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 |
![Image](https://www.geogebra.org/resource/qqrequpm/6CjN0L2Od5WZbXVo/material-qqrequpm.png)
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
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
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?