Slope of the Tangent and Normal Line

BASIC CONCEPTS OF GRAPHICAL ANALYSIS

When studying functions and their graphs, we are often interested in the orientation of the line that "touches" the graph at a specific point. This line is called the tangent. The line perpendicular to the tangent at the same point is called the normal. Both lines are defined by their slope (or gradient), which measures the inclination of the line with respect to the horizontal axis.

SLOPE OF THE TANGENT LINE

The tangent to the graph of a function at a certain point is a line that touches the graph at that point and has the same slope as the function at that point. The slope of the tangent line at a point x₀ is equal to the value of the derivative of the function f at x₀, therefore:

k_t = f′(x₀)
(where k_t is the slope of the tangent)

This means that if we have a given function f and we calculate its derivative, the value of the derivative at the chosen point gives us the slope of the tangent.

SLOPE OF THE NORMAL LINE

The normal line is a line that passes through the same point as the tangent but is perpendicular to it. If the tangent is increasing, the normal is decreasing, and vice versa. Mathematically, the slope of the normal line is given by:

k_n = -1 / f′(x₀), provided f′(x₀) ≠ 0.
(where k_n is the slope of the normal)

Thus, the normal's slope is defined only at points where the tangent is defined and its slope is not equal to 0. If f′(x₀) = 0, then the tangent line is horizontal, and the normal line is vertical (and does not have a defined slope in the form y=mx+b).

EXAMPLE: FUNCTION F(X) = X² AT POINT X = 1

First, we calculate the derivative:
f′(x) = 2x.
At x = 1, the derivative is f′(1) = 2(1) = 2.

Therefore, the slope of the tangent line is k_t = 2.
The slope of the normal line is:
k_n = -1 / 2.

The point on the graph is (1, f(1)) = (1, 1² ) = (1, 1).
The equation of the tangent line at (1, 1) is:
y – y₁ = k_t(x – x₁)
y – 1 = 2(x – 1) → y = 2x – 2 + 1 → y = 2x – 1.

The equation of the normal line at (1, 1) is:
y – y₁ = k_n(x – x₁)
y – 1 = (-1/2)(x – 1) → y = (-1/2)x + 1/2 + 1 → y = -0.5x + 1.5.

CONCLUSION

The slope of the tangent line is equal to the value of the derivative of the function at a specific point. The normal line, being perpendicular to the tangent, has a slope that is the negative reciprocal of the tangent's slope, provided the tangent's slope exists and is non-zero. Both slopes are key in describing the local geometry of a function's graph.