Linear Functions and Rate of Change Preview

Learning Objectives

By the end of this section you will be able to:

1. State what makes a function linear and recognise the form $y = m\,x + b$.
2. Compute slope from any two points using $m = \dfrac{\Delta y}{\Delta x}$ — and explain why the answer never depends on which two points you pick.
3. Translate between three forms of a line: slope–intercept, point–slope, and two-point.
4. Generalise the slope formula to average rate of change for any function — the secant slope $\dfrac{f(b)-f(a)}{b-a}$.
5. See how the secant rotates onto the tangent as the two points collapse — the first taste of the derivative.
6. Implement all of the above in plain Python and then in PyTorch, and confirm the limit numerically against autograd.

The Big Picture: Constant Rate of Change

A linear function is a function whose rate of change never changes. Pick any starting point, walk any distance to the right, and the function will always step up (or down) by the same multiple of how far you walked.

Imagine you are filling a bathtub from a tap that pours at exactly the same speed every second. After one second the water is up by some amount; after two seconds it is up by twice that amount; after three seconds, three times. The water level is a linear function of time. The story is boring in the best possible way — every second is exactly like the last.

Why is this the right place to start a calculus book? Because all of differential calculus is built on a single dream: zoom in close enough to any smooth curve and it looks like a straight line. The slope of that local straight line is what we will eventually call the derivative. So before we earn the right to study curves, we must master the lines that approximate them.

What is a Linear Function?

A linear function is a function of the form

where $m$ and $b$ are fixed real numbers. The graph is a straight line in the $xy$-plane.

Slope: The Soul of a Linear Function

Take any two points on a line, call them $P_1 = (x_1, y_1)$ and $P_2 = (x_2, y_2)$. The slope of the line is

The symbol $\Delta$ (Greek capital delta) is the standard mathematical notation for "the change in". $\Delta y$ means "the change in y", $\Delta x$ means "the change in x". The slope is rise over run.

Why is the slope constant? A short proof you can do in your head

Take any two points $(x_1, y_1)$ and $(x_2, y_2)$ on the line $y = m\,x + b$. Then by definition $y_1 = m\,x_1 + b$ and $y_2 = m\,x_2 + b$. Subtract:

The two $b$'s cancel. Divide both sides by $x_2 - x_1$ and you get $(y_2 - y_1)/(x_2 - x_1) = m$. The slope formula always returns $m$, independent of which two points you picked. This is the algebraic shadow of the geometric fact that the line is perfectly straight.

Interactive Explorer: y = m·x + b

Drag the two coloured points along the line. The dashed amber leg is $\Delta x$ and the dashed red leg is $\Delta y$. Watch the readout in the side panel: $\Delta y / \Delta x$ never moves from $m$, no matter where you drag.

• Make $\Delta x$ huge, then tiny — the ratio does not budge.
• Drag $P_1$ to the same place as $P_2$: the formula goes to $0/0$ (undefined). To talk about slope you must always pick two different points.
• Slide $m$ to $0$: the line is horizontal — "y never changes when x changes".
• Set $m$ negative: the line slopes down ("y decreases as x increases").

Worked Example — Pizza Delivery by Hand

Let's leave the abstract and ground everything in a realistic scenario. You deliver pizzas. Your pay model is simple: a flat $\$15$ daily stipend plus $\$3$ per delivery.

Let $x$ be the number of deliveries on a given day and $y$ the dollars you take home. Try to derive the linear law, slope, intercept, and predict your earnings — all by hand — before opening the answer below.

The y-Intercept b: Where the Line Starts

The constant $b$ in $y = m\,x + b$ answers one simple question: what is y when x is zero? Plug it in: $y(0) = m \cdot 0 + b = b$.

Geometrically, $b$ is the height at which the line crosses the vertical axis. The interactive explorer above marks it with the green dot. Slide the $b$ slider and the entire line moves up or down without tilting — slope does not depend on intercept.

