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(x) is a function and the point (a,f(a)) lies on the curve, the equation of the tangent line at x=a is: yf(a)=f(a)(xa)

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(x) is a function and the point (a,f(a)) lies on the curve, the equation of the normal line at x=a is: yf(a)=1f(a)(xa)

Calculation Using Derivative Rules:

  1. Tangent Line Calculation:

    • Step 1: Find the derivative f(x) of the function.
    • Step 2: Substitute the x-coordinate a into f(x) to find f(a).
    • Step 3: Use the formula Tangent Line Equation=yf(a)=f(a)(xa) to determine the equation of the tangent line.
  2. Normal Line Calculation:

    • Step 1: Find the derivative f(x) of the function.
    • Step 2: Substitute the x-coordinate a into f(x) to find f(a).
    • Step 3: Use the formula Normal Line Equation=yf(a)=1f(a)(xa) to determine the equation of the normal line.

Strategies:

  • Calculation of Derivatives: Accurately compute the derivative f(x) 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(x)=x23x+2:

  1. Tangent Line at x=2:

    • Step 1: Find the derivative f(x): f(x)=2x3.
    • Step 2: Substitute x=2 into f(x): f(2)=2×23=1.
    • Step 3: Use the tangent line formula: yf(2)=f(2)(x2) y(223×2+2)=1×(x2)
  2. Normal Line at x=2:

    • Step 1: f(x)=2x3.
    • Step 2: f(2)=2×23=1.
    • Step 3: Use the normal line formula: yf(2)=1f(2)(x2) y(223×2+2)=11(x2) y3=(x2)  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(x), the length of the subtangent at a point x=a is given by: Subtangent length=1f(a)
  2. Subnormal Length:

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

Example:

Example 1: Subtangent and Subnormal of a Curve

Given f(x)=x33x2+2x5:

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

Solution:

  1. Subtangent Length:

    • Find f(x) to determine the derivative at x=2: f(x)=ddx(x33x2+2x5)=3x26x+2
    • Calculate f(2) to get the derivative at x=2: f(2)=3(2)26(2)+2=4
    • Subtangent length formula: Subtangent length=1f(2)=14=14
  2. Subnormal Length:

    • Find f(2) to determine the function value at x=2: f(2)=(2)33(2)2+2(2)5=2
    • Subnormal length formula: Subnormal length=f(2)f(2)=24=12

Strategies:

  • Derivative Calculation: Accurately compute the derivative f(x) 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.