Front

What is the differential equation for the slope field represented by dy/dx = y/x?

Back

dy/dx = y/x is a first-order differential equation that describes the slope of the tangent line at any point (x, y) in the plane.

Front

What does the slope field for dy/dx = 2x represent?

Back

The slope field for dy/dx = 2x shows that the slope of the tangent line increases linearly with x, indicating that the function is quadratic.

Front

What is the differential equation represented by the slope field with constant slope -1?

Back

The differential equation is dy/dx = -1, indicating that the slope of the tangent line is constant and negative.

Front

What does the slope field for dy/dx = -x represent?

Back

The slope field for dy/dx = -x indicates that the slope of the tangent line decreases linearly as x increases.

Front

What is the significance of the slope field in understanding differential equations?

Back

Slope fields provide a visual representation of the solutions to differential equations, helping to understand the behavior of solutions without solving them analytically.

Front

How does the slope field for dy/dx = x^2 differ from linear slope fields?

Back

The slope field for dy/dx = x^2 shows that the slope increases quadratically with x, leading to curves that steepen as x moves away from zero.

Front

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

Back

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

Front

What does a horizontal line in a slope field indicate?

Back

A horizontal line indicates that the slope dy/dx = 0, meaning the function is constant at that y-value.

Front

What is the relationship between slope fields and initial value problems?

Back

Slope fields can be used to visualize the solutions to initial value problems, showing how solutions behave given specific initial conditions.

Front

What is the slope at the point (1, 2) for the equation dy/dx = 2x?

Back

At the point (1, 2), the slope is dy/dx = 2(1) = 2.