Tangent and normals at a point (AA/AI SL 5.4)

Keywords

English	Japanese	Korean	Chinese Simplified
Tangent Line Equation	接線の方程式	접선의 방정식	切线方程
Normal Line	法線	법선	法线
Differential Calculus	微分計算法	미분 계산법	微分学
Derivative	導関数	도함수	导数
Slope	傾き	기울기	斜率
Negative Reciprocal	負の逆数	음의 역수	负倒数
Coordinates	座標	좌표	坐标

Factual Inquiry Questions	Conceptual Inquiry Questions	Debatable Inquiry Questions
What is the formula for finding the equation of a tangent line to a curve at a given point?	Why is the derivative of a function at a point used to find the slope of the tangent line at that point?	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?
How is the normal line to a curve at a particular point defined in differential calculus?	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?	Can the study of tangents and normals provide insights into non-mathematical fields such as economics and social sciences? How?
		How might advancements in computational methods impact the application and significance of tangents and normals in solving real-world problems?

Tangents and normals

Scenario: The Mysterious Hills of Functionland Background: In the fantastical realm of Functionland, the landscape is shaped by mathematical functions. A legendary hill shaped like the function

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. Objective: 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.

2. Unveiling the Tangent Line: - Use the derivative of the hill's function to find the slope of the tangent line at your chosen point. - Calculate the y-intercept of the tangent line using the point-slope formula. - Reveal the equation of the tangent line and describe its magical properties.

3. Revealing the Normal Line: - Use the negative reciprocal of the tangent's slope to find the slope of the normal line. - Calculate the y-intercept of the normal line using the point-slope formula. - State the equation of the normal line and explain its mystical significance.

Questions for Investigation: 1. Discovery Question: - If you were to pick another point on the hill, how would the equations of the tangent and normal lines change?

Investigation Steps: 1. Finding the Path's Coordinate: - Choose a point on the hill's path and determine its coordinates.

2. Understanding Slopes: - Why does the normal line have a negative reciprocal slope compared to the tangent line?

Part 2 - Checking understanding

Check out these two videos to see it with the calculator

Reflection - How might the equations of the tangent and normal lines help a traveler navigate the hidden paths of Functionland? - In what ways do the concepts of tangents and normals connect to the real world?

Tangent and normals at a point (AA/AI SL 5.4)

Keywords

Tangents and normals

Part 2 - Checking understanding

[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)

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Tangent and normals at a point (AA/AI SL 5.4)

Keywords

Tangents and normals

Part 2 - Checking understanding

[MAA 5.4] TANGENT AND NORMAL LINES

[MAA 5.4] TANGENT AND NORMAL LINES.pdf

[MAA 5.4] TANGENT AND NORMAL LINES_solutions

[MAA 5.4] TANGENT AND NORMAL LINES_solutions.pdf

Tangent and normals at a point- Intuition pump (thought experiments and analogies)

Tangent and normals at a point.pdf

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