Tangent and normals at a point (AA/AI SL 5.4)
Keywords
[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]Tangent Line Equation[/td][br][td]接線の方程式[/td][br][td]접선의 방정식[/td][br][td]切线方程[/td][br][/tr][br][tr][br][td]Normal Line[/td][br][td]法線[/td][br][td]법선[/td][br][td]法线[/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]導関数[/td][br][td]도함수[/td][br][td]导数[/td][br][/tr][br][tr][br][td]Slope[/td][br][td]傾き[/td][br][td]기울기[/td][br][td]斜率[/td][br][/tr][br][tr][br][td]Negative Reciprocal[/td][br][td]負の逆数[/td][br][td]음의 역수[/td][br][td]负倒数[/td][br][/tr][br][tr][br][td]Coordinates[/td][br][td]座標[/td][br][td]좌표[/td][br][td]坐标[/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 formula for finding the equation of a tangent line to a curve at a given point?[/td][br][td]Why is the derivative of a function at a point used to find the slope of the tangent line at that point?[/td][br][td]In the context of mathematical modeling, is the analysis of tangent lines more critical than that of normal lines, or do they hold equal importance?[/td][br][/tr][br][tr][br][td]How is the normal line to a curve at a particular point defined in differential calculus?[/td][br][td]How does the concept of a normal line relate to the tangent line at the same point on a curve, and what does it signify about the curve's geometry?[/td][br][td]Can the study of tangents and normals provide insights into non-mathematical fields such as economics and social sciences? How?[/td][br][/tr][br][tr][br][td][/td][br][td][/td][br][td]How might advancements in computational methods impact the application and significance of tangents and normals in solving real-world problems?[/td][br][/tr][br][/table][br]
Tangents and normals
Scenario: The Mysterious Hills of Functionland[br][br]Background: In the fantastical realm of Functionland, the landscape is shaped by mathematical functions. A legendary hill shaped like the function[math]f(x)=x^2[/math] has a path that runs straight to the top, but the locals believe it's enchanted because no one can see the path directly—only its tangent and normal lines at any given point.[br][br]Objective:[br]As an aspiring mathematical mage, you're tasked with uncovering the secrets of the hill's path using the magic of calculus to find the equations of these invisible lines.[br]
2. Unveiling the Tangent Line:[br] - Use the derivative of the hill's function to find the slope of the tangent line at your chosen point.[br] - Calculate the y-intercept of the tangent line using the point-slope formula.[br] - Reveal the equation of the tangent line and describe its magical properties.
3. Revealing the Normal Line:[br] - Use the negative reciprocal of the tangent's slope to find the slope of the normal line.[br] - Calculate the y-intercept of the normal line using the point-slope formula.[br] - State the equation of the normal line and explain its mystical significance.
Questions for Investigation:[br][br]1. Discovery Question:[br] - If you were to pick another point on the hill, how would the equations of the tangent and normal lines change?[br]
Investigation Steps:[br][br]1. Finding the Path's Coordinate:[br] - Choose a point on the hill's path and determine its coordinates.
2. Understanding Slopes:[br] - Why does the normal line have a negative reciprocal slope compared to the tangent line?[br]
Part 2 - Checking understanding
Check out these two videos to see it with the calculator[br]
Reflection[br] - How might the equations of the tangent and normal lines help a traveler navigate the hidden paths of Functionland?[br] - In what ways do the concepts of tangents and normals connect to the real world?
[MAA 5.4] TANGENT AND NORMAL LINES
[MAA 5.4] TANGENT AND NORMAL LINES_solutions
Tangent and normals at a point- Intuition pump (thought experiments and analogies)
