Final answer:
a) Sketch the slope field for the given differential equation at the 8 given points. b) Write the equation for the line tangent to the graph of y=f(x) at x=1 and use it to approximate f(1.5). c) Find the particular solution y=f(x) to the given differential equation with the initial condition f(1)=2.
Step-by-step explanation:
a) To sketch a slope field for the given differential equation at the given points, we can consider different values of x and y to find the slope. For example, when x=0 and y=0, the slope is 0((0) - 1) = 0. Repeat this process for all 8 given points and sketch the slopes as small line segments at those points.
b) To find the equation for the line tangent to the graph of y=f(x) at x=1, we can use the derivative of f(x). Differentiating the given differential equation, we get dy/dx = y - 1. At x=1, y=2, so the slope of the tangent line is 2 - 1 = 1. The equation of the tangent line is y - 2 = 1(x - 1), which simplifies to y = x + 1. Using this equation, we can approximate f(1.5) by substituting x=1.5.
c) To find the particular solution y=f(x) to the given differential equation with the initial condition f(1)=2, we can use separation of variables. Rearrange the differential equation as dy/(y-1) = dx/x and integrate both sides. We get ln|y-1| = ln|x| + C, where C is the constant of integration. Substitute in the initial condition f(1) = 2 to find the value of C. Then solve for y in terms of x.