[table][br][tr][br][td]English[/td][br][td]Japanese[/td][br][td]Korean[/td][br][td]Chinese Simplified[/td][br][/tr][br][tr][br][td]Second Derivative[/td][br][td]二次導関数[/td][br][td]2차 도함수[/td][br][td]二阶导数[/td][br][/tr][br][tr][br][td]Concavity[/td][br][td]凹み[/td][br][td]오목함[/td][br][td]凹性[/td][br][/tr][br][tr][br][td]Inflection Points[/td][br][td]変曲点[/td][br][td]변곡점[/td][br][td]拐点[/td][br][/tr][br][tr][br][td]Second Derivative Test[/td][br][td]二次導関数テスト[/td][br][td]2차 도함수 검정[/td][br][td]二阶导数检验[/td][br][/tr][br][tr][br][td]Local Maximum[/td][br][td]局所最大値[/td][br][td]극대값[/td][br][td]局部最大值[/td][br][/tr][br][tr][br][td]Local Minimum[/td][br][td]局所最小値[/td][br][td]극소값[/td][br][td]局部最小值[/td][br][/tr][br][tr][br][td]Polynomial Function[/td][br][td]多項式関数[/td][br][td]다항 함수[/td][br][td]多项式函数[/td][br][/tr][br][tr][br][td]Maxima and Minima[/td][br][td]最大値と最小値[/td][br][td]최대값과 최소값[/td][br][td]极大值与极小值[/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 does the second derivative of a function represent?[/td][br][td]Why is the sign of the second derivative critical in determining whether a point is a local maximum, local minimum, or an inflection point?[/td][br][td]Is the analysis of second derivatives more significant in theoretical mathematics than in practical applications, or does it hold substantial practical value?[/td][br][/tr][br][tr][br][td]How is the second derivative test used to determine the concavity of a function and identify inflection points?[/td][br][td]How can the concept of the second derivative enhance our understanding of a function's graph and its geometric properties?[/td][br][td]Can the principles underlying second derivatives be effectively applied in fields outside of mathematics, such as economics or physics, to predict changes and trends? How?[/td][br][/tr][br][/table][br]

Scenario: The Great Calculus Coaster[br][br]Background:[br]Welcome to Mathemagic Land, where the latest attraction is the Great Calculus Coaster! This thrilling roller coaster is designed based on the principles of calculus, with its path modeled by a polynomial function. The coaster's engineers use the second derivative to ensure the safety and excitement of the ride by identifying maximum and minimum points of the coaster's hills and valleys.[br][br]Objective:[br]As a junior roller coaster designer, your task is to use the applet provided to understand how the second derivative affects the design of the roller coaster's path and ensures a thrilling yet safe ride for all guests.[br][br]Investigation Steps:[br][br]1. Understanding the Function:[br] - Examine the original function that models the path of the roller coaster.[br] - Identify the points where the first derivative is zero to find potential maxima and minima.[br][br]2. Applying the Second Derivative Test:[br] - Use the second derivative to determine whether each point is a maximum or a minimum.[br] - Explain how the second derivative provides this information.[br][br]3. Designing for Thrills and Safety:[br] - Discuss how maxima and minima points contribute to the thrill of a roller coaster.[br] - Consider how these points also relate to the safety of the ride.[br][br]4. Presenting Your Findings:[br] - Create a presentation for the Mathemagic Land board of directors, showcasing how calculus ensures both the excitement and security of the Great Calculus Coaster.[br][br]Questions for Investigation:[br][br]1. Discovery Question:[br] - If you were to change the function of the coaster's path, how would that affect the location and nature of its maxima and minima?[br][br][br]2. Engineering Challenges:[br] - What are some potential design challenges you might face when applying calculus to roller coaster design?[br][br]4. Reflection:[br] - Reflect on the importance of calculus in engineering and design.[br][br]

Part 2 - Formalising findings and what it all means

It's also useful to know that your calculator has the ability to numerically find the derivative at a point on a graph, and also to plot the derivative.[br]