1. Math 3C Practice Final Problems Solutions Prepared by Vince Zaccone For Campus Learning Assistance Services at UCSB

2. Prepared by Vince Zaccone For Campus Learning Assistance Services at UCSB 1a) (find a general solution) First we solve the homogeneous equation: Now find a particular solution – it will be a constant, say y p =A. Plug this into the equation and solve for A:

3. Prepared by Vince Zaccone For Campus Learning Assistance Services at UCSB 1b) (find a general solution) First we solve the homogeneous equation: Now we use variation of parameters to find a particular solution: Quotient rule Multiply both sides by t 8

4. Prepared by Vince Zaccone For Campus Learning Assistance Services at UCSB 1c) (solve the initial value problem) First we solve the homogeneous equation: Now we use variation of parameters to find a particular solution: Now plug in the initial value to find C. Solution to IVP

5. Prepared by Vince Zaccone For Campus Learning Assistance Services at UCSB 1d) (solve the initial value problem) First we solve the homogeneous equation: Now we use variation of parameters to find a particular solution: Now plug in the initial value to find C. Solution to IVP

6. Prepared by Vince Zaccone For Campus Learning Assistance Services at UCSB 2) Farmer Fred opens a bank account with an initial deposit of $10,000. The account earns a monthly interest rate of 0.5%, compounded continuously. In addition, Fred will deposit $100 each month. How many months will it take for the account value to double? Define variables: y(t) = account value at time t (months) The differential equation is We also have an initial value y(0)=10,000 We could solve this equation by separation, but let’s use the excellent guess method instead. First we solve the homogeneous equation: For the particular solution, notice that we should get a constant. Plug in and solve: Now plug in the initial value to find C. Finally we can use our formula to find the answer. Just set y(t)=20,000 and solve for t.

7. Prepared by Vince Zaccone For Campus Learning Assistance Services at UCSB 3) A lukewarm beverage (initially at 70°F) is placed into a cold (40°F) refrigerator. Five minutes later the temperature of the still-not-cold-enough beverage is 60°F. It is replaced in the fridge to continue the chilling process. How many minutes will it take for the beverage to reach the temperature of optimal refreshment (43°F)? Assume that the temperature follows Newton’s law of cooling, i.e. the rate of change of the temperature is proportional to the difference between the temperature of the beverage and the temperature of the surroundings. Define variables: T(t)=temperature of beverage (in ° F) at time t (minutes) The differential equation isHere k is a constant of proportionality Let’s solve this one by separation. We need to find the 2 constants C and k. From the given information we know y(0)=70 and y(5)=60. Now we can rewrite our formula with the constants we just found, then use it to solve the problem.

8. Prepared by Vince Zaccone For Campus Learning Assistance Services at UCSB 4) Consider the following autonomous differential equation: a) Sketch a slope field for this equation. b) Find any equilibrium solutions and classify them as stable or unstable. c) Find an explicit formula for the solution that passes through the initial value y(0) = 1. a) The slope field shows the equilibrium solutions, and their stability. We will calculate them algebraically as well. b) To find equilibrium solutions, set y’=0: Equilibrium solutions are the horizontal lines y=2 and y=-1 To assess the stability, plug in values on either side to find if the slope is positive or negative.

9. Prepared by Vince Zaccone For Campus Learning Assistance Services at UCSB 4) Consider the following autonomous differential equation: c) Find an explicit formula for the solution that passes through the initial value y(0) = 1. We will solve this by separation: We use partial fractions to simplify the integral: Use the initial value to find C This is an implicit solution – we can do some algebra and solve for y to get an explicit solution Careful with the absolute values – in this case we have y(0)=1, so we can drop the abs value bars on the bottom, but we have to switch the top of the fraction. This solution is valid for -1<y<2

10. Prepared by Vince Zaccone For Campus Learning Assistance Services at UCSB 5) Consider the following system of differential equations: a) Find and graph the nullclines of this system. b) What are the equilibria of this system? c) Classify any equilibria as stable or unstable. d) Sketch a reasonable solution to this system of differential equations. Set x’=0 to find v-nullcline: Set y’=0 to find h-nullcline: Equlibrium Points are (0,0) – unstable (6,2) - stable h-nullclines in yellow v-nullclines in red Equilibria at intersections Unstable equil. Stable equil. Solution curve

11. Prepared by Vince Zaccone For Campus Learning Assistance Services at UCSB 6) Solve the following system of linear equations:

12. Prepared by Vince Zaccone For Campus Learning Assistance Services at UCSB 6) Solve the following system of linear equations: Augmented matrix form

13. Prepared by Vince Zaccone For Campus Learning Assistance Services at UCSB 6) Solve the following system of linear equations: Augmented matrix form Row reduction:

14. Prepared by Vince Zaccone For Campus Learning Assistance Services at UCSB 6) Solve the following system of linear equations: Augmented matrix form Row reduction: The 3 rd row represents the equation 0x+0y+0z = -5. This equation has no solution, so the original system of equations is inconsistent (not solvable).

15. Prepared by Vince Zaccone For Campus Learning Assistance Services at UCSB 7) Consider the following matrix: a)Find the determinant of this matrix. b)Is this matrix invertible? If so, find the inverse. c)What is the determinant of the inverse of this matrix? d)Using this information, solve the following system of linear equations:

16. Prepared by Vince Zaccone For Campus Learning Assistance Services at UCSB 7) Consider the following matrix: a)Find the determinant of this matrix. b)Is this matrix invertible? If so, find the inverse. c)What is the determinant of the inverse of this matrix? d)Using this information, solve the following system of linear equations: a) Since A is a triangular matrix, the determinant is the product of the diagonal elements – det(A) = -312

17. Prepared by Vince Zaccone For Campus Learning Assistance Services at UCSB 7) Consider the following matrix: a)Find the determinant of this matrix. b)Is this matrix invertible? If so, find the inverse. c)What is the determinant of the inverse of this matrix? d)Using this information, solve the following system of linear equations: a) Since A is a triangular matrix, the determinant is the product of the diagonal elements – det(A) = -312 b) Since det(A) is not zero, A is invertible, and we can use row reduction to find the inverse.

18. Prepared by Vince Zaccone For Campus Learning Assistance Services at UCSB 7) Consider the following matrix: a)Find the determinant of this matrix. b)Is this matrix invertible? If so, find the inverse. c)What is the determinant of the inverse of this matrix? d)Using this information, solve the following system of linear equations: c) Since A -1 is a triangular matrix, the determinant is the product of the diagonal elements: det(A -1 ) = -1/312

19. Prepared by Vince Zaccone For Campus Learning Assistance Services at UCSB 7) Consider the following matrix: a)Find the determinant of this matrix. b)Is this matrix invertible? If so, find the inverse. c)What is the determinant of the inverse of this matrix? d)Using this information, solve the following system of linear equations: c) Since A -1 is a triangular matrix, the determinant is the product of the diagonal elements: det(A -1 ) = -1/312 d) Notice that we have the inverse for the coefficient matrix, so we just need to multiply:

20. Prepared by Vince Zaccone For Campus Learning Assistance Services at UCSB 7) Consider the following matrix: a)Find the determinant of this matrix. b)Is this matrix invertible? If so, find the inverse. c)What is the determinant of the inverse of this matrix? d)Using this information, solve the following system of linear equations: c) Since A -1 is a triangular matrix, the determinant is the product of the diagonal elements: det(A -1 ) = -1/312 d) Notice that we have the inverse for the coefficient matrix, so we just need to multiply:

21. 8) Consider the following set of vectors in R 3 a)What is the dimension of the span of this set of vectors? b)Is the span of this set of vectors R 3 ? c)Is this a basis for R 3 ? d)Is each of the following vectors within the span of this set? If so, express them as a linear combination of the vectors in the set. Prepared by Vince Zaccone For Campus Learning Assistance Services at UCSB

22. 8) Consider the following set of vectors in R 3 a) Put the column vectors in a matrix and perform elementary row operations: Prepared by Vince Zaccone For Campus Learning Assistance Services at UCSB a)What is the dimension of the span of this set of vectors? b)Is the span of this set of vectors R 3 ? c)Is this a basis for R 3 ? d)Is each of the following vectors within the span of this set? If so, express them as a linear combination of the vectors in the set.

23. 8) Consider the following set of vectors in R 3 a) Put the column vectors in a matrix and perform elementary row operations: This has 2 pivot columns, so there are 2 linearly independent vectors in the set. In other words, the span of this set is 2-dimensional. Prepared by Vince Zaccone For Campus Learning Assistance Services at UCSB a)What is the dimension of the span of this set of vectors? b)Is the span of this set of vectors R 3 ? c)Is this a basis for R 3 ? d)Is each of the following vectors within the span of this set? If so, express them as a linear combination of the vectors in the set.

