Wemazi

Topic 7.3

Recognize patterns across a field

Builds on locating zero-slope sets. 7 minutes.

Rows that repeat

The field shows \(\dfrac{dy}{dx} = \dfrac{y - 1}{2}\). The orange segments lie on one row, which is a horizontal line of the field.

Move the slider to change the row. On every row the orange segments are the same, so the row repeats. On the row \(y = 3\) every slope is \(\dfrac{3 - 1}{2} = 1\).

Press "Show a column". A column is a vertical line of the field, and the orange segments on a column differ.

Along a row only \(x\) changes. The equation has no \(x\), so the slope cannot change along the row. Along a column \(y\) changes, so the slope does.

A missing variable

A variable that is missing from the equation cannot change the slope. The equation of this field is not shown, and the orange segments form one column and one row.

Press "Only y in the equation". Along a row y does not change, so an equation with y alone gives one slope on the row.

Press "Only x in the equation". Along a column x does not change, so the orange segments on the column are the same.

Press "Both x and y". Neither pattern holds, because both variables change the slope.

Notation

  1. Compare two segments on one row. If every row repeats, the equation has \(y\) but not \(x\).

    \[\dfrac{dy}{dx} = g(y)\]

  2. Compare two segments on one column. If every column repeats, the equation has \(x\) but not \(y\).

    \[\dfrac{dy}{dx} = h(x)\]

  3. If neither pattern holds, both variables appear. Then find the flat segments, which lie on the dashed orange curves, and the sign of the slope in each region between them.

  4. An equation fits the field when it gives the slope of the field at every point that you test.

    Test two points far apart, because two equations can give the same slope at one point.

When it fails

In this field no row repeats and no column repeats, because both variables appear. The slope depends on \(y - x\), so the segments repeat along each diagonal line \(y = x + c\), such as the orange one, \(y = x - 1\).

Both tests fail, so the flat segments decide. They lie on the line \(y = x\), where \(y - x = 0\), with positive slopes above it and negative slopes below.

Example 1: columns that repeat

In a field, the segments on each column are the same. They fall to the left of the \(y\)-axis, are flat on it, and rise to the right of it. Which equation fits: A, \(\dfrac{dy}{dx} = \dfrac{y}{2}\); B, \(\dfrac{dy}{dx} = \dfrac{x}{2}\); or C, \(\dfrac{dy}{dx} = \dfrac{x + y}{2}\)?

  1. Compare two segments on the column \(x = 2\), such as the ones at \((2, -2)\) and \((2, 3)\). They are the same, so \(y\) does not change the slope.

  2. Equations A and C contain \(y\), so they do not fit. Equation B has \(x\) only.

  3. Test B at three columns. The slope is \(-1\) at \(x = -2\), 0 at \(x = 0\), and 1 at \(x = 2\), as in the field.

    \[\dfrac{-2}{2} = -1, \quad \dfrac{0}{2} = 0, \quad \dfrac{2}{2} = 1\]

Example 2: neither pattern

In a field, the segment at \((2, 2)\) has slope 4, and the segment at \((0, 2)\) is flat. Which equation fits: A, \(\dfrac{dy}{dx} = x + y\), or B, \(\dfrac{dy}{dx} = xy\)?

  1. Test the point \((2, 2)\). Both equations give 4, the slope in the field.

    \[2 + 2 = 4, \qquad 2 \cdot 2 = 4\]

  2. A tempting wrong move: Equation A gives 4 at \((2, 2)\), so A is the equation.

    \[\dfrac{dy}{dx} = x + y\]

    Equation B also gives 4 there, so one point cannot decide. At \((0, 2)\) the field is flat, but A gives \(0 + 2 = 2\), the red segment.

  3. Test the second point, \((0, 2)\), where the field is flat. Equation A gives 2 and equation B gives 0, so B fits.

    \[\text{A: } 0 + 2 = 2, \qquad \text{B: } 0 \cdot 2 = 0\]

Practice

In a field, the segments on each row are the same. The row \(y = 0\) is flat, the rows above it fall, and the rows below it rise. Which equation fits: A, \(\dfrac{dy}{dx} = -\dfrac{y}{2}\); B, \(\dfrac{dy}{dx} = \dfrac{y}{2}\); C, \(\dfrac{dy}{dx} = -\dfrac{x}{2}\); or D, \(\dfrac{dy}{dx} = xy\)?

Move the row slider. Which equations give the same slope along a whole row?

Solution

  1. Every row repeats, so the equation has \(y\) but not \(x\). Equations C and D contain \(x\), so they do not fit.

  2. Above the \(x\)-axis, \(y\) is positive and the segments fall, so the slope is negative there. Equation B gives a positive slope, so it does not fit.

  3. Equation A fits. It gives \(-1\) at \(y = 2\), 0 at \(y = 0\), and 1 at \(y = -2\).

    \[\begin{gathered} -\dfrac{2}{2} = -1, \quad -\dfrac{0}{2} = 0, \\ -\dfrac{-2}{2} = 1 \end{gathered}\]

Questions

The figure shows the slope field of \(\dfrac{dy}{dx} = \dfrac{3 - x}{2}\). One column is orange.

  1. The segments on the column \(x = 1\) are all the same. What does this tell you about the equation?

    • It has \(x\) but not \(y\).
    • It has \(y\) but not \(x\).
    • It has both \(x\) and \(y\).
    Answer

    It has \(x\) but not \(y\).

  2. Which line has all of the flat segments?

    • \(x = 3\)
    • \(y = 3\)
    • \(x = 0\)
    • \(x = 6\)
    Answer

    \(x = 3\)

  3. A field has segments that are the same on each row, and the row \(y = 4\) is flat. Which equation fits?

    • \(\dfrac{dy}{dx} = y - 4\)
    • \(\dfrac{dy}{dx} = x - 4\)
    • \(\dfrac{dy}{dx} = x(y - 4)\)
    • \(\dfrac{dy}{dx} = 4y\)
    Answer

    \(\dfrac{dy}{dx} = y - 4\)

Summary

You compare segments along a row and along a column, and match the pattern with the variables in the equation.

This works because a variable that is missing from the equation cannot change the slope.

You use it when a problem gives a field and asks for its equation, or gives an equation and asks for its field.

This is one of four skills in topic 7.3 of Which & Why: AP Calculus.