/td][br][td]Is the Maclaurin series more practical for computational purposes than other forms of series expansions? Why or why not?[/td][br][/tr][br][tr][br][td]Can you list the Maclaurin series for basic functions like [math]e^x[/math], [math]sin(x)[/math], and [math]cos(x)[/math]?[/td][br][td]How does the Maclaurin series help in approximating functions near the point [math]x=0[/math]?[/td][br][td]Can the limitations of the Maclaurin series in approximating functions over a wide range be effectively mitigated? If so, how?[/td][br][/tr][br][tr][br][td][/td][br][td][/td][br][td]How might the understanding and application of the Maclaurin series change with further advancements in mathematical theory and computational technology?[/td][br][/tr][br][/table][br]

Exploration Title: The Maclaurin Series Mystery[br][br]Objective:[br]Delve into the infinite realms of calculus by decoding the Maclaurin series. Transform functions into their power series form, and predict the shape of curves with nothing but coefficients![br][br]Mission Steps:[br][br]1. Series Start-up:[br] - Identify the pattern in the derivatives of the sine function and how these are used in the Maclaurin series.[br] - Calculate the first four non-zero terms of the Maclaurin series for cos(x).[br][br]2. Polynomial Probing:[br] - Use the Maclaurin series to approximate sin(x) and cos(x) to the 4th order. Graph these approximations and the original functions to compare their accuracy.[br][br]3. Factorial Fun:[br] - Notice the factorial in the Maclaurin series. Discuss why factorials are used and the impact they have on the function as n increases.[br][br]Questions for Investigation:[br][br]1. Application Adventure:[br] - How do Maclaurin series help us in real-world calculations, like those used in engineering or physics?[br][br]2. Convergence Challenge:[br] - For which values of x does the series for sin(x) converge quickly to the actual value? Set up a graph to test your hypothesis.[br][br]3. Trig Transformation:[br] - Can you derive the Maclaurin series for tan(x) using the series for sin(x) and cos(x)?[br][br]4. Creative Coefficients:[br] - Create a new function by altering the coefficients of the Maclaurin series for sin(x). What does the graph of this new function look like?[br][br]Engagement Activities:[br][br]- "Series Showdown": Compete to see who can calculate higher orders of Maclaurin series the fastest.[br]- "Graph Guessing Game": Given a series expansion, guess the original function and graph it to see if you're correct.[br][br]Through this investigation, uncover the magic of infinite series and their power to unveil the continuous nature of the universe, one coefficient at a time.[br]

Watch the below video[br][br]If you want to see graphically what is happening in the video for y=e^(2x), you can try it in the applet above exp(2x) as the function.