The most difficult part about slope fields was really visualizing. Is a slope field really suppose to simplify things? How I see it is that at each point you have to calculate the slope of each y coordinate in the plane. It just appears to be inconvenient. What I had difficulty with was solving differential equations to find families of solution curves, I'm not sure but if there are multiple solutions, are there limits to what the solutions can be? I guess I am trying to figure why slope fields are useful and what they can show us besides the rate of change at specific points for functions; or is that the point and I am not seeing its importance?
What I did find interesting was how the idea of differential equations with initial values are incorporated into slope fields. Initial values equations represent real situations that have a real (unique) answer. Its gratifying to have math presented in reality so there is substance to the ideas we discuss.
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