dy vs Delta y Comparison

Run the dy vs Delta y Comparison MicroSim Fullscreen Edit with the p5.js Editor

About This MicroSim

The symbols \(dy\) and \(\Delta y\) look almost the same, and for a small step they are almost the same number. But they measure two different things:

Symbol	Formula	What it measures	Color
\(dy\)	\(f'(x)\cdot dx\)	Rise along the tangent line	Orange
\(\Delta y\)	\(f(x + dx) - f(x)\)	Actual rise along the curve	Blue
error	\(\Delta y - dy\)	The gap between the two	Purple

Both start at the same point \((x, f(x))\) and both use the same horizontal step \(dx\). One follows the straight tangent line, the other follows the curve. The shaded sliver between the curve and the tangent line is exactly where they disagree.

For small \(dx\):

\[\Delta y \approx dy = f'(x)\cdot dx\]

Delta Moment

"If I hold my tilt perfectly steady and roll forward, I end up at \(dy\). If I follow the actual ground, I end up at \(\Delta y\). For a teeny tiny step, I can't tell the difference. For a big step? Oh, I can tell."

How to Use

Look at the default picture: \(f(x) = x^2\), \(x = 1\), \(dx = 1\). Find the orange \(dy\) bracket, the blue \(\Delta y\) bracket and the purple error bracket.
Drag Step dx toward zero. Watch the error bracket shrink much faster than \(dx\) does.
Drag Point x to move the point of tangency. The tangent line gets steeper or flatter, and \(dy\) changes with it.
Click f(x) = √x to switch to a concave down curve. Which is bigger now, \(dy\) or \(\Delta y\)?
Read the panel under the graph. It shows both calculations with the current numbers plugged in.

What to Observe

On the concave up curve (\(x^2\)), the tangent line runs below the curve, so \(dy\) is smaller than \(\Delta y\) and the error is positive.
On the concave down curve (\(\sqrt{x}\)), the tangent line runs above the curve, so \(dy\) is larger than \(\Delta y\) and the error is negative.
Halving \(dx\) cuts the error to roughly one quarter. The error shrinks like \(dx^2\), which is why the approximation is so good for small steps.
For \(f(x) = x^2\) the error is exactly \(dx^2\), because \(\Delta y = 2x\,dx + dx^2\) and \(dy = 2x\,dx\).

Iframe Embed Code

You can add this MicroSim to any web page by adding this to your HTML:

<iframe src="https://dmccreary.github.io/calculus/sims/dy-vs-delta-y-comparison/main.html"
        height="587px"
        width="100%"
        scrolling="no"></iframe>

Lesson Plan

Grade Level

Grades 11-12 (college placement Calculus)

Duration

10-15 minutes

Learning Objectives

After using this MicroSim, students will be able to:

Differentiate between the actual change \(\Delta y\) along the curve and the differential \(dy\) along the tangent line (Bloom Level 4)
Compare \(dy\) and \(\Delta y\) for the same step \(dx\) and identify which one is larger
Relate the sign of the error to the concavity of the curve
Explain why \(\Delta y \approx dy\) becomes more accurate as \(dx\) shrinks

Prerequisites

The derivative as the slope of the tangent line
Linear approximation: \(L(x) = f(a) + f'(a)(x - a)\)
Concave up and concave down

Activities

Name the segments (3 min): With the default settings, students write the length of each bracket and verify the numbers against the panel: \(dy = 2\), \(\Delta y = 3\), error \(= 1\).
Shrink the step (4 min): Students record the error for \(dx = 1,\ 0.5,\ 0.25,\ 0.1\) on \(f(x) = x^2\) at \(x = 1\). They look for the pattern (the error is \(dx^2\)).
Flip the concavity (4 min): Students predict whether \(dy\) will be larger or smaller than \(\Delta y\) for \(\sqrt{x}\), then click the button to check. They explain the result using the position of the tangent line relative to the curve.
Connect to approximation (4 min): Students use \(dy\) to estimate \(\sqrt{1.1}\) starting from \(x = 1\) and compare with the panel values at \(dx = 0.1\).

Discussion Questions

\(dx\) and \(\Delta x\) are the same number. Why are \(dy\) and \(\Delta y\) not?
On which curve does \(dy\) overestimate the true change? What feature of the graph tells you that without any calculation?
When \(dx\) is cut in half, the error drops to about one fourth. What does that suggest about how the error depends on \(dx\)?
An engineer uses \(dy\) to estimate how much a measurement error changes a computed volume. When is that a safe shortcut?

Assessment

Students have met the objective when they can:

label \(dx\), \(dy\), \(\Delta y\) and the error on a blank sketch of a curve and its tangent line
compute \(dy\) and \(\Delta y\) for a given function, point and step
state whether \(dy\) overestimates or underestimates \(\Delta y\) from the concavity of the curve