Slope of Tangent and Normal

Slope of a tangent

Slope of a tangent to the curve y = f(x) is given by dy / dx

Slope of a normal

Slope of the normal to the curve y = f(x) is given by -1 / (dy/dx)

Equation of Tangent and Normal

Equation of tangent at (x₀, y₀)

Equation of the tangent at the point (x₀, y₀) is given by
                                  (y - y₀) = dy / dx (x - x₀)
                                              OR
                                  (y - y₀) = f’(x) (x - x₀)

Equation of normal at (x₀, y₀)

 
Equation of the normal at the point (x₀, y₀) is given by
                               (y – y₀) = -1 / (dy / dx) (x - x₀)
                                               OR
                               (y - y₀) = [-1 / f’(x₀)] (x - x₀)

Condition for perpendicularity and parallelism

•    If the lines are parallel then the slopes are equal
•    If the lines are perpendicular the product of their slopes is -1
•    If the tangent is parallel to x-axis then dy / dx = 0
•    If the tangent is parallel to y-axis then dy / dx = 1 / 0
•    Slope of the line segment joining two points (x₁, y₁) and (x₂, y₂) is given by (y₂ - y₁) / (x₂ - x₁)
The following examples will help us to understand the concept more thoroughly:
Example 1: Find the point on the curve y = x³ - 11x + 5 at which tangent is y = x - 11
Solution: Slope of the tangent to the curve, dy / dx = 3x² - 11
Since y = x - 11, dy / dx = 1
3x² – 11 = 1
x² = 4
x = 2
When x = 2, y = 23 – 11 x 2 + 5 = 8 – 22 + 5 = -9, so the point is (2, -9)

Example 2: Find the equation of the tangent line to the curve y = x² - 2x + 7 which is parallel to the line 2x – y + 9 = 0
Solution: Slope of the tangent = 2x - 2
Slope of the line = 2
Since they are parallel, 2x – 2 = 2
x =2
When x = 2, y = 4 – 4 + 7 = 7
So the point is (2, 7)
Equation of the tangent is, y – 7 = 2 (x - 2)
                               y = 2x + 3

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Reference Links:

• http://en.wikipedia.org/wiki/Slope
• http://en.wikipedia.org/wiki/Tangent
• http://ref.subwiki.org/wiki/Normal_(mathematics)
• http://en.wikipedia.org/wiki/Equation
• http://en.wikipedia.org/wiki/Parallel_(geometry)
• http://en.wikipedia.org/wiki/Perpendicular