24. 8) Consider the following set of vectors in R 3 a) Put the column vectors in a matrix and perform elementary row operations: This has 2 pivot columns, so there are 2 linearly independent vectors in the set. In other words, the span of this set is 2-dimensional. b)No, the span is 2-dimensional, and R 3 is 3-dimensional. c)No, this is not a basis because is does not span R 3. Prepared by Vince Zaccone For Campus Learning Assistance Services at UCSB a)What is the dimension of the span of this set of vectors? b)Is the span of this set of vectors R 3 ? c)Is this a basis for R 3 ? d)Is each of the following vectors within the span of this set? If so, express them as a linear combination of the vectors in the set.

25. 8) Consider the following set of vectors in R 3 a) Put the column vectors in a matrix and perform elementary row operations: This has 2 pivot columns, so there are 2 linearly independent vectors in the set. In other words, the span of this set is 2-dimensional. b)No, the span is 2-dimensional, and R 3 is 3-dimensional. c)No, this is not a basis because is does not span R 3. d) We have several options for this part. One way is to make a good guess. If you notice that v 2 is just 2 times a 3 then we have our answer for that one: Prepared by Vince Zaccone For Campus Learning Assistance Services at UCSB a)What is the dimension of the span of this set of vectors? b)Is the span of this set of vectors R 3 ? c)Is this a basis for R 3 ? d)Is each of the following vectors within the span of this set? If so, express them as a linear combination of the vectors in the set.

26. 8) Consider the following set of vectors in R 3 a) Put the column vectors in a matrix and perform elementary row operations: This has 2 pivot columns, so there are 2 linearly independent vectors in the set. In other words, the span of this set is 2-dimensional. b)No, the span is 2-dimensional, and R 3 is 3-dimensional. c)No, this is not a basis because is does not span R 3. d) We have several options for this part. One way is to make a good guess. If you notice that v 2 is just 2 times a 3 then we have our answer for that one: For v 1 there is no obvious answer, so we need to try to solve for it. Since we know that our span is only 2-dimensional, we only need two basis vectors. Choose any 2 independent vectors from our set, say a 1 and a 2, and try to write v 1 as a linear combination of them. Prepared by Vince Zaccone For Campus Learning Assistance Services at UCSB a)What is the dimension of the span of this set of vectors? b)Is the span of this set of vectors R 3 ? c)Is this a basis for R 3 ? d)Is each of the following vectors within the span of this set? If so, express them as a linear combination of the vectors in the set.

27. 8) Consider the following set of vectors in R 3 a) Put the column vectors in a matrix and perform elementary row operations: This has 2 pivot columns, so there are 2 linearly independent vectors in the set. In other words, the span of this set is 2-dimensional. b)No, the span is 2-dimensional, and R 3 is 3-dimensional. c)No, this is not a basis because is does not span R 3. d) We have several options for this part. One way is to make a good guess. If you notice that v 2 is just 2 times a 3 then we have our answer for that one: For v 1 there is no obvious answer, so we need to try to solve for it. Since we know that our span is only 2-dimensional, we only need two basis vectors. Choose any 2 independent vectors from our set, say a 1 and a 2, and try to write v 1 as a linear combination of them. Prepared by Vince Zaccone For Campus Learning Assistance Services at UCSB a)What is the dimension of the span of this set of vectors? b)Is the span of this set of vectors R 3 ? c)Is this a basis for R 3 ? d)Is each of the following vectors within the span of this set? If so, express them as a linear combination of the vectors in the set.

28. 8) Consider the following set of vectors in R 3 a) Put the column vectors in a matrix and perform elementary row operations: This has 2 pivot columns, so there are 2 linearly independent vectors in the set. In other words, the span of this set is 2-dimensional. b)No, the span is 2-dimensional, and R 3 is 3-dimensional. c)No, this is not a basis because is does not span R 3. d) We have several options for this part. One way is to make a good guess. If you notice that v 2 is just 2 times a 3 then we have our answer for that one: For v 1 there is no obvious answer, so we need to try to solve for it. Since we know that our span is only 2-dimensional, we only need two basis vectors. Choose any 2 independent vectors from our set, say a 1 and a 2, and try to write v 1 as a linear combination of them. We get an inconsistent set of equations, so v 1 is not in the span. Prepared by Vince Zaccone For Campus Learning Assistance Services at UCSB a)What is the dimension of the span of this set of vectors? b)Is the span of this set of vectors R 3 ? c)Is this a basis for R 3 ? d)Is each of the following vectors within the span of this set? If so, express them as a linear combination of the vectors in the set.

