< Previous Section Home Next Section >

Section 1.4Infinitesimal Variations and Zooming a Graph

A central idea in calculus is to quantify relationships between quantities' values as they vary through small intervals.

However, it is sometimes counter-intuitive to anticipate a quantity varying through a large interval by varying smoothly through tiny intervals that compose it.

An activity might make the centrality of anticipating variation over small intervals more real to you. Complete, Activity 1.4.1 before returning to this page.

We hope Activity 1.4.1 conveyed two messages to you:

• Always anticipate that variation in a quantity's value can happen over smaller intervals than you are presently contemplating.
• Looking at variation over smaller intervals often reveals more of a relationship's overall behavior.

Figure 1.4.1 also illustrates needing to think about behavior over small intervals even when it appears there is nothing unusual about a graph. Looking at variation in x over small intervals reveals "bumpy" behavior we could not see originally in the relationship between x and y.

Figure 1.4.1 At normal scale we see nothing irregular about the graph of $y=x^2+0.0001\sin(10000x)$.

What is an infinitesimally small quantity?

You dealt in prior sections with exceedingly small and exceedingly large quantities. But they were finite nevertheless. You could think of numbers even larger or smaller than the largest or smallest we dealt with and pretend they are measures of something.

In answering Exercise 1.1.2 ("How far away is 1?"), you found that the distance from the 0 to 1 on your screen's y-axis, given the distance from 0 to $10^{-64}$ was about 4 inches, was approximately 1 billion trillion billion trips across the visible universe.

This is an impossibly large physical distance. But we can imagine it and talk about it. Here is a wonderful YouTube video that addresses this very issue—unimaginably large and small numbers in human experience.

In another chapter you will find situations where where you talk about rates of change over "infinitesimally small" intervals, or about the accumulation of variations over intervals having "infinitesimal width".

The phrases "infinitesimally small" and "infinitely large" have been controversial in the history of mathematics. We will not ask you to address that controversy.

Instead, we ask you to form a personal image that, in practice, gives meaning to "infinitesimally small" and "infinitely large" quantities.

We will speak about intervals that are small enough so that, were they any smaller, it would make no essential difference relative to the problem being addressed.

If you must have a definite length to consider as "infinitesimally small", then think of the Planck Length. It is approximately $1.6 \cdot 10^{-35}$ meters. Theoretically, a Planck Length is the smallest length that can be measured.

For practical purposes, you can think of an interval of length one Planck Length on a number line in place of the phrase "infinitesimally small". You can also think of $6.5 \cdot 10^{34}$ meters, the reciprocal of one Planck Length, in place of "infinitely large".

Zooming a Graph

We include this section on zooming in or out on a graph in GC for practical purposes.

A function’s graph sometimes has details that are difficult to see. This is often because the graph’s details are too small or too large to see at the axes’ current scale.

Figure 1.4.1 showed a case where details were too small to see. Figure 1.4.2, below, is a case where details are too large to see. It shows the graph of $y = 2^x$ overlaid with the graph of $y = x^4$.

Many high school textbooks state that exponential growth always overtakes polynomial growth. But in Figure 1.4.2 we see only two intersections, suggesting that $x^4$ is always larger than $2^x$ for values of x larger than some number between 1 and 2.

Therefore, either these textbooks were wrong or there must be some number N so that $2^x\gt x^4$ for all values of $x\gt N$.

In other words, the two graphs must have a third intersection point if exponential growth always outpaces polynomial growth.

If there is a third intersection point, it is off the screen when the axes are scaled as in Figure 1.4.2. We need to rescale the axes to look for a third intersection of the two graphs.

Figure 1.4.2. Using GC’s "zoom" buttons to change axes’ scales. Clicks of the mouse appear as black rings in the video. Clicks while holding the "shift" key ("command" key on Mac) show in red rings. Move your pointer away from the animation to hide the control bar.

Figure 1.4.2 suggests that $2^x\gt x^4$ for $x\gt 16$.

What GC Does When It Zooms

When you click on one of GC’s zoom icons, it expands the visible axes (when zooming in) or it contracts the visible axes (when zooming out).

Zooming in is like getting closer to a graph. Your field of vision includes more fine-grained detail in a smaller region of the coordinate plane. Zooming out is like moving away from a graph. Your field of vision includes less fine-grained detail in a larger region of the coordinate plane.

Either way, GC always keeps the origin, or the location of trace crosshairs, in the same place, zooming in or out from it.

You also can rescale vertical and horizontal axes independently. Holding the shift key (command key on Mac) when you click a zoom button re-scales only the vertical axis.

Holding the alt key (option key on Mac) when you click a zoom button re-scales only the horizontal axis.

Zoom by "boxing"

Sometimes you’ll want to zoom in on a section of a graph instead of zooming the entire viewing area. Zoom in a section of a graph by holding the shift key as you click and drag to draw a box around the section you wish to expand.

Figure 1.4.3 shows a box being drawn around a section of a graph and what GC does after you release the mouse button.

Figure 1.4.3. Shift-click-drag to draw a box. GC fills the viewing pane with the region you boxed.

How you draw a box will determine how you affect the proportions of your zoomed region.

The animation in Figure 1.4.4 shows the sections of a graph being scaled differently according to how the box is drawn. If you want to stretch a section of a graph vertically, draw a short wide box. If you want to compress a section vertically, draw a tall narrow box.

The function’s graph is unaffected. Only the scale of its display is affected.

Figure 1.4.4. Shift-click-drag to draw a box. How you draw a box determines whether the scale will be stretched or compressed vertically. The graph itself is unaffected. Only the scale of its display is affected.

Exercise Set 1.4

1. The variable x varies through the interval $[2,7]$ by varying through intervals of length $\Delta x=0.0001$.
1. Through how many subintervals of length $\Delta x$ will the value of x vary as it varies from $x=2$ to $x=7$?
2. Suppose x is a number of miles, and that the value of x varies at a rate of 0.57 mi/sec. How many seconds will it take for the value of x to vary through one interval of length $\Delta x$ miles?
3. Suppose x is a number of miles, and that the value of x varies through one interval of length $\Delta x$ in 0.0000004 seconds. At what rate, in mi/sec is the value of x varying?

Exercises 2-5: A section of each displayed graph is boxed. Predict how GC’s display of the graph will appear after releasing the mouse button. Then click the solution link to see how GC actually zoomed that section.

2. solution

3. solution

4. solution

5. solution

6. Exercises 6-9: Open each GC file. Click the zoom buttons, with or without modifier keys, to get the view shown for that file.