Three Ways to Write the Same Line

The same straight line can be described in different algebraic outfits. Knowing all three lets you start from whatever information you happen to have.

All three are algebraically equivalent. The point–slope form is especially important in calculus because it is the form of the tangent line: once you know the slope at a point (the derivative) and the point itself, point–slope is how you write the approximation.

Compare this with the slope formula $m = \Delta y / \Delta x$: point–slope is literally the same equation, just multiplied out. The two are rearrangements of one another. Never memorise both — memorise $m = \Delta y / \Delta x$ and derive the rest.

The Staircase: Watching Slope Step by Step

Here is a way to feel the slope in your body. Imagine walking along the line one unit at a time on the x-axis. Each step rises by exactly $m$ on the y-axis. Press Play and watch the climber zigzag right and up, right and up, always the same horizontal foot and the same vertical rise.

If you flip the slope slider to $m = 0$ the rises disappear — the climber walks on flat ground. Flip to $m = 2$ and each step jumps up two. Flip to $m = -1$ and the climber descends one unit per step. The staircase is the slope made physical.

Average Rate of Change for Any Function

We have been very strict so far: the slope formula belongs to straight lines. But the formula is so good that we use it on curves too — we just rename it.

For any function $f$ and any two points $a$ and $b$ in its domain, the average rate of change of $f$ on the interval $[a, b]$ is

Geometrically this is the slope of the straight line — called a secant — drawn through the two graph points $(a, f(a))$ and $(b, f(b))$.

An analogy you can drive home

Average rate of change is exactly the same idea as your average speed on a road trip. If you drive 240 km in 4 hours, your average speed is $240/4 = 60$ km/h. But your speedometer almost certainly was not pinned at 60 km/h the whole time — at any instant your instantaneous speed was probably higher or lower. The speedometer reads what calculus will eventually call the derivative; the average from trip start to trip end is the secant slope.

Secant → Tangent: A Preview of the Derivative

Below is the central animation of the entire book. A curve $f$, an anchor point $P = (a, f(a))$, and a movable second point $Q = (a+h, f(a+h))$. The orange line through them is the secant; the dashed green line is the tangent at $P$.

Press Shrink h → 0. Watch the orange secant rotate until it nearly overlaps the green tangent. Read the side panel: the average rate $(f(a+h)-f(a))/h$ approaches $f'(a)$.

This is the only idea in differential calculus, stripped to its skeleton:

Read it slowly. $f'(a)$ is the instantaneous rate of change of $f$ at $a$. It equals the limit of the average rate of change, as the interval shrinks to a single point. We will spend the whole of Chapter 2 explaining what $\lim$ means rigorously, but you already have the picture: rotate the secant until both endpoints merge.

Python: Computing Rates of Change

Let's pin down everything we just said in code. First we verify that for a linear function the slope is the same for any two points (so our formula really is well-defined). Then we look at the famously non-linear function $f(x) = x^2$ at $a = 1$ and watch the discrete difference quotient converge to $2$.

1def slope(x1, y1, x2, y2):
2    """Slope of the straight line through (x1, y1) and (x2, y2)."""
3    return (y2 - y1) / (x2 - x1)
4
5# A linear law: hourly wage. Earnings y = 25*x + 10
6# (the 10 is a flat sign-on bonus you get even at x = 0 hours).
7def earnings(hours):
8    return 25 * hours + 10
9
10# Sample THREE arbitrary pairs of hours. The slope must be the same.
11pairs = [(0, 1), (2, 7), (3.5, 10.25)]
12for (h1, h2) in pairs:
13    y1 = earnings(h1)
14    y2 = earnings(h2)
15    m  = slope(h1, y1, h2, y2)
16    print(f"hours {h1:>5} -> {h2:<5}   "
17          f"Δy/Δx = ({y2:>6.2f} - {y1:>6.2f}) / ({h2} - {h1}) = {m:.4f}")
18
19# Now do the same for a NON-linear function f(x) = x^2 at a = 1.
20def f(x):
21    return x * x
22
23a = 1
24print("\\nAverage rate of f(x) = x^2 on [1, 1 + h]:")
25print(f"{'h':>8}  {'avg rate':>10}  {'distance to 2':>14}")
26for h in [1.0, 0.5, 0.1, 0.01, 0.001, 1e-5]:
27    avg = (f(a + h) - f(a)) / h
28    print(f"{h:>8}  {avg:>10.6f}  {abs(avg - 2):>14.6f}")

