Slope Fields (warm-up 2 #6-10)

Front

What is a slope field?

Back

A slope field is a graphical representation of the solutions to a first-order differential equation, showing the slope of the solution curves at various points in the plane.

Front

What does dy/dx represent in a slope field?

Back

dy/dx represents the derivative of y with respect to x, indicating the slope of the tangent line to the curve at any given point.

Front

How do you determine the slope at a point in a slope field?

Back

To determine the slope at a point, substitute the coordinates of the point into the derivative function (dy/dx) to find the corresponding slope value.

Front

What is the significance of the slope in a slope field?

Back

The slope indicates the direction in which the solution curve is heading at that point, helping to visualize the behavior of the solutions.

Front

What is the derivative for the slope field represented by dy/dx = 1/x?

Back

The derivative for this slope field is dy/dx = 1/x, which indicates that the slope decreases as x increases.

Front

What is the derivative for the slope field represented by dy/dx = cos x?

Back

The derivative for this slope field is dy/dx = cos x, which oscillates between -1 and 1 as x varies.

Front

What does dy/dx = y represent in a slope field?

Back

dy/dx = y represents an exponential growth slope field, where the slope at any point is equal to the y-coordinate.

Front

How does dy/dx = x^2 behave in a slope field?

Back

In the slope field for dy/dx = x^2, the slope increases as x moves away from zero, indicating steeper slopes for larger x values.

Front

What does dy/dx = -x/y indicate about the slope?

Back

dy/dx = -x/y indicates that the slope is negative when x is positive, suggesting that the solution curves decrease as x increases.

Front

What is the general form of a first-order differential equation?

Back

The general form of a first-order differential equation is dy/dx = f(x, y), where f is a function of x and y.