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## AP®︎/College Calculus AB

### Course: AP®︎/College Calculus AB>Unit 7

Lesson 3: Sketching slope fields

# Worked example: equation from slope field

Given a slope field and a few differential equations, we can determine which equation corresponds to the slope field by considering specific slopes.

## Want to join the conversation?

• If you had no answer choices, how would you determine this? It seems like you are just guessing and checking?

If you had infinite slope field lines, could you calculate the function? • Observe that you can draw infinitely many possible graphs for a given slope field. A slope field doesn't define a single function, rather it describes a class of functions which are all solutions to a particular differential equation. For instance, suppose you had the differential equation:

𝑦' = 𝑥

By integrating this, we would obtain 𝑦 = (1/2)𝑥² + 𝐶. Observe that there are an infinite number of functions 𝑦 that satisfy the differential equation (because we can choose any constant of integration). This is also reflected by the fact that the slope field permits infinitely many curves to be drawn through it.

So if you had all of the slope field lines, chances are you could narrow down the what the graphs of all of the solutions to a particular differential equation must look like. However, if you wish to know the explicit form of the solutions, you are most likely out of luck. You run into the same problem you get when integrating many functions. That is, sometimes there simply does not exist an elementary form for the solutions.

For instance, if you wanted to find the antiderivative of 𝑒^(-𝑥²), you'd be hard-pressed to find an elementary closed form solution. Because there is none. In fact, we define a function called the error function to be directly related to the antiderivative of 𝑒^(-𝑥²) because there is no elementary function (a function in terms of polynomials, rational functions, logarithms, exponentials, trigonometric functions, etc.), that is an antiderivative of 𝑒^(-𝑥²).

Likewise, when solving a differential equation, often times, the solution to the equation does not have an elementary form. In those cases, the best we can do is to compute things numerically, or just define a whole new function to be the solution to a particular differential equation.

Comment if you have questions!
• What does Sal mean when he said the solution depends on the points it contains? • how can i know the slopes (( the shape of this lines )) ?? is there a kind of equation or law ?? • It's just the slope of lines, so there are some "rules" that are derived from the equation of lines:

A horizontal line represents a slope of `0`
A line that "grows" (like "/") represents a positive slope, the steeper it is, the greater the slope.
A line that "falls" (like "\") represents a negative slope, the steeper it is, the more negative the slope.

A line that grows that is exactly at 45° represents a slope of `1`.
A line that falls that is exactly at 45° represents a slope of `-1`.

With this guidelines you can approximate the value of the slope in the slope field.
• Aren't you able to rule out the rational options off the bat, since (0,0) has a slope defined? 0/0 wouldn't result in a slope on the slope field right? • How do you solve the differential equation if we can't separate the variables?
dy/dx = x+y • could you solve this more analytically by looking at the line, whose equation is x+y=-1, and then realising that dy/dx=-1 at the points which lie on the line, so straight away, substituting dy/dx for -1, the differential equation x+y=dy/dx pops out?
(1 vote) • I just happen to remember what function would create a field like this, but I suppose it won't work every-time, especially in ℝ≥4?
(1 vote) • At , Sal said that the slope at x=-1 and y=-1 appears to be negative, which matches the yellow equation, which is -2. However, the yellow dy/dx = 2, not -2, so I'm wondering where he picked up the negative.
(1 vote) • the shape of the graph is depended on the starting point and anything else ? i still cannot get how sal draw that? • Can the slope of a function be defined where the function does not exist? For example the slope of `f(x) = x^2` at x = 1 and y = 5 and what does this value mean in relation to the function? 