Differential Equations

What is the primary use of differential equations in mathematical modeling, as described in the provided text?

Which of the following scenarios is best modeled using a differential equation?

A scientist observes that the rate of decay of a radioactive substance is proportional to the amount of the substance remaining. Which type of equation would be most suitable to model this phenomenon?

Which of the following differential equations corresponds to the slope field where the slopes are always equal to the x-coordinate at any given point?

For the differential equation $\frac{dy}{dx} = x + y$, in which region of the xy-plane are the slopes in the slope field positive?

A slope field is shown with horizontal line segments along the line y = x. Which of the following differential equations could the slope field represent?

Using Euler's method with a step size of 0.1, approximate y(1.1) for the differential equation $\frac{dy}{dx} = x$ with the initial condition y(1) = 2.

Consider the differential equation $\frac{dy}{dx} = y$ with initial condition y(0) = 1. Using Euler's method with a step size of h = 0.2, approximate y(0.4).

Given the differential equation $\frac{dy}{dx} = x + y$ with initial condition y(0) = 1, you want to approximate y(1) using Euler's method. To ensure your approximation is within 0.1 of the actual value, which strategy is most appropriate?