REVIEW PROBLEMS FOR MIDTERM I MATH 2373, FALL 2016 ANSWER KEY

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1 REVIEW PROBLEMS FOR MIDTERM I MATH 2373, FALL 2016 ANSWER KEY This list of problems is not guaranteed to be an absolutely complete review. For a complete review make sure that you know how to do all the homework assigned in the course up to and including that due on October 4, and all the worksheets up to and including those of week 4. Strive to solve all the problems on this review sheet by hand, except perhaps for the Alice, Bob and Carol problem which has not-so-pretty numbers. In general you are expected to solve problems on an exam by hand unless specific instructions to the contrary are supplied on the exam. On the first midterm exam we will permit use of the calculator only sparingly (just for scientific calculator functions like +,,,,, log, exp) but on later exams we will ease up and permit calculator use for row reduction and determinants in many cases. However, on all exams, for algebra and calculus work not involving matrices, we insist that you perform and show all the steps needed to justify your final answer by hand calculation. An answer key for this review sheet will be posted on the course web page gwanders/math2373 at 10:30am on Wednesday, October 5. There is also a link to this URL on the Moodle page. Problem 1 Standing in line at the supermarket I see Alice, Bob and Carol ahead of me in the express check-out lane. Alice buys 2 bags of chips, 3 diet sodas, 1 bag of circus peanuts and spends $5.54. Bob buys 3 bags of chips, 1 diet soda, 3 bags of circus peanuts and spends $7.13. Carol buys 1 bag of chips, 1 diet soda, 1 bag of circus peanuts and spends $2.97. Find the price for a bag of chips, for a diet soda and for a bag of circus peanuts. You may use your calculator for the numerical work. Use the x A 1 b method of solution. Identify A, x and b clearly. A x x x y z x y z price of bag of chips price of can of diet soda price of bag of circus peanuts x y z Alice s bill Bob s bill Carol s bill b Date: September 30,

2 2 REVIEW FOR MT1: ANSWER KEY Remark. Another way to solve these equations, different from the method we asked for in this problem, is the following: and hence rref Solve the system of equations price of chips price of diet soda price of circus peanuts Problem x y +5w 1 2x +z +w 3 using the standard method taught in class, which means: write out the augmented matrix for the system, reduce the augmented matrix to reduced row echelon form, and present the final answer in the format of Example 7 on pp (see the next-to-last displayed line in the example). Do your work entirely by hand. Explain each row operation using notation like R 2 R 2 3R 3, as in the textbook. The augmented matrix of the system is [ Applying R 1 R 1 /2, we get [ 1 1/2 0 5/2 1/ Applying R 2 R 2 2R 1, we get [ 1 1/2 0 5/2 1/ Applying R 2 R 2, we get [ 1 1/2 0 5/2 1/ which is in row echelon form. Finally, applying R1 R 1 R 2 /2, we get [ 1 0 1/2 1/2 3/2, which is in reduced row echelon form. The last matrix above corresponds to the pair of equations... x +z/2 +w/2 3/2 y z 6w 4,.,

3 REVIEW FOR MT1: ANSWER KEY 3 The variables z and w are free since they appear in columns without pivots in the reduced row echelon matrix. Let z s and w t. Then, solving for the remaining variables x and y in terms of the parameters s and t, we get x s/2 t/2 + 3/2 and y s + 6t 4. The final answer written in the officially acceptable manner is x s/2 t/2 + 3/2 1/2 1/2 y z s + 6t 4 s s t 6 0 w t 0 1 where s and t may take any (real) values. Problem 3 The following matrix is not quite in row echelon form (i) Reduce the matrix to row echelon form. (At most two more steps are required.) Clearly identify which matrix is supposed to be in row echelon form. (ii) Then reduce the matrix all the way to reduced row echelon form. The last matrix you write down should be exactly in reduced row echelon form. Furthermore, every step of row-reduction should be explained by using notation like R2 R 2 3R 1, as in the textbook. (We are only asking for you to do row operations. You are not being asked to solve any equations.) 3/ , (given matrix) R4 R 4 R 3, R4 R (row echelon form) R3 R 3 R 4, R2 R 2 R 4, R1 R 1 R R 2 R 2 R 3, R 1 R 1 R 3

