How do I graph slope fields for dy/dt = t² − t and dy/dt = y²?

[abacus]

I'm still having trouble with figuring out slope fields.

If I have a function dy/dt= f(t,y), how would I go about graphing the slope field for my solutions without the use of a calculator.

An example would be how would I graph the slope field of dy/dt = t^2-t and dy/dt = y^2? Thanks.

 

[HallsofIvy]

Choose a number of (t,y) points on your graph. Calculate f(t,y) for each of those. Draw a short line through (t,y) with slope f(t,y).

For example, at (1,1) with dy/dt= t²- t, dy/dt= 0 so we draw a horizontal line- in fact, since y is not explicitely in that formula, it is obvious that dy/dt= 0 for all points on the vertical line t=1. At each point on that line, draw a short horizontal line. At t= 0.5, t²= .25-.5= -.25 so at every point on the vertical line t= 0.5, draw a short line with slope -.25. Try to connect those to the lines at t= 1.

For dy/dt= y², since t does not appear explicitely, you can calculate y² for each y and draw short lines with that slope all along the horizontal line at y.

 

[M98Ranger]

Graphing slope fields can seem intimidating at first, but with a few key steps, you can easily create accurate and informative graphs without the use of a calculator.

First, let's break down the given example of dy/dt = t^2-t and dy/dt = y^2. The first step is to identify the variables in the equation. In this case, we have t and y as our independent and dependent variables, respectively.

Next, we need to plot a few points on our graph to represent different values of t and y. For example, we can choose values such as t = -2, -1, 0, 1, 2 and y = -2, -1, 0, 1, 2. This will help us visualize the behavior of the slope at different points on the graph.

Now, we can use the given equations to calculate the slope at each of these points. For the first equation, dy/dt = t^2-t, we can substitute the values of t and y to get the slope at each point. For instance, at t = -2 and y = 1, the slope would be (-2)^2-(-2) = 6. Similarly, we can calculate the slope for the other points.

Once we have calculated the slope for each point, we can plot them on our graph. For example, at t = -2 and y = 1, we would plot a small line segment with a slope of 6. We can repeat this process for all the other points we plotted earlier.

By connecting these line segments, we can create a slope field that represents the behavior of the slope at different points on the graph. The slope field for the equation dy/dt = t^2-t would look something like this:

[Insert slope field graph here]

The same process can be followed for the second equation, dy/dt = y^2. We would plot points with different values of t and y, calculate the slope at each point, and then connect them to create a slope field.

I hope this explanation helps you understand how to graph slope fields without a calculator. Remember to choose a variety of points to get a better understanding of the overall behavior of the slope. Practice makes perfect, so keep trying and don't hesitate to ask for help if needed. Best of luck!