IB Mathematics AA SL 5.4 Tangents and normals at a given point Study Notes- New Syllabus
IB Mathematics AA SL 5.4 Tangents and normals at a given point Study Notes
IB Mathematics AA SL 5.4 Tangents and normals at a given point Study Notes Offer a clear explanation of Use of Tangents and normals at a given points , including various formula, rules, exam style questions as example to explain the topics. Worked Out examples and common problem types provided here will be sufficient to cover for topic Tangents and normals at a given points.
Tangents and Normals at a Given Point — Analytic Approach
Tangents and Normals at a Given Point — Analytic Approach
Given a function \( y = f(x) \):
Tangent at a point: The slope of the tangent at \( x = x_0 \) is \( f'(x_0) \).
Normal at a point: The normal is perpendicular to the tangent. Its slope is \( -\frac{1}{f'(x_0)} \) (if \( f'(x_0) \neq 0 \)).
Equation of the tangent line:
\( y – y_0 = f'(x_0)(x – x_0) \) where \( y_0 = f(x_0) \)
Equation of the normal line:
\( y – y_0 = -\frac{1}{f'(x_0)}(x – x_0) \)
Example:
Find the equations of the tangent and normal to \( y = x^3 – 3x \) at the point where \( x = 1 \).
▶️ Answer/Explanation
\( y_0 = 1^3 – 3(1) = 1 – 3 = -2 \)
\( f'(x) = 3x^2 – 3 \)
\( f'(1) = 3(1)^2 – 3 = 3 – 3 = 0 \)
The tangent is horizontal: \( y = -2 \)
The normal is vertical (since the tangent is horizontal): \( x = 1 \)
Tangents and Normals Using Technology
Tangents and Normals Using Technology (GDC Steps)
You can use a graphing calculator or graphing software (e.g. TI-Nspire, Casio fx-CG50, Desmos) to find the tangent or normal at a point on a curve. Here’s how:
- Enter the function
- Input your function \( f(x) \) into the graphing calculator’s graph mode.
- Plot the graph
- Set an appropriate viewing window and display the graph of \( f(x) \).
- Select the tangent or normal tool
- On TI-Nspire: Menu → Analyze Graph → Tangent Line or Normal Line
- On Casio: Interactive → Geometry → Tangent / Normal
- On Desmos: Type derivative at point or use tools to find tangent line.
- Specify the point
- Move the cursor along the graph to the desired \( x \)-value or enter it directly (e.g., \( x = 2 \)).
- Read off the equation
- The GDC displays the equation of the tangent (or normal) line in the form: \( y = m(x – x_0) + y_0 \) or simplified to \( y = mx + c \).
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The technology computes \( f'(x_0) \) to find the slope of the tangent and derives the normal slope as \( -\frac{1}{f'(x_0)} \).
Example:
Use your GDC to find the tangent and normal lines to \( f(x) = x^2 + 2x \) at \( x = 1 \).
▶️ Answer/Explanation
- Enter \( f(x) = x^2 + 2x \) into the graphing calculator or Desmos.
- Plot the function on a suitable window (e.g., \( -5 \le x \le 5 \)).
- Select:
TI-Nspire → Menu → Analyze Graph → Tangent Line → click or enter \( x=1 \).
Casio → Interactive → Geometry → Tangent → click \( x=1 \). - The GDC shows the tangent line: Compute manually for confirmation: \( f'(x) = 2x + 2 \) \( f'(1) = 2(1) + 2 = 4 \) \( f(1) = 1 + 2 = 3 \) Tangent: \( y – 3 = 4(x – 1) \Rightarrow y = 4x – 1 \)
- Now find the normal line using: TI-Nspire → Menu → Analyze Graph → Normal Line → \( x=1 \) Casio → Interactive → Geometry → Normal → \( x=1 \) Normal: \( y – 3 = -\frac{1}{4}(x – 1) \Rightarrow y = -\frac{1}{4}x + \frac{13}{4} \)
