### Topics

## Introduction to Differential Equations - At A Glance:

We know that the slope of a line is given by

or by

Since *y* is usually the dependent variable and *x* is usually the independent variable, you may also see

or

The symbol Δ is the Greek capital letter "Delta" (have button to click to hear delta pronounced), which mathematicians use to mean "change."

We usually use the slope formula to calculate the slope of a line given Δ *y* and Δ *x*. If we know the slope and Δ *x* we can instead use the slope formula to find Δ *y*.

Using algebra, if

then multiplying both sides by Δ *x* we get

Therefore if we know the slope of a line and we move over by Δ *,* then we know *y* is needs to to change by (slope × Δ *x*).

#### Example 1

The line below has a slope of 5. Fill in the indicated number. | |

The missing number is Δ *y* and we're told Δ *x* = 2. This means the missing number is Δ *y* = slope × Δ *x* = (5)(2) = 10. | |

#### Example 2

The line below has a slope of 4. Fill in the indicated number. | |

The line has slope 4 and Δ *x* = 1. So Δ *y* = slope × Δ *x* = 4 × 1 = 4. The number we want is the sum of Δ *y* and the value of *y* at the first point. therefore the missing number is 4 + 1 = 5. When doing problems like this, it can be helpful to think about the values as y_{old} and y_{new}. y_{old} is usually the value of *y* we're given in the problem. If we add y_{old} and the change in *y*, we get y_{new}. y_{new} = *y*_{old} + Δ *y*. | |

#### Example 3

The line below has a slope of -0.7. Find the indicated number. | |

We have *y*_{old} = 3 and we want to find *y*_{new}. We know *y*_{new} = *y*_{old} + Δ *y*,
so we just need to find Δ *y*, which we know how to do: Δ *y* = slope × Δ *x* = (-0.7)(4.5) = -3.15. Now we know *y*_{new} = 3 + (-3.15) = -0.15.
| |

#### Example 4

The function *f* is a line with slope . If *f* (*a*) = 3, what is ? | |

The function *f* is a line with slope that passes through the point (*a*,3). The old value of *f* is 3 and we want to know the new value of *f* if we change *x* by : We don't know what a is, but that's ok. We know , so To get the new value of *f*, we add Δ *y* to the old value of *f*: | |

#### Example 5

The line below has a slope of 2. Find the indicated number. | |

We use the same formula as before. The slope is 2 and Δ *x* = -0.5, so Δ *y* = 2(-0.5) = -1. Then *y*_{new} = *y*_{old} + (-1) = 4 - 1 = 3.
This makes sense, if you think about it. Having a slope of 2 means for every step of size 1 that *x* moves right, *y* should move up 2. This means if *x* moves to the right by 0.5, *y* should move up by 1. So the value of *y* when *x* = 4 should be one more than the value of *y* when *x* = 3, which is what happens. | |

#### Exercise 1

The line below has slope 3. Find the indicated number.

Answer

The slope is 3 and Δ *x* = 2, so

Δ *y* = slope × Δ *x* = 3(2) = 6.

To find the missing value of *y*, we add y_{old} = -1 and Δ *y* = 6 to get y_{new} = 5.

#### Exercise 2

The line below has slope . Find the indicated number.

Answer

The slope is and Δ *x* = 3.

This means

Then we add y_{old} = 5 and to get

#### Exercise 3

The line below has slope 0.2. Find the indicated number.

Answer

The missing value is Δ *y* (we aren't told y_{old}, so we couldn't find y_{new} anyway).

The slope is 0.2 and Δ *x* = 3, so

Δ *y* = (0.2)(3) = 0.6.

#### Exercise 4

Find the indicated number.

Answer

This problem has one more step than the others, because first we need to figure out the slope of the line.From the first two points we can calculate

Now let's ignore the point (1,2). Look only at these two points, and think y_{old} = 5 and Δ *x* = 3.

Then

and

#### Exercise 5

Find the indicated number.

Answer

Again, we need to find the slope of this line. Looking at the outermost two points we can calculate rise and run.

So

We get

and

#### Exercise 6

The line below has slope . Find the indicated number.

Answer

The picture in this case is a little misleading. Since the point whose *y*-value we want to know is to the left of the point whose *y*-value we know, Δ *x* is negative.

Since both the slope and Δ *x* are negative, their product Δ *y* is positive:

so

y_{new} = y_{old} + Δ *y* = 7 + 10 = 17.

This makes sense: given the picture, we would expect y_{new} to be larger than y_{old}.

#### Exercise 7

The line below has slope . Find the indicated number.

Answer

We have y_{old} = 3 and Δ *x* = -3.

From the picture, we expect y_{new} to be less than y_{old}.

#### Exercise 8

The line below has slope -7. Find the indicated number.

Answer

In this problem we are asked to find Δ y. Look carefully at the picture, and make sure to use -2, not + 2, for Δ x.

We get

Δ *y* = slope × Δ *x* = (-7)(-2) = 14.