1. What Tangent and Normal Lines Represent
A tangent is the straight line that just touches the curve at a point and moves in the same direction as the curve there. The normal is the line perpendicular to the tangent at that point. In personal-notes style: tangent follows the curve’s direction, normal cuts across it at 90 degrees.
2. Slope of Tangent Using the Derivative
The derivative gives the slope of the tangent at any point on the curve.
2.1. Formula
m_{\text{tangent}} = f'(a)
Here, (a, f(a)) is the point of contact.
3. Equation of Tangent Line
Once the slope is known, the tangent line is written using the point–slope form.
3.1. Formula
y - f(a) = f'(a)(x - a)
3.2. Example
For f(x) = x^2 at x = 1:
f'(x) = 2x → f'(1) = 2
y - 1 = 2(x - 1)
4. Slope and Equation of the Normal Line
The normal line is perpendicular to the tangent, so its slope is the negative reciprocal of the tangent slope.
4.1. Slope of Normal
m_{\text{normal}} = -\frac{1}{f'(a)}
4.2. Equation of Normal
y - f(a) = -\frac{1}{f'(a)} (x - a)
4.3. Example
Using the earlier example f(x) = x^2 at x = 1:
m_{\text{normal}} = -\frac{1}{2}
y - 1 = -\frac{1}{2}(x - 1)
5. Finding Tangent and Normal at a Specific Point — Steps
- Find the derivative f'(x).
- Evaluate f'(a) to get tangent slope.
- Use point–slope form to write the tangent equation.
- Take negative reciprocal of slope to write the normal equation.
6. More Examples
These examples show how the tangent and normal are found in different types of functions.
6.1. Example 1 — Trigonometric Function
Find tangent and normal for f(x) = sin x at x = π/3.
f'(x) = \cos x → f'(π/3) = \frac{1}{2}
Tangent slope = 1/2
y - \frac{ \sqrt{3}}{2} = \frac{1}{2}(x - \frac{\pi}{3})
Normal slope = −2.
6.2. Example 2 — Exponential Function
For f(x) = e^x at x = 0:
f'(x) = e^x → f'(0) = 1
Tangent: y - 1 = 1(x - 0)
Normal: y - 1 = -1(x - 0)
7. Why Tangent and Normal Are Important
Tangent and normal lines help in curve analysis, motion problems, reflections, optimization geometry, and approximating curves using straight lines close to a point.