AP Physics C: Mechanics · Unit 0: Calculus Primer · Lesson 0.1

Deep Dive: Limits and Instantaneous Rate of Change

🔬 Deep Dive
This is your textbook for this topic. Take your time. Read it more than once.
Back to Lesson 0.1
Unit 0 · 0.1Concept

Average vs. Instantaneous Rate of Change

If a car travels 100 meters in 5 seconds, its average velocity over that interval is 20 m/s. But its speedometer at any single instant inside that 5-second window might read something completely different — faster at one moment, slower at another. Average rate of change tells you what happened over an interval. It says nothing about what's happening at a single point in time.

Graphically, an average rate of change is the slope of a secant line — a line connecting two points on a curve. To find the rate of change at a single instant, you need a different idea: shrink the interval between those two points down to nothing, and watch what the secant line's slope approaches.

f(t) = t², with a fixed point A at t = 1. Drag h to move point B toward A and watch the secant line (slate) rotate toward the true tangent line at A (platinum, dashed).

h = 1.00
AB
secant slope = 3.00tangent slope = 2.00

A few checkpoints along the way, all exact — secant slope = 2 + h:

💡Notice the secant slope in the tool above is always exactly 2 + h. As h gets smaller, 2 + h gets closer and closer to 2 — and 2 is also the slope of the actual tangent line at A. That's not a coincidence: the tangent slope is the number the secant slope is heading toward.
Unit 0 · 0.1Math

The Limit Definition of a Derivative

What you just watched happen visually has a formal name and a formal definition. The derivative of a function f at a point x is the limit of the secant slope as the gap between the two points shrinks to zero:

f'(x) = lim(h→0) [f(x+h) − f(x)] / h

Here, h plays the role that "Δt" or "Δx" played earlier — it's the size of the gap between the two points. As h shrinks toward (but never quite reaches) zero, the expression [f(x+h) − f(x)] / h approaches a single number. That number is f'(x), the instantaneous rate of change of f at x.

🔑This is exactly where notation like dx/dt comes from. When you see v = dx/dt in Lesson 1.2, it's shorthand for v = lim(Δt→0) Δx/Δt — the exact same limiting process you just watched, applied to position instead of a generic function f.
Unit 0 · 0.1Math

Computing a Derivative from the Definition

The limit definition isn't just a formula to memorize — it's a procedure you can actually carry out by hand. Here it is, done in full, for a simple function.

ExampleWorked Example — Differentiating f(x) = x²

Use the limit definition of a derivative to find f'(x) for f(x) = x².

The exact same four moves — substitute, expand, cancel an h, then let h vanish — work on other polynomials too. Try walking through one yourself.

ExampleGuided Example — Differentiating f(x) = x³

Use the limit definition to find f'(x) for f(x) = x³.

Step 1 — Write the definition and substitute
f'(x) = lim(h→0) [(x+h)³ − x³] / h
💡Look at the pattern: the derivative of x² is 2x, and the derivative of x³ is 3x². That's not a coincidence either — it's the pattern Lesson 0.2 turns into a shortcut rule so you never have to run this limit by hand again.
Unit 0 · 0.1Concept

Why This Matters for Physics

Unit 1 opens with a definition that will look completely ordinary if you've made it this far: velocity is the derivative of position, v = dx/dt. Acceleration is the derivative of velocity, a = dv/dt. Neither of those statements is just notation — each one is a limit, exactly like the one you built by hand in this lesson.

🔑You will almost never take a derivative from the limit definition again after today — Lesson 0.2's power rule makes that unnecessary for the vast majority of functions you'll meet this year. But every time you write v = dx/dt without a second thought, you're leaning on the limit you just proved actually works.
← Back to Lesson 0.1Ready for 0.2? The Power Rule for Derivatives turns everything you just did by hand into a two-second shortcut.