Notes on Secant/Tangent Lines, Linear Approximation, and Derivatives

Secant line, tangent line, and intuition

Secant line: when you connect two points on a graph with a straight line, that line is called a secant line. Secant means cutting across the graph.
Tangent line intuition: as the two points used to form the secant line get closer together (the interval shrinks) and hone in on a, the line approaches the tangent line at x = a. The tangent line is the line that 'touches' the graph at a point and approximates the graph locally.
Important nuance: the tangent line does not have to touch the graph at exactly one point. In some contexts (e.g., certain oscillatory engineering graphs), the line can touch the graph in multiple places or even resemble the graph closely over an extended region.

The tangent line provides a very good approximation to the curve over a small region around the point a.
If you zoom in on a small piece of the curve around a, the piece looks nearly straight, and the tangent line is the line that best describes that local straightened piece.
The concept of instantaneous rate of change is tied to this tangent line: the slope of the tangent line is the instantaneous rate of change at a.

Given a function f, at x = a the tangent line has slope f'(a) and passes through the point (a, f(a)). Using point-slope form, the tangent line can be written as:
y - f(a) = f'(a)\,(x - a)
Rewriting in slope-intercept form gives:
y = f(a) + f'(a)\,(x - a)
The linear (tangent) approximation to f near a is:
f(x) \approx f(a) + f'(a)\,(x - a)
This can also be phrased in terms of changes:
\Delta y \approx f'(a)\,\Delta x \quad \text{where} \quad \Delta x = x - a, \; \Delta y = f(x) - f(a)
Important caveat: this is a good approximation when x is close to a; as you move far from a, the tangent line may no longer be a good approximation.

Set up: let y = f(t) be the distance traveled in meters, with time t in seconds.
Given:
- After t = 4 seconds, the object has traveled 100 meters: f(4) = 100
- The instantaneous rate of change at t = 4 is 30 meters per second: f'(4) = 30
Tangent line (around t = 4):
y = f(4) + f'(4)\,(t - 4) = 100 + 30\,(t - 4)
This line describes the motion locally around t = 4. It is not the exact motion unless the speed were constant.
To predict distance at t = 4.2 seconds using the tangent line:
f(4.2) \approx 100 + 30\,(4.2 - 4) = 100 + 30\,(0.2) = 106
Therefore, the approximate increase in distance over that 0.2 s interval is \Delta y \approx 6 \text{ meters} .
Relationship to the change notation:
\Delta y \approx f'(a)\,\Delta x
Rearranged for the derivative:
f'(a) \approx \frac{f(x) - f(a)}{x - a}

When given a table of x and f(x), you can approximate the derivative at a point by using slopes between nearby points.
Example setup (conceptual, not numeric exact values from transcript): to approximate f'(9) , you can use:
- Slope between (9, f(9)) and (12, f(12)):
  \frac{f(12) - f(9)}{12 - 9}
- Slope between (6, f(6)) and (9, f(9)):
  \frac{f(9) - f(6)}{9 - 6}
Because the function may bend, these two estimates can differ. Averaging them (a symmetrical interval approach) can reduce bias and give a better estimate of the derivative at x = 9.
Practical note: real-world data often contain noise. The simple rule of connecting adjacent points to form a straight line can be a poor approximation in noisy data. In practice, statistical methods (regression, smoothing) are used to extract the underlying trend.
A qualitative takeaway: data = model + noise. The underlying model (the trend) is what we try to recover; noise represents random fluctuations.
In this course, we often work with idealized, precisely measured data to focus on the models and methods rather than statistical noise.

Replacing the constant a with a variable x yields the derivative function, also called f'(x) or the derived function.
The derivative function f'(x) gives the instantaneous rate of change of f at any x where the derivative exists:
f'(x) = \text{the instantaneous rate of change of } f \text{ at } x.
If f is linear, the derivative f'(x) is constant. For most functions studied in this class, the derivative varies with x.

The derivative at a point x is defined as the limit of the average rate of change as the interval shrinks to zero:
f'(x) = \lim_{h \to 0} \frac{f(x+h) - f(x)}{h}
This is the limit of the average rate of change between x and x + h as h tends to 0.
Example computation: let f(x) = x^2. Then
f'(x) = \lim{h \to 0} \frac{(x+h)^2 - x^2}{h} \;=\, \lim{h \to 0} \frac{2xh + h^2}{h} \;=\, \lim_{h \to 0} (2x + h) = 2x.
The operation that maps f to f' is called differentiation; the derived function f' is a new function defined wherever the limit exists.

The derivative measures instantaneous rate of change.
The derivative function f'(x) generalizes the slope information to all x in the domain.
If f is linear: f'(x) is constant.
In general, the derivative varies with x and captures how the rate of change evolves along the graph.