Differential Equations

5. Use the given inverse matrices to find (AB)¹ and (AT). Be sure to show your work. A-1 Coffee = 0 4 -4 27 1 2 231 B-1 65-37 -1 4 -2 4 1 3

Let A 4 be the R-vector space V = {p € R[x] / degree (p) 3). (a) Show that B = ((1+x)°, (1+x)1, (1+x)2, (1 + x)3) an ordered basis of V shall. What is the dimension of V? (b) Calculate the coordinate vector of p = 2x3 + 5x2 + 4x - 1 with respect to the Base B. (3 + 2 Points)

2. Two chemicals are mixed together and as they react the temperature of the mixture is measured. One, two and three hours after the mixing takes place, the temperatures of the chemicals are, respectively, 156.8, 161.7 and 177.2 degrees. (a) Create a system of linear equations for the data to fit the curve T(t) = at² + bt+c, where t is the time in hours and T' is the temperature. You essentially need to find a parabola that passes through the three points (1, 156.8), (2, 161.7), (3, 177.2). (b) Write the system in the form Ax = b. (c) Use Cramer's Rule to solve the system. Be sure to at least show the set up for using Cramer's Rule. You can use a computer to compute the determinants if so desired. (d) Write the equation of the parabola. 1 le

5. Consider the differential equation y' + (cost)y = 0 and the vector space C¹(R) (the vector space of functions having continuous first derivatives on the entire real line). (a) Does the solution set of the differential equation form a subspace of C¹ (R)? If it does, justify your answer. If not, explain why not. (b) If the solution set is a subspace, find a basis and determine the dimension.

4. Determine whether the set S = {-2t+t²,8+t³, -t² + t³, −4+ t²} spans P3.

1. Use the inverse matrix to solve the linear following linear systems. Look carefully at the linear systems to significantly reduce the workload. (a) 2x + 3y + z = 4 3x + 3y + 2 = 8 2x + 4y + z = 5 (b) 2x + 3y+z=0 3x + 3y + z = 0 2x + 4y+z=0

5. Graph the following function showing all relative extreme points and inflection points: f(x) = x³ + 3x² - 9x - 7

15. Find the integral using integration by parts for the following: f(x-2)(x 2) (x + 1) 5 dx

3) Determine whether the following systems are memoryless: (a) y(t-1) = 2x(t-1) and (b) y(t) = d/dt x(t)

3. Let A and B denote nxn matrices. For each of the following, prove the statement in general or give a counterexample that shows it is not true. (a) (A + B) (A - B) = A²-B² (b) (I+A)² = 1 + 2A + A²