4 4 REVIEW FOR MT1: ANSWER KEY R1 R 1 R 2 Problem 4 (reduced row echelon form) Expand the following determinant along the third row. You need not evaluate the resulting 3 by 3 determinants, just get all the numbers in the right places, and get all the signs right. Then answer the same question except now expand along the second column checkerboard: expansion along row 3: expansion along column 2: ( 1) ( 6) +4 Solve the initial value problem column row ( 6) Problem t dt 4y + t5 sin t and y(1)

5 REVIEW FOR MT1: ANSWER KEY 5 We start by rewriting the equation in the standard form and then calculate in standard fashion: dt 4 t y t4 sin t ( ) exp 4 dt/t exp ( 4 log t) t 4 d dt t 4 y sin t t 4 y cos t + C 2 y(1) cos(1) + C C 2 + cos(1) t 4 y cos t cos(1) y t 4 cos t + (2 + cos(1))t 4. Solve the initial value problem dt y + 1 2t + 8 Problem 6 and y(0) 3. We use the technique of separation of variables. y + 1 dt 2(t + 4) log y log t C 2 y + 1 e C t + 4 1/2 y + 1 C 1 (t + 4) 1/2 (C 1 ±e C, t + 4 > 0) 4 y(0) /2 C 1 C 1 2 y + 1 2(t + 4) 1/2 y 2(t + 4) 1/2 1. To double-check our work, note that y(0) 2(0 + 4) 1/

6 6 REVIEW FOR MT1: ANSWER KEY using old math professor s trick We always get rid of the absolute value bars the same way. So instead of repeating those moves every time, we organize the calculation as follows: y + 1 dt 2(t + 4) log(y + 1) 1 log(t + 4) + ln C 2 y + 1 C(t + 4) 1/2 4 y(0) /2 C C 2 y + 1 2(t + 4) 1/2 y 2(t + 4) 1/2 1. The old math professor s trick of writing ln C is just a shorthand for skipping over some steps that are always the same when you have logarithms on one or both sides of the equation after integrating. To further double-check our work let us verify that the function we found does indeed satisfy the given differential equation. dt y + 1 2t + 4 Indeed, this checks out (t + 4)1/2 1 (t + 4) 1/2. (2(t + 4)1/2 1) + 1 2(t + 4) Problem 7 (t + 4) 1/2. Find k such that the following system of homogeneous equations has infinitely many solutions: x y 0. 1 k 2 z 0 It is the same thing to solve the equation k 2 0 because in general, for a square matrix A, the system of equations A x 0 has infinitely many solutions if and only if det A 0. Using basketweave to evaluate the left side we get the equation 2 4 ( 2) ( 5) ( 3) k ( 5) ( 3) ( 2) 2 7 k 0 which simplifies to 7 + k 0. Thus the solution is k 7. Another way to solve the problem is to expand the

7 REVIEW FOR MT1: ANSWER KEY 7 determinant along the third row: (1) k and then solve for k as before. + ( 2) k k 0 checkerboard: row 3 Find a, b, c, d, w, x, y and z. Problem d w x y z a b 12 c You must do all the work by hand, and justify the steps. Note: This problem is only long because there are so many variables. It is not particularly difficult to solve for any one particular variable. If you are in a hurry just solve for two or three of the variables. 0 [ a b 0 [ [ a 12 2a a a b y 27 c a + b b c c 27.

8 8 REVIEW FOR MT1: ANSWER KEY 0 [ d d + 12 d 6. 0 [ w z w 27 2w w 27/2 0 [ x x 54 2x x 27. (watch out: next pitch is a curveball!) 1 [ y 27 c y 82 6y y y [ w z 2z z + 41 z Finally, here s the puzzle with the blanks all filled in: / /2 41/

9 REVIEW FOR MT1: ANSWER KEY 9 Problem 9 (i) Find the general solution of the autonomous differential equation y (1 + y)(5 y). (ii) Also find the solution which satisfies the initial condition y(0) 7. (iii) Find the equilibrium solutions. (iv) Determine the stability of each equilibrium solution. (v) Sketch the direction field for the differential equation well enough to justify your determination of stability. First find the general solution. Start by separating variables (1 + y)(5 y) dt Partial fractions: 1 (1 + y)(5 y) A y B y 5 Multiply on both sides by (y + 1)(y 5) to get 1 A(y 5) + B(y + 1) Plug in y 1 to get A 1/6. Plug in y 5 to get B 1/6. We can rewrite the separated differential equation as 1 6 y y 5 dt Clear denominators: y + 1 y 5 6 dt Integrate on both sides and use the old math professor s trick: log(y + 1) log(y 5) 6t + ln C y + 1 ( ) y 5 Ce6t Solve for y: y + 1 yce 6t 5Ce 6t y(1 Ce 6t ) 5Ce 6t 1 y 5Ce6t + 1 Ce 6t 1 Next we solve the indicated initial value problem. Plug t 0 in earlier line marked with ( ) to get C. Thus y 20e6t + 1 4e 6t 1 is the solution satisfying y(0) 7. Equilibrium solutions are y 1 and y 5. (The numbers 1 and 5 are just the roots of (1 + y)(y 5) 0.) Looking at the dfield plot, we can see that y 5 is stable and y 1 is unstable. Using dfield here was overkill. Any legible sketch of the direction field which establishes where the slopes are positive and where the slopes are negative is good enough.

