1. In the last assignment you reflected on how to approach a difficult problem. Another key component of problem solving is to reflect on your solution after you have solved the problem and check if your answer makes sense. Looking back on the previous three assignments, identify a specific problem in which you could have checked that your answer made sense based on the other information in the assignment, your answers to other questions in the assignment, or with any other tools at your disposal (Desmos, etc.). In one or two paragraphs, describe some of the ways in which you could have checked (or did check) your answer for this specific problem, as well as some of the ways that you check your answers to problems in general.

2. Consider the differential equation below, in which a,b,c,k are real constants and c2 + k2  > 0:

(a) Find 4 specific numbers a,b,c,k for which the function y = x2  − 3x provides a solution for equa-

tion (∗) on the interval (0, +∞). Briefly explain your method.

(b) Is there only one correct answer to part (a)? If so, explain why; if not, describe the set of all choices

for (a,b,c,k) that should be accepted.

(c) Consider the ODE (∗) with the constants found in part (a). Find a solution satisfying y(2) = − 49

Hint: Start by postulating y = xu(x) for some new unknown function u. Find u, then recover y .

3. 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) = \at 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 ti  = a + i∆t; Column C lists the corresponding function values g(ti ); and Column D lists the Trapezoidal

Rule approximations for G(ti ) = \ati  g(x) dx based on the tabulated values g(ti ).

In the sample spreadsheet provided, the values of G(ti ) shown in Column D are literal numbers, not active formulas, so they do not update when the values of g(ti ) shown in Column C change. Fix this by filling in Column E with formulas that calculate G(ti ) 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)2 , and H(t) =

\at h(x)dx. 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 α > 0 is a constant and f(t) is a given function.

(∗∗)              αy\ (t) + y(t) = f(t),        y(0) = 0.

Notice that when α 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(t) 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(t).

(For plotting, use α = 0.1 and consider 0 ≤ t ≤ 1.  You may scan a hand-drawn picture or use a package like Desmos.)

(c) Find a nonzero function y1 (t) satisfying αy +y1  = 0. Then postulate a solution for (∗∗) in the form y(t) = y1 (t)u(t) for some new unknown function u.  Derive a new initial-value problem describing u(t).

(d) Write the solution for (∗∗) in the form y(t) = v(t) \0 t g(r)dr, where the functions v and g are defined in terms the ingredients of the original problem (i.e., the constant α, the function f, etc.). Suggestion: Do this first for the function u of part (c), then upgrade your result from u to y .

(e) Confirm your results in the previous part by taking f(t) = 1 and evaluating your integral expression.

Verify that the result agrees with the one you found in part (b).

This  is  the point  where  your spreadsheet  returns .   The  solution formula  derived  in part  (d)  expresses y(t) as the product of a function you know and an integral-so-far style factor that your spreadsheet can quickly evaluate for any given function f(t) .

(f) Adapt your spreadsheet from part  (a) to produce approximate solutions for the IVP  (∗∗).   In

particular, arrange it so that populating one column with the values of a given “input function” f(t) will quickly update another column with the corresponding “output function”y(t) and plot the graphs of both f(t) and y(t) on the same axes.  Use your new tool to plot pairs of input and output functions in each of the situations below.  (In all cases, use the interval 0 ≤ t ≤ 1 and step size ∆t = 0.002.)

i. α = 0.1, f(t) = 1        (Compare your exact solution in part (b).)

ii. α = 0.1, f(t) = sin(2πt)

iii. α = 0.1, f(t) = sin(10πt)

iv. α = 0.1, f(t) = sign(sin(4πt))

(Notation: sign(x) is the “sign of x”, with the value +1 if x > 0, −1 if x < 0, and 0 if x = 0. Some spreadsheets, including the three named in the Discussion section below, have this function built in.)

(g) Make 3 more plots like the ones from part (f), but this time choose your own values for the time

constant α > 0 and/or the function f(t). State your choices clearly, and briefly explain what makes your results interesting and/or informative. (For example, you could explore the effect of changing α 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.)

Discussion: