Tangent lines and Normal lines

A tangent line to a curve at a specific point is a straight line that touches the curve at exactly that point and has the same instantaneous slope as the curve at that point. In calculus, the slope of the tangent line at a point x = a is given by the value of the function's derivative at that point, f'(a).

A normal line to a curve at a specific point is a straight line that passes through the same point as the tangent line and is perpendicular to the tangent line at that point. It essentially represents the direction directly "away" from the curve at that location.

The tangent line and the normal line at the same point on a curve are always **perpendicular** to each other (unless the tangent is horizontal or vertical, in which case the normal is vertical or horizontal, respectively).

Since the normal line is perpendicular to the tangent line, their slopes are negative reciprocals of each other. If the slope of the tangent line at a point is mtangent, then the slope of the normal line at the same point is:

mnormal = -1 ÷ mtangent

**Important Cases:**

• If mtangent = 0 (the tangent line is horizontal), the normal line is vertical, and its slope is undefined.
• If the tangent line is vertical (its slope is undefined), the normal line is horizontal, and its slope is mnormal = 0.

To find the equation of the normal line to a function f(x) at a point where x = a:

1. Find the y-coordinate of the point: Calculate y0 = f(a). The point is (a, y0). This point is on both the curve and the normal line.
2. Find the derivative of the function: Calculate f'(x).
3. Find the slope of the tangent line at the point: Calculate mtangent = f'(a).
4. Find the slope of the normal line: Calculate mnormal = -1 ÷ mtangent (handling the horizontal/vertical cases mentioned above).
5. Write the equation of the normal line using the point-slope form: y - y0 = mnormal(x - a). You can then rearrange this into slope-intercept form (y = mx + b) if desired.

Follow these steps for a function f(x) at x = a:

1. Find the point of tangency: Calculate y0 = f(a). The point is (a, y0).
2. Find the derivative: Calculate f'(x).
3. Find the slope of the tangent: Calculate mtangent = f'(a).
4. Write the Tangent Line equation: Use y - y0 = mtangent(x - a).
5. Find the slope of the normal: Calculate mnormal = -1 ÷ mtangent (addressing horizontal/vertical cases).
6. Write the Normal Line equation: Use y - y0 = mnormal(x - a).