Quotient rule (AASL/HL)
Keywords
[table][br][br][tr][br][td]Quotient rule[/td][br][td]商の法則[/td][br][td]분수 규칙[/td][br][td]商规则[/td][br][/tr][br][tr][br][td]Differential calculus[/td][br][td]微分計算法[/td][br][td]미분 계산법[/td][br][td]微分学[/td][br][/tr][br][tr][br][td]Derivative[/td][br][td]微分[/td][br][td]미분[/td][br][td]导数[/td][br][/tr][br][tr][br][td]Functions quotient[/td][br][td]関数の商[/td][br][td]함수의 몫[/td][br][td]函数的商[/td][br][/tr][br][tr][br][td]Rates of change[/td][br][td]変化率[/td][br][td]변화율[/td][br][td]变化率[/td][br][/tr][br][tr][br][td]Numerator and denominator functions[/td][br][td]分子と分母の関数[/td][br][td]분자 및 분모 함수[/td][br][td]分子和分母函数[/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]1. What is the quotient rule in differential calculus?[/td][br][td]1. Why is the quotient rule necessary for differentiating quotients of functions, and how does it differ from simply dividing their individual derivatives?[/td][br][td]1. Is the quotient rule more prone to errors in application than the product rule due to its complexity?[/td][br][/tr][br][tr][br][td]2. How is the quotient rule applied to find the derivative of the quotient of two functions?[/td][br][td]2. How does the quotient rule illustrate the relationship between the rates of change of the numerator and denominator functions?[/td][br][td]2. Could the principles of the quotient rule be simplified or improved to make calculus more accessible to beginners?[/td][br][/tr][br][tr][br][td][/td][br][td][/td][br][td]3. How might the role of the quotient rule in teaching calculus evolve with the increasing use of technology in education?[/td][br][/tr][br][/table][br]
Inquiry questions
Factual Inquiry Questions[br]What is the quotient rule in differential calculus?[br]How is the quotient rule applied to find the derivative of the quotient of two functions?[br][br]Conceptual Inquiry Questions[br]Why is the quotient rule necessary for differentiating quotients of functions, and how does it differ from simply dividing their individual derivatives?[br]How does the quotient rule illustrate the relationship between the rates of change of the numerator and denominator functions?[br][br]Debatable Inquiry Questions[br]Is the quotient rule more prone to errors in application than the product rule due to its complexity?[br]Could the principles of the quotient rule be simplified or improved to make calculus more accessible to beginners?[br]How might the role of the quotient rule in teaching calculus evolve with the increasing use of technology in education?
Mini-Investigation: The Quotient Quest[br][br]Welcome, brave Calculus Explorer! Today, we set sail on a mathematical voyage to uncover the mysteries of the Quotient Rule. Are you ready to unravel the enigma of dividing functions?
Chapter 1: Discovery of the Quotient Realm[br][br]In the land of Numeratoria and Denominatoria, two functions u(x) and v(x) live in harmony. Your first discovery is their quotient, y = (x + 3) / x.[br][br]1. Imagine u(x) and v(x) are two different territories. How do they come together to form the landscape of y?[br]2. Without using the Quotient Rule, can you find an alternative path to travel from u(x)/v(x) to dy/dx?
Chapter 2: The Quotient Rule Riddle[br][br]A legendary scroll reveals the Quotient Rule, a powerful formula: dy/dx = (v du/dx - u dv/dx) / v^2. It's time to decipher this riddle![br][br]1. Apply the Quotient Rule to u(x) and v(x). Document each magical transformation step by step.[br]2. Compare your journey using the Quotient Rule to the alternative path you found earlier. Which was more perilous and which was more straightforward?
Chapter 3: The Duel of Derivatives[br][br]A challenge is issued! You must use both the Quotient Rule and the alternative method to find the derivative of a new territory, [math]y=(x+5)/(x-1)[/math].[br][br]1. Calculate [math]dy/dx[/math] using both methods. Which one brings you to the solution faster?[br]2. Is there treasure to be found in understanding both methods? What insights do they offer about the changing landscape of y?
Question 1:[br]If[math]f(x)=x^3[/math] and [math]g(x)=x^2+1[/math], what is the derivative of [math]h(x)=f(x)/g(x)[/math]?[br][br]
Chapter 4: Realms of Application[br][br]Beyond the theoretical world lies a vast expanse of practical applications, where the Quotient Rule helps navigate complex terrains.[br][br]1. In the realm of the real world, where might the Quotient Rule be essential for understanding rates of change?[br]2. Can you find a situation where the alternative method might provide deeper understanding, despite potentially being more complex?
Part 2 - Checking your understanding
See how you do with these questions. Attempt these questions. [br]Watch the video to see the quotient rule in action
Question 2:[br]Given [math]u(x)=e^x[/math] and [math]v(x)=cos(x)[/math], what is the derivative of [math]w(x)=u(x)/v(x)[/math]?[br][br][br]
Question 3:[br]If [math]p(t)=ln(t)[/math] and [math]q(t)=t^2[/math], what is the derivative of [math]r(t)=p(t)/q(t)[/math]?[br][br][br]
Question 4:[br]Consider [math]f(x)=sin(x)[/math] and [math]g(x)=e^x[/math]. What is the derivative of the quotient [math]h(x)=f(x)/g(x)[/math]?[br][br]
Question 5:[br]If [math]m(x)=x^4[/math]and [math]n(x)=2x-1[/math], what is the derivative of [math]o(x)=m(x)/n(x)[/math]?
Part 3 - Exam style questions
Question 1-9 - Practice questions[br]Question 10-25 - Section A short response exam-style[br]Question 26-27 - Section B long response exam-style[br][br][b]Question 5, 12, 14, 18 specifically require use of quotient rule[/b]
[MAA 5.2] DERIVATIVES - BASIC RULES
[MAA 5.2] DERIVATIVES - BASIC RULES_solutions
Lesson plan - Exploring the Quotient Rule in Differential Calculus
Quotient rule -- Intuition pump (thought experiments and analogies)