29. 9) Consider the following set of vectors in P 2 : a)What is dimension of the span of this set? b)Does this set span P 2 ? c)Is this set a basis for P 2 ? d)Express each of the standard basis vectors {1, x, x 2 } as a linear combination of the vectors in the set. Prepared by Vince Zaccone For Campus Learning Assistance Services at UCSB

30. 9) Consider the following set of vectors in P 2 : a) These are 3 independent vectors. We can see this if we try to form a linear combination of them, and find that there is no way to cancel them out: Prepared by Vince Zaccone For Campus Learning Assistance Services at UCSB a)What is dimension of the span of this set? b)Does this set span P 2 ? c)Is this set a basis for P 2 ? d)Express each of the standard basis vectors {1, x, x 2 } as a linear combination of the vectors in the set.

31. 9) Consider the following set of vectors in P 2 : a) These are 3 independent vectors. We can see this if we try to form a linear combination of them, and find that there is no way to cancel them out: Prepared by Vince Zaccone For Campus Learning Assistance Services at UCSB a)What is dimension of the span of this set? b)Does this set span P 2 ? c)Is this set a basis for P 2 ? d)Express each of the standard basis vectors {1, x, x 2 } as a linear combination of the vectors in the set.

32. 9) Consider the following set of vectors in P 2 : a) These are 3 independent vectors. We can see this if we try to form a linear combination of them, and find that there is no way to cancel them out: b) P 2 is a 3-dimensional space, and we have 3 independent vectors in P 2 – yes they span P 2. Prepared by Vince Zaccone For Campus Learning Assistance Services at UCSB a)What is dimension of the span of this set? b)Does this set span P 2 ? c)Is this set a basis for P 2 ? d)Express each of the standard basis vectors {1, x, x 2 } as a linear combination of the vectors in the set.

33. 9) Consider the following set of vectors in P 2 : a) These are 3 independent vectors. We can see this if we try to form a linear combination of them, and find that there is no way to cancel them out: b) P 2 is a 3-dimensional space, and we have 3 independent vectors in P 2 – yes they span P 2. c) P 2 is a 3-dimensional space, and we have 3 independent vectors in P 2 – yes they form a basis for P 2. Prepared by Vince Zaccone For Campus Learning Assistance Services at UCSB a)What is dimension of the span of this set? b)Does this set span P 2 ? c)Is this set a basis for P 2 ? d)Express each of the standard basis vectors {1, x, x 2 } as a linear combination of the vectors in the set.

34. 9) Consider the following set of vectors in P 2 : a) These are 3 independent vectors. We can see this if we try to form a linear combination of them, and find that there is no way to cancel them out: b) P 2 is a 3-dimensional space, and we have 3 independent vectors in P 2 – yes they span P 2. c) P 2 is a 3-dimensional space, and we have 3 independent vectors in P 2 – yes they form a basis for P 2. d) Form a linear combination of a 1, a 2 and a 3 : Prepared by Vince Zaccone For Campus Learning Assistance Services at UCSB a)What is dimension of the span of this set? b)Does this set span P 2 ? c)Is this set a basis for P 2 ? d)Express each of the standard basis vectors {1, x, x 2 } as a linear combination of the vectors in the set.

35. 9) Consider the following set of vectors in P 2 : a) These are 3 independent vectors. We can see this if we try to form a linear combination of them, and find that there is no way to cancel them out: b) P 2 is a 3-dimensional space, and we have 3 independent vectors in P 2 – yes they span P 2. c) P 2 is a 3-dimensional space, and we have 3 independent vectors in P 2 – yes they form a basis for P 2. d) Form a linear combination of a 1, a 2 and a 3 : Prepared by Vince Zaccone For Campus Learning Assistance Services at UCSB a)What is dimension of the span of this set? b)Does this set span P 2 ? c)Is this set a basis for P 2 ? d)Express each of the standard basis vectors {1, x, x 2 } as a linear combination of the vectors in the set.

36. 9) Consider the following set of vectors in P 2 : a) These are 3 independent vectors. We can see this if we try to form a linear combination of them, and find that there is no way to cancel them out: b) P 2 is a 3-dimensional space, and we have 3 independent vectors in P 2 – yes they span P 2. c) P 2 is a 3-dimensional space, and we have 3 independent vectors in P 2 – yes they form a basis for P 2. d) Form a linear combination of a 1, a 2 and a 3 : Similar analysis yields the other 2 combinations. Prepared by Vince Zaccone For Campus Learning Assistance Services at UCSB a)What is dimension of the span of this set? b)Does this set span P 2 ? c)Is this set a basis for P 2 ? d)Express each of the standard basis vectors {1, x, x 2 } as a linear combination of the vectors in the set.