Zooming In

A claim is on the table: every graph, zoomed far enough in at a point, becomes a straight line. Your group’s job is to make the claim rich — and then to break it.

The task

In Using Desmos, graph $y=x^2$ and zoom in at $(1,1)$ until the curve looks straight. Measure the slope of what you see — pick two on-screen points and compute rise over run. Repeat at $(2,4)$ and at $(3,9)$; record all three slopes and conjecture the pattern. Now the hunt: find a graph and a point where zooming never straightens anything. Try $y=|x|$. Try $y=\sin x$. Try one of your own invention. At which points do these graphs straighten, and at which do they refuse? Finish with a rule the class can vote on: which functions earn a slope at a point, and what, exactly, disqualifies the others?

Facilitation notes — for the teacher

The slope measured at $x=3$ should collide with the number the secant tables chased on Day 2 — same value, different road; wait for someone to notice rather than pointing it out. At the origin, $y=|x|$ keeps its corner at every magnification — “it is Vs all the way down” is the sentence you are listening for, and it is the whole idea of non-differentiability arriving early. Fast groups: $y=x^{1/3}$ at the origin straightens into a vertical line — a tangent whose slope no number can state — and piecing together where $y=|x|$ does have slopes builds the derivative function of a non-smooth function, a Unit 3 idea a fortnight early.

What mathematics tends to surface

Local linearity: smooth curves are secretly straight up close, and the slope of that hidden line is the instantaneous rate the secants were chasing. Corners are the price of admission — where a graph has one, no single line fits, and no slope exists there at all. The measured slopes at $1$, $2$, and $3$ line up in a pattern begging to be named — which is exactly what The Derivative does next class.

Where it leads

The tangent’s slope becomes the derivative’s definition next class, and the zoomed-in view returns all semester: Curve Sketching reads whole graphs through these local slopes, and every “is it smooth here?” question in the course is settled by today’s zoom.

The answer is not on this page

The pattern, the counter-example, and the rule all happen at the boards — bring your best broken graph.

Curriculum connection

A1.2

describe connections between the average rate of change of a function that is smooth (i.e., continuous with no corners) over an interval and the slope of the corresponding secant, and between the instantaneous rate of change of a smooth function at a point and the slope of the tangent at that point Sample problem: Given the graph of $f(x)$ shown below, explain why the instantaneous rate of change of the function cannot be determined at point $P$. [The document prints the graph here: a function with a corner at point P.]

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A1.3

make connections, with or without graphing technology, between an approximate value of the instantaneous rate of change at a given point on the graph of a smooth function and average rates of change over intervals containing the point (i.e., by using secants through the given point on a smooth curve to approach the tangent at that point, and determining the slopes of the approaching secants to approximate the slope of the tangent)

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