Math 250B Midterm I Information Fall 2018

Transcription

1 Math 250B Midterm I Information Fall 2018 WHEN: Wednesday, September 26, in class (no notes, books, calculators I will supply a table of integrals) EXTRA OFFICE HOURS: Sunday, September 23 from 8:00 PM 12:00 MIDNIGHT Monday, September 24 from 11:00 AM 11:50 AM, 3:00 3:50 PM, and 7:30 9:00 PM Tuesday, September 25 from 4:00 6:00 PM Wednesday, September 26 from 10:00 AM 11:50 AM and 3:00 3:50 PM REVIEW SESSIONS: Sunday, September 23 from 6:00 8:00 PM, in MH-565 Tuesday, September 25 from 7:00 9:00 PM, Room T.B.A. COVERAGE: The midterm will cover the material discussed in lecture from Sections 1.1,1.2,1.4, , , and STUDYING: Here is an overview of the topics we have covered. You should be comfortable with all of the following words below: Chapter 1: Differential equation, Newton s second law of motion, Distance/Velocity/Acceleration problems, Newton s law of cooling, order of a DE, linear DE, general solution to a DE, initial conditions, initial value problem, particular solution to a DE, separable DE, integrating factor, mixing problem (chemical concentration and amount), Bernoulli equation, homogeneous DE, change of variables Chapter 2: Matrix, row vector, column vector, square matrix, main diagonal, diagonal matrix, lower triangular, upper triangular, symmetric, skew-symmetric, operations on matrices: addition, subtraction, scalar multiplication, multiplication, transpose, zero matrix, identity matrix, associative, distributive, non-commutative properties of matrices, linear equation, linear system, matrix of coefficients, augmented matrix, elementary row operations, row-echelon matrix, pivots, rank of a matrix, Gaussian elimination, free variable, bounded/leading variable, row-equivalent matrices, invertible matrix, Gauss-Jordan method, elementary matrix, Invertible Matrix Theorem

2 Chapter 3: Determinants, Properties of determinants, minors, cofactors, Cofactor Expansion Theorem, matrix of cofactors, adjoint of a matrix, adjoint method for A 1 THINGS TO BE ABLE TO DO: Verify that a given solution satisfies a DE and/or initial conditions. Solve separable DE (such as the one associated with Newton s Law of Cooling) Solve Newton s Law of Cooling Problems Solve 1st-order linear DE through the use of integrating factors Chemical mixing problems Solve 1st-order homogeneous DE by change of variables, using V := y x y = xv ) (i.e. Solve Bernoulli equations Add, subtract, scalar multiply, multiply, and take transposes of matrices Be able to decide whether a given matrix is symmetric and/or skew-symmetric Translate back and forth between linear systems and matrices Convert a matrix to an equivalent row echelon form matrix by performing ERO Find the rank of a matrix Use Gaussian elimination to find the solution set to a linear system using free variables Determine if a matrix is invertible, and find the inverse if it is Be able to identify the theoretical advantages of working with an invertible matrix: Invertible Matrix Theorem Know the properties of invertible matrices (Theorem , etc) Know the determinants for 1 1, 2 2, 3 3 matrices especially, and for upper (or lower) triangular matrices Know the properties (P1)-(P10) of determinants

3 Know the determinant test for invertibility of a square matrix A Be able to use the Cofactor Expansion Theorem to find det(a) Be able to compute the matrix of cofactors of a matrix A, the adjoint of A, and be able to use it find A 1 in cases where A is invertible ADVICE: I suggest reviewing the group work. If you have time, it might be best to try re-working those exercises from scratch and looking up the answers afterwards. You will not have time to re-do all of the homework, but you might try some of them again, especially the ones that will assist you with the THINGS TO BE ABLE TO DO listed above. TRY TO RE-DO PROBLEMS FROM SCRATCH, RATHER THAN JUST REVIEWING YOUR ALREADY-COMPLETED SOLUTIONS. Also, quiz each other in study groups. Finally, you can ask me questions as much as you want, and I will be happy to review or pop-quiz a topic with you if you feel shaky. Basically, I m here to help and I want everybody to do well, so please don t be shy :=)!! Below is a list of some sample problems, in a random order, that may be similar to midterm questions. Solutions will be posted. Practice Problems: Problem 1. Find the inverse of each matrix below if it is invertible (use the Gauss- Jordan technique), and express each invertible matrix as a product of elementary matrices: (a): (b): Problem 2. Solve the initial value problem y y 2ln y x = 1 (1 2 ln x) x subject to y(1) = e by using the change of variables u = ln y. Problem 3. Use Gaussian elimination to solve the system of equations: x + 3y z = 1 3x + 4y 4z = 7 3x + 6y + 2z = 3.

4 Problem 4. Let A and B be n n matrices. Under what conditions is it true that (A B)(A + B) = A 2 B 2? Find a specific example to show that this equation is not always true. Problem 5. Do Problem #40 on page 193. Problem 6. Solve the differential equation x dy dx y = x3 y e y x. Problem 7. Show that if A is an invertible matrix and B can be obtained from A by a series of elementary row operations, then B is invertible. Problem 8. Solve the DE subject to y(1) = 0. Problem 9. Solve the DE x dy dx + y = 2x xy 4 dx + (y 2 + 2)e 3x dy = 0. Problem 10. Do Problem #48 on p Problem 11. Initially, 50 pounds of salt is dissolved in a large tank holding 300 gallons of water. A solution containing 2 pounds of salt per gallon is pumped into the tank at 3 gallons per minute and the well-stirred mixture is pumped out of the tank at 3 gallons per minute, how much salt is present after 50 minutes? [You may use a calculator here.] Problem 12. Let A be an n n matrix with A 8 = 0. Show that I n A 3 is invertible with (I n A 3 ) 1 = I n + A 3 + A 6. Problem 13. If A, B, and C are invertible matrices, show that A 1 C 1 B 2 is invertible. Problem 14. Use the adjoint method to find the inverse of each matrix below, if it is invertible (a): (b):

5 Problem 15. Use row reduction to show that det a b c = (b a)(c a)(c b). a 2 b 2 c 2 Problem 16. Find the determinant of each matrix below by direct calculation: (a): (b): Problem 17. Repeat each part of Problem 16 by using elementary row operations to bring the matrix to row-echelon form.