1. This problem is designed to help you understand the very important concept that "integration is a smoothing operator" while differentiation is just the reverse the derivative is often rougher than the function you are differentiating. Keep in mind that integration and differentiation are inverses of each other, so if integration smooths things out then differentiation must "roughen" things up. Consider the differential equation where dy dt = f(t), y(0) = 0, - (1) f(t) = 1, 0≤t≤1, f(t) = -1, 1 < t <2. For this problem you are only interested in the solution y(t) for 0 < t≤2. Note that it does not matter how you define f(1) or even whether you define it at all. = Determine y(t) for 0 ≤ t≤ 2 and show that y(t) is continuous, but not continuously differentiable, at t 1. Show that unlike the case for f(t), you now have a specific value for y(1) - state what it is. Thus, you integrated a discontinuous function and got a continuous function - integration smoothed out the function f(t). MA This is a general property of differential equations - the derivatives are generally rougher (more discontinuous in this case) than the solution. While for most engineering ap- plications you have differential equations where the (unknown) dependent variable y appears on the right hand side (so that you cannot solve them by just doing an inte- gral) since differentiation is the opposite of integration the term "integrate" is often used to refer to the solution process. The fact that integration is a smoothing opera- tor (or differentiation is a "roughening" operator) means that generally it is a harder task to approximate the solution to a differential equation, than to just approximate a function. You will study ways of approximating differential equations shortly.