Using Desmos

Desmos is free, runs in any browser, and is the fastest way in the world to ask a graph a question. Treat it as a thinking tool — a place to test conjectures — not an answer machine you obey.

A slider on a secant

Type $f(x)=x^2$, then the secant slope $\frac{f(3+h)-f(3)}{h}$ with a slider for $h$. Before you drag anything, predict aloud: what should the slope read at $h=1$? What happens as $h$ slides toward zero — and what happens at zero? Then slide, and watch the number settle toward 6 while the input it wants is the one input it cannot have. Two minutes of sliding builds the intuition behind The Limit faster than an hour of tables — and the gap at $h=0$ is not a bug in Desmos; it is the exact hole The Derivative was invented to speak about. Prediction first is what keeps Desmos a thinking tool; sliding without predicting is just television.

Check the angle mode

This course thinks in radians, and Desmos should too. Graph $y=\sin x$ alongside $\frac{d}{dx}\sin x$ — the derivative curve should be $\cos x$ exactly, peaking at 1. If it comes out crushed nearly flat, something is measuring in degrees, and every sinusoidal derivative it touches will be wrong by a factor nobody ordered.

A function and its derivative, stacked

Desmos will graph $f^{\prime }(x)$ if you simply type it. Put $f(x)=x^3-3x$ in one line and $f^{\prime }(x)$ in the next, then interrogate the pair: where the original peaks, its derivative should cross zero heading down; where the original bottoms out, the derivative crosses zero heading up; where the original falls, the derivative should be underwater. Every claim in Curve Sketching can be cross-examined this way — and when the two graphs surprise you, you have found the exact edge of your understanding.

When your head is faster

Desmos shines when the question is about shape and behaviour: a family of cubics under a slider, a descent profile against its acceleration, a conjectured derivative checked against the built-in one in ten seconds. It is slower than your head for the power rule, for the dot product of $⟨3,4⟩$ and $⟨5,0⟩$, for reading where a sign chart changes, and for everything Estimation Duels trains. Reaching for a tool you do not need is its own kind of slow — the skill is knowing which moment you are in.