Maclaurin series

Keywords

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

Inquiry questions

Factual Inquiry Questions	Conceptual Inquiry Questions	Debatable Inquiry Questions
What is the Maclaurin series, and how is it defined for a function?	Why is the Maclaurin series considered a special case of the Taylor series?	Is the Maclaurin series more practical for computational purposes than other forms of series expansions? Why or why not?
Can you list the Maclaurin series for basic functions like , , and ?	How does the Maclaurin series help in approximating functions near the point ?	Can the limitations of the Maclaurin series in approximating functions over a wide range be effectively mitigated? If so, how?
		How might the understanding and application of the Maclaurin series change with further advancements in mathematical theory and computational technology?

The Maclaurin Series Mystery

Exploration Title: The Maclaurin Series Mystery Objective: 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! Mission Steps: 1. Series Start-up: - Identify the pattern in the derivatives of the sine function and how these are used in the Maclaurin series. - Calculate the first four non-zero terms of the Maclaurin series for cos(x). 2. Polynomial Probing: - 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. 3. Factorial Fun: - Notice the factorial in the Maclaurin series. Discuss why factorials are used and the impact they have on the function as n increases. Questions for Investigation: 1. Application Adventure: - How do Maclaurin series help us in real-world calculations, like those used in engineering or physics? 2. Convergence Challenge: - 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. 3. Trig Transformation: - Can you derive the Maclaurin series for tan(x) using the series for sin(x) and cos(x)? 4. Creative Coefficients: - 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? Engagement Activities: - "Series Showdown": Compete to see who can calculate higher orders of Maclaurin series the fastest. - "Graph Guessing Game": Given a series expansion, guess the original function and graph it to see if you're correct. 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.

Part 2 - Further examples and explanation

Watch the below video 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.

Maclaurin series

Keywords

Inquiry questions

The Maclaurin Series Mystery

Part 2 - Further examples and explanation

[MAA 5.23] MACLAURIN SERIES - EXTENSION OF BINOMIAL THEOREM

[MAA 5.23] MACLAURIN SERIES - EXTENSION OF BINOMIAL THEOREM_solutions