# Tangents and Normals

### Tangents and Normals:

Tangent Line:

• A tangent line to a curve at a given point shares the same slope as the curve at that specific point.
• Formula for Tangent Line Equation: If $f\left(x\right)$ is a function and the point $\left(a,f\left(a\right)\right)$ lies on the curve, the equation of the tangent line at $x=a$ is: $y-f\left(a\right)={f}^{\mathrm{\prime }}\left(a\right)\left(x-a\right)$

Normal Line:

• A normal line to a curve at a given point is perpendicular to the tangent line at that point.
• Formula for Normal Line Equation: If $f\left(x\right)$ is a function and the point $\left(a,f\left(a\right)\right)$ lies on the curve, the equation of the normal line at $x=a$ is: $y-f\left(a\right)=-\frac{1}{{f}^{\mathrm{\prime }}\left(a\right)}\left(x-a\right)$

### Calculation Using Derivative Rules:

1. Tangent Line Calculation:

• Step 1: Find the derivative ${f}^{\mathrm{\prime }}\left(x\right)$ of the function.
• Step 2: Substitute the x-coordinate $a$ into ${f}^{\mathrm{\prime }}\left(x\right)$ to find ${f}^{\mathrm{\prime }}\left(a\right)$.
• Step 3: Use the formula  to determine the equation of the tangent line.
2. Normal Line Calculation:

• Step 1: Find the derivative ${f}^{\mathrm{\prime }}\left(x\right)$ of the function.
• Step 2: Substitute the x-coordinate $a$ into ${f}^{\mathrm{\prime }}\left(x\right)$ to find ${f}^{\mathrm{\prime }}\left(a\right)$.
• Step 3: Use the formula  to determine the equation of the normal line.

### Strategies:

• Calculation of Derivatives: Accurately compute the derivative ${f}^{\mathrm{\prime }}\left(x\right)$ to determine slopes for tangents and normals.
• Point of Interest: Identify the point on the curve ($x=a$) to calculate function values and derivatives for the tangent and normal equations.

### Example:

Consider the function $f\left(x\right)={x}^{2}-3x+2$:

1. Tangent Line at $x=2$:

• Step 1: Find the derivative ${f}^{\mathrm{\prime }}\left(x\right)$: ${f}^{\mathrm{\prime }}\left(x\right)=2x-3$.
• Step 2: Substitute $x=2$ into ${f}^{\mathrm{\prime }}\left(x\right)$: ${f}^{\mathrm{\prime }}\left(2\right)=2×2-3=1$.
• Step 3: Use the tangent line formula: $y-f\left(2\right)={f}^{\mathrm{\prime }}\left(2\right)\left(x-2\right)$ $y-\left({2}^{2}-3×2+2\right)=1×\left(x-2\right)$
2. Normal Line at $x=2$:

• Step 1: ${f}^{\mathrm{\prime }}\left(x\right)=2x-3$.
• Step 2: ${f}^{\mathrm{\prime }}\left(2\right)=2×2-3=1$.
• Step 3: Use the normal line formula: $y-f\left(2\right)=-\frac{1}{{f}^{\mathrm{\prime }}\left(2\right)}\left(x-2\right)$ $y-\left({2}^{2}-3×2+2\right)=-\frac{1}{1}\left(x-2\right)$ $y-3=-\left(x-2\right)$  $y=-x+5$

### Subtangents and Subnormals:

• Subtangent: A line segment drawn from a point on the curve to the x-axis parallel to the tangent at that point.
• Subnormal: A line segment drawn from a point on the curve to the x-axis perpendicular to the tangent at that point.

### Formulas and Calculations using Derivative Rules:

1. Subtangent Length:

• For a curve $y=f\left(x\right)$, the length of the subtangent at a point $x=a$ is given by:
2. Subnormal Length:

• For a curve $y=f\left(x\right)$, the length of the subnormal at a point $x=a$ is given by:

### Example:

#### Example 1: Subtangent and Subnormal of a Curve

Given $f\left(x\right)={x}^{3}-3{x}^{2}+2x-5$:

1. Find the lengths of the subtangent and subnormal at $x=2$.

Solution:

1. Subtangent Length:

• Find ${f}^{\mathrm{\prime }}\left(x\right)$ to determine the derivative at $x=2$: ${f}^{\mathrm{\prime }}\left(x\right)=\frac{d}{dx}\left({x}^{3}-3{x}^{2}+2x-5\right)=3{x}^{2}-6x+2$
• Calculate ${f}^{\mathrm{\prime }}\left(2\right)$ to get the derivative at $x=2$: ${f}^{\mathrm{\prime }}\left(2\right)=3\left(2{\right)}^{2}-6\left(2\right)+2=4$
• Subtangent length formula:
2. Subnormal Length:

• Find $f\left(2\right)$ to determine the function value at $x=2$: $f\left(2\right)=\left(2{\right)}^{3}-3\left(2{\right)}^{2}+2\left(2\right)-5=2$
• Subnormal length formula:

### Strategies:

• Derivative Calculation: Accurately compute the derivative ${f}^{\mathrm{\prime }}\left(x\right)$ to determine values for calculating subtangent and subnormal lengths.
• Utilize Absolute Values: Ensure absolute values are used when finding lengths to consider both positive and negative values.