An important pairing in the Fundamental Theorem of Calculus links a given function g with its integral- so-far function G, as shown here: G(t) = / g(x) dx. Follow the steps below to build and use a spreadsheet-powered approximate version of this relationship. To get started, acquire a personal copy of the sample spreadsheet linked here. It supplies 4 columns: Column A lists the values of an integer index, i; Column B lists the equally-spaced input values t, = a + iAt; Column C lists the corresponding function values g(t;); and Column D lists the Trapezoidal Rule approximations for G(t,) = / g(x) dx based on the tabulated values g(t;). In the sample spreadsheet provided, the values of G(t;) shown in Column D are literal numbers, not active formulas, so they do not update when the values of g(t,) shown in Column C change. Fix this by filling in Column E with formulas that calculate G(t,) dynamically. Verify your formulas by comparing the computed outputs with the known-correct values in Column D. Once these match, you can safely delete Column D and start building something to hand in. (a) Let g(t) be the function tabulated on the given spreadsheet. Define h(t) = g(t) , and H(t) = h(x) dr. Adapt your spreadsheet to plot the graphs of h and H on the same axes. Submit both the labelled figure and the number H(1), along with a description of how your spreadsheet works. (See the section labelled "Discussion" below for more advice on presentation.)In parts (b)-(e) below, traditional hand calculations set up an opportunity to do something serious with your spreadsheet from part (a). Complete these parts without using that spreadsheet. Consider the following initial-value problem (IVP) defining a function y. Here a > 0 is a constant and f (t) is a given function. (* * ) ay' ( t ) + y(t ) = f (t), y(0) = 0. Notice that when o is extremely small, the ODE here is not much different from the identity y(t) = f (t). In what follows we will think of the function f(t) as an "input" and the solution y() as an "output" and investigate why this setup is described as a "low-pass filter". (b) Assuming f(t) = 1 for all t in (**), find the unique solution y(t). Hand in both the exact formula, with supporting work, and a reasonable plot of the graph of y(f). (For plotting, use o = 0.1 and consider 0) 0, -1 if x 0 and/or the function f(#). State your choices clearly, and briefly explain what makes your results interesting and/or informative. (For example, you could explore the effect of changing a on the same input signal; or you could validate the "low pass filter" terminology by providing different input signals; or you could find another interesting question that these methods could address.)