10 10 REVIEW FOR MT1: ANSWER KEY 10 Figure 1. Direction field y (1 + y) (5 y) y t Problem 10 Find the least squares line y k + mx for the following data points. (1, 4), (3, 8), (4, 9), (6, 12). Do so by first expressing the two equations for the unknowns k and m in matrix form. You can do the numerical work with your calculator.. The inconsistent system of equations we wish that we could solve exactly (but cannot) is the following: [ k m In order to do the best we can by the method of least squares, we use the trick to left-multiply both sides by the transpose. [ [ k m [ After multiplying out using MATLAB or hand calculator, we get [ [ [ 4 14 k m

11 REVIEW FOR MT1: ANSWER KEY 11 Finally, [ k m Thus the least squares line is [ [ [ y x. Here is a plot of the data points and the least squares line that fits these data points:. Figure 2. Least squares line and data points 15 y x y x (We just included this figure as an illustration. It is not an official part of the answer to this line-fitting problem.) Problem 11 The rate at which the amount of a radioactive substance changes is proportional to the amount present. At a certain time, a sample of a certain radioactive substance contains 40 mg. The same sample only contains 25 mg of the substance 50 years later. (i) How much of the substance will remain after 120 years? (ii) What is the half-life of the substance? Start with the differential equation.

12 12 REVIEW FOR MT1: ANSWER KEY Let y denote the amount of the substance in milligrams and let t denote the time in years measured from the time observations of the substance began. We are given that ky, y(0) 40, y(50) 25. dt Here k is a negative constant we need to determine. The general solution of the differential equation is y Ce kt. We have C 40 because y(0) 40. We have 40e 50k 25 because y(50) 25. Thus e 50k 25/40 5/8 and hence y 40(5/8) t/50. Some students may prefer to solve for k, so we present that route as well. From, say, the equation e 50k 5/8 above we get and hence k ln(5/8) y 40e t, which is an equivalent expression for the amount of substance at time t. We then have y(120) 40(5/8) 120/50 40 exp( ( )(120)) mg. which answers part (i) in several different formats any of one of which is acceptable but with the last being preferred since easiest to read. To answer part (ii) of the question we have to calculate the half-life T of the substance. This is determined by either of the following two equations: Solving the former we have Solving the latter we get 40(5/8) T/50 20 and 40e T 20 T 50 ln(1/2) ln(5/8) ln(1/2) years T ln( 1 2 ) years, which (no surprise) is the same answer. I solved this problem in two ways but in practice of course you only have to pick one way of solving and then stick with it. Problem 12 Initially an 800 gallon tank contains 400 gallons of water in which is dissolved 50 lbs of salt. Brine containing 4 lbs of salt per gallon enters the tank at the rate of 5 gallons per minute. The well mixed brine leaves the tank at the slower rate of 3 gallons per minute. (i) Find an expression for the number of pounds of salt in the tank at time t, until the tank overflows. (ii) How much salt is in the tank at the moment the tank overflows?