hours     0 -> 1       Δy/Δx = ( 35.00 -  10.00) / (1 - 0)     = 25.0000
hours     2 -> 7       Δy/Δx = (185.00 -  60.00) / (7 - 2)     = 25.0000
hours   3.5 -> 10.25   Δy/Δx = (266.25 -  97.50) / (10.25 - 3.5) = 25.0000

Average rate of f(x) = x^2 on [1, 1 + h]:
       h    avg rate   distance to 2
     1.0    3.000000        1.000000
     0.5    2.500000        0.500000
     0.1    2.100000        0.100000
    0.01    2.010000        0.010000
   0.001    2.001000        0.001000
   1e-05    2.000010        0.000010

Notice the structure. The wage example has a flat slope of 25, period. The curve example has a slope of $2 + h$ that you can see converging to 2 in the column on the right. Every time you halve $h$, the distance to 2 halves with it. That is what "the limit equals 2" means in plain Python.

PyTorch: Vectorising the Rate of Change

Python for-loops are fine for an interactive demonstration but slow once we want to apply the same formula to millions of points or differentiate a neural network. PyTorch does two things for us:

1. Vectorisation: compute the difference quotient at many $h$ values in parallel on the CPU or GPU.
2. Autograd: hand us the exact derivative $f'(a)$ with no numerical error — perfect for cross-checking that the limit really is what we claim.

1import torch
2
3def f(x):
4    return x ** 2
5
6a = torch.tensor(1.0)
7h = torch.tensor([1.0, 0.5, 0.1, 0.01, 0.001, 1e-5])
8
9# Vectorised: a + h broadcasts across the whole h tensor.
10avg_rate = (f(a + h) - f(a)) / h
11print("h        :", h.tolist())
12print("avg rate :", [round(v, 6) for v in avg_rate.tolist()])
13
14# Truth: autograd gives f'(1) exactly.
15x = torch.tensor(1.0, requires_grad=True)
16y = f(x)
17y.backward()
18print("f'(1) by autograd =", x.grad.item())

h        : [1.0, 0.5, 0.1, 0.01, 0.001, 1e-05]
avg rate : [3.0, 2.5, 2.1, 2.01, 2.001, 2.00001]
f'(1) by autograd = 2.0

The two answers agree. Numerical limit and symbolic derivative converge to the same number. This is the recipe we will reuse in every later chapter: derive a formula on paper, sanity-check it with a one-line PyTorch experiment.

Why This Matters — Applications

Common Pitfalls

Summary

Linear functions are the simplest functions whose rate of change is meaningful — and the rate is constant. This single property forces the graph to be a straight line, makes the slope formula return the same answer for every pair of points, and seeds every later idea in calculus.

Key Takeaways

1. A function is linear iff its rate of change is the same on every interval.
2. The slope formula $m = \Delta y / \Delta x$ works on any two points of a line and returns the same answer.
3. For a curve, that same formula gives an average rate of change — the secant slope.
4. As the two endpoints collapse to one, the secant rotates onto the tangent line. Its slope is the derivative.
5. The tangent itself is a linear function — point–slope form $y = f(a) + f'(a)\,(x - a)$.
6. Both plain Python and PyTorch make this concrete: tabulate the difference quotient, watch it converge, double-check with autograd.

Coming next: Section 1.3 zooms out from straight lines to polynomial functions, where the rate-of-change idea finally has something to chew on — the slope of $x^2$, $x^3$, and their combinations will be different at every point, foreshadowing Chapter 4's derivative rules.