HKUST

MATH1003 Calculus and Linear Algebra

Make-up Exam (Version B) Name:

12nd December 2013 Student ID:

08:30-11:30 Lecture Section:

Directions:

• Do NOT open the exam until instructed to do so.

• Please turn off all phones and pagers, and remove headphones.

• Please write your name, ID number, and Tutorial Section in the space provided above.

• When instructed to open the exam, please check that you have ?? pages in addition to thecover page.

• Answer all questions. Show an appropriate amount of work for each problem. If you do notshow enough work, you will get only partial credit.

• You may use an ordinary scientific calculator, but calculators with graphical, or symboliccalculation functions are NOT allowed.

• This is a closed book examination.

• Cheating is a serious offense. Students caught cheating will receive a zero scorefor the midterm exam, and will also be subjected to further penalties imposedby the University.

Question No. Points Out of

Q. 1-7 35

Q. 8 15

Q. 9 15

Q. 10 15

Total Points 80

1

Part I: Multiple Choice Questions.

Each of the following MC questions is worth 2 points. No partial credit. Put your MCquestion answers in the MC answer sheet provided.

1. (2015-midterm-1, not used) A couple has $240,000 invested in a saving account with annual nominalrate 8% compounded monthly. If the couple will withdraw monthly $2,000 from the account, startingfrom the next month, how much will remain in the account immediately after they have made the120th withdraw?

(a) $0 (b) $240231 (c) $350761 (d) $6817 (e) * $166822

2. (2015-midterm-1) John has started a saving plan in which a fixed amount of money will be depositedinto an account at the end of every month in the coming 36 months. The annual interest rate onthe account is 6%, compounded monthly. If John wants to have a total sum of $36,000 at the endof the 36 month period, what is the amount of the monthly deposit John would need to make,rounded to the nearest dollar?

(a) *$915 (b) $241 (c) $765 (d) $890 (e) $2341

3. (2015-midterm-1) You can afford monthly deposits of $200 into an account that pays 5.7% com-pounded monthly. How long will it be until you have $7,000? (Round to the next-higher month ifnot exact.)

(a) 30 (b) * 33 (c) 41 (d) 52 (e) 20

4. (2015-midterm-1) Given the augmented matrix A =

1 0 3 50 1 2 70 0 m n

, for what values of m and n,

the corresponding system of linear equations is consistent and has infinitely many solutions?

(a) * m = n = 0 (b) m 6= 0, n 6= 0 (c) m = 0, n 6= 0

(d) m 6= 0, n 6= 0 (e) None of the previous

2

5. (2015-midterm-1) Among the four matrices below: 1 2 −3 10 0 1 40 0 0 0

,

1 0 −1 30 2 1 10 0 0 0

,

1 1 0 10 0 1 10 0 0 0

,

1 0 −2 30 0 0 00 1 −1 4

,

how many are in reduced form?

(a) 0 (b) 1 (c) * 2 (d) 3 (e) 4

6. (2015-midterm-1) Let M1 =

[1 −2−2 3

], M2 =

[−1 1

]. Then compute 3M2M

−11

(a) [−3,−3] (b) [3,−3] (c) [2, 3] (d) [1,−1] (e) * None of the previous

7. (2015-midterm-1) For the matrix equation XA = B where

A =

1 −1 10 2 −12 3 0

, B = [1, 1, 0] , X = [x1, x2, x3] .

If it is known that A−1 =

3 3 −1−2 −2 1−4 −5 2

, what is the value of x2?

(a) 0 (b) * 1 (c) 2 (d) 3 (e) 4

8. (2015-midterm-1) Given the matrix A =

1 −1 21 1 11 0 1

, what is second row of A−1?

(a) * [0, 1,−1] (b) [0, 1, 1] (c) [1, 1, 1] (d) [−1, 1,−1] (e) [0, 1, 0]

3

9. (2014-midterm-1) Solve the system of linear equationsx− 2y + 2z = 33x− 7y + 5z = 42x− 4y + 5z = 9

We find z =?

(a) z = 1

(b) z = 2

(c) ** z = 3

(d) z = 0

(e) None of the above

10. (2014-midterm-1) A note will pay $15,000 at maturity 5 years from now. How much should you bewilling to pay for the note now if money is worth 5.2% compounded quarterly?

(a) $13,640.6 (b) $13,257.8 (c) $12,305.7 (d) ** $11,585.2 (e) $10,650.2

11. (2014-midterm-1) A =

2 −5 0−1 3 −40 1 −2−3 0 9

, B =

4 −67 13 2

. Find the entries in the third row of

AB.

(a) [5, 1].

(b) [7, 2]

(c) [15, 36]

(d) ** [1,−3]

(e) [2, 3]

12. (2014-midterm-1) How many days will it take money to double if it is invested at 7.5% compoundeddaily? (Use a 365-day year.)

(a) 9.24 (b) 3200 (c) ** 3374 (d) 10.53 (e) 3650

4

13. (2014-midterm-1) What is the equation of the tangent line of the curve

y =2√x− 3x

x2

at the point x = 1?

(a) x + 2y + 1 = 0

(b) ** y + 1 = 0

(c) x− 2y − 3 = 0

(d) 3x− 2y − 5 = 0

(e) None of the above

14. (2014-midterm-2) Find the equation of the tangent line to the curve x3 + y3 = 2xy at the point(1, 1).

(a) x + y = 2 **

(b) x + y = 1

(c) y = x

(d) y = −x(e) y = 0

15. (2014-midterm-2) A point is moving on the graph y2−4x2 = 12 so that its x coordinate is decreasingby 2 units per second when (x, y) = (1, 4). Find the rate of change of the y coordinate at thatmoment.

(a) 1 unit per second

(b) -1 unit per second

(c) 2 units per second

(d) -2 units per second **

(e) -1/2 unit per second

5

16. (2014-midterm-2) Suppose f is a differentiable function with derivative f ′(x) = x(x − 2)(x + 1)2.Determine the number of local extrema of f(x).

(a) 0 (b) 1 (c) ** 2 (d) 3 (e) 4

17. (2014-midterm-2) Evaluate

∫ 2

0

x

(x2 + 1)2dx (hint: substitution u = x2 + 1)

(a) ln 5

(b) −2

5

(c) **2

5

(d) −4

5

(e)4

5

18. (2014-midterm-2) Evaluate

∫ 1

0xe2xdx.

(a)e2 − 2

4

(b)e2 + 1

2

(c)e2 − 1

2

(d) **e2 + 1

4

(e)e2 + 2

4

19. (2013-final) If you invest $5,650 in an account paying 8.65% compounded continuously, how muchmoney will be in the account at the end of 10 years?

(a) $10,245.5 (b) $23,046.7 (c) $8,621.6 (d) ** $13,418.8 (e) $12,673.5

6

20. (2013-final) If you borrow $4,000 from an online lending firm for the purchase of a computer andagree to repay it in 48 equal installments at 0.9% interest per month on the unpaid balance, howmuch total interest will be paid?

(a) $102.99 (b) **$943.71 (c) $925.60 (d) $763.24 (e) $642.56

21. (2013-final) Find the annual percentage yield on a CD earning 6.25% if interest is compoundedmonthly.

(a) **6.43% (b) 6.30% (c) 6.21% (d) 7.16% (e) 6.26%

22. (2013-final) Find the entries of the first row of the inverse of A =

[3 28 5

].

(a) [3,−8].

(b) [−3, 8]

(c) **[−5, 2]

(d) [5,−2]

(e) [−5,−2]

23. (2013-final) A particle located at the origin when t = 0 moves along the x-axis with velocityv(t) = 1

2 t2− t feet/second. Let s(t) be its position at time t. Which one of the following represents

s(t)?

(a) s(t) = t− 1

(b) **s(t) = 16 t

3 − 12 t

2

(c) s(t) = 32 t

3 − 2t2

(d) s(t) = 12 t

3 − t2

(e) None of the above

7

24. (2013-final) If A =

[1 −2−2 5

], AB =

[−1 2 −16 −9 3

], then what is the first column of B?

(a)

[11

]

(b)

[−8−5

]

(c) **

[74

]

(d)

[53

](e) None of the above

25. (2013-final) Suppose f ′′(x) = (2− x)(x− 4)2. On what interval(s) is the graph of y = f(x) concavedownward?

(a) (2, 4) and (4,∞)

(b) (−∞, 2)

(c) (2, 4)

(d) (−∞, 2) and (4,∞)

(e) **None of the above

26. (2013-final) Find all critical numbers of h(x) = x2

x−3 .

(a) 0, 2

(b) **0, 6

(c) 0, 2, 3

(d) 0, 3, 6

(e) None of the above

8

27. (2013-final) A farmer wants to enclose a rectangular area with 400 feet of fence. In order to maximizethe area within the fence what function do we wish to maximize?

(a