13 REVIEW FOR MT1: ANSWER KEY 13 Let y y(t) be the amount (lbs) of salt in the tank at time t (min). We have to solve the initial value problem y(0) 50 and dt 4 lbs gal 5 gal y lbs min (5 3)t gal 3 gal. min }{{}}{{} rate in rate out After simplifying algebraically we get 3y t 20. y(0) 50 and dt + We now use the integrating factor method to finish up. ( ) ( ) 3 dt 3 dt exp exp exp t t ( 3 log(t + 200) 2 d dt (t + 200)3/2 y 20(t + 200) 3/2 (t + 200) 3/2 y 8(t + 200) 5/2 + C ) (t + 200) 3/ / /2 + C C 200 3/ /2 4, 384, 062. y 8(t + 200) 4, 384, 062/(t + 200) 3/2. The tank overflows when t 800 and hence t minutes. Plugging in t 200 we get y(200) 8( ) 4, 384, 062/( ) 3/ pounds of salt in the tank at the moment of overflow. Note: On an exam, to answer part (i) of the question, it would be okay to write, say, y 8(t + 200) + C/(t + 200) 3/2, where C 200 3/ /2, without simplifying C further. Problem 13 A steel ball with an initial temperature of 60 o F is placed in an oven which is maintained at 420 o F. After 4 minutes the temperature of the ball is 180 o F. (i) Find an expression for the temperature of the ball at time t. (ii) Find the surface temperature of the ball after 12 minutes. (iii) Find the time at which the ball will have a temperature of 350 o F. Start with the differential equation. Let y y(t) be the temperature in degrees Fahrenheit of the ball at time t in minutes after being placed in the oven. We know that k(420 y) dt by Newton s law of heating, where k > 0 is a constant we have to determine using the information given in the problem. We are also given that y(0) 60 and y(4) 180.

14 14 REVIEW FOR MT1: ANSWER KEY We solve using the integrating factor method. We start by rewriting the differential equation as + ky 420k. dt The integrating factor is exp( k dt) e kt. We then have d dt ekt y 420ke kt e kt y 420e kt + C y Ce kt. To get the constant C we use the initial value information: Thus we have y(0) C C 360. y e kt. To answer part (i) of the question, we are going to replace e kt by an expression in which the only variable is t. One way to do that is to solve for k and another way is to use algebra to get such an expression. We will show both ways, and we will show below how to pursue both ways to answer parts (ii) and (iii) of the question as well. We have It follows that 180 y(4) e 4k e 4k e kt y ( ) t/4 2, 3 ( ) t/ which answers part (i) of the question. We can also solve the equation e 4k 2 3 for k, obtaining k 1 ( ) 2 4 ln Then we get another (equivalent) answer for part (i), namely We then have y(12) y e t. ( ) 12/ exp( ) which answers part (ii) of the question both ways. Finally, to find the time T asked for in part (iii) we have to solve the equation ( ) T/4 2 y(t ) exp( T ) This can be rewritten ( ) T/4 2 exp( T ). 3 Solving by the usual precalculus methods, we get T 4 ln(7/36) ln(2/3) ln(7/36) minutes, which answers part (iii) of the question (in both ways).

15 REVIEW FOR MT1: ANSWER KEY 15 Remark. Just so there is no misunderstanding: I solved the problem two different ways. When you are solving problems like this you only have to pick one of the ways and then stick with it. Find the determinant Problem by hand. Use whatever combination of methods seems the easiest. Explain each step of your calculation by a word or phrase (use the row operation R4 R 4 + 2R 1 ) (expand along third column) (expand along first column) (This is not the only possible solution.) Problem 15 Find the inverse of the following matrix You must show all work. You must use the textbook notation for row operations to explain each step. Check at least three times that you wrote out the correct starting matrix for your calculation.

16 16 REVIEW FOR MT1: ANSWER KEY We use the side-by-side method , R 2 R 2 + 2R 1, R 3 R 3 2R , hence:, R 3 R 3, R 2 R 2 + R 3, R 1 R 1 R 3, R 1 R 1 R Solve the system of equations Problem 16 5x 4 x 3 7, 4x 3 2x 2 + x 4 9, 2x 2 + x 4 x 3 17, x 1 + x 2 + x 3 + x 4 14 for x 3 using Cramer s Rule. You are expected first to rewrite the system of equations neatly, and then to express x 3 as a ratio of 4-by-4 determinants. You do not have to evaluate any determinants. The emphasis is on demonstrating that you know the pattern of Cramer s Rule. Note that the equations were written sloppily. You have to begin by rewriting the equations neatly, with each variable placed in its own column. x 3 +5x 4 7 2x 2 +4x 3 +x x 2 x 3 +x x 1 +x 2 +x 3 +x 4 14 Having written things out neatly, I can read off the required ratio of determinants easily: / x

17 REVIEW FOR MT1: ANSWER KEY 17 Problem 17 Find the inverse of the following matrix: [ Express all the entries of the inverse matrix as fractions, not as decimals. Solve this problem entirely by hand. Show all your work. [ [ [ 5/3 1/3 17/3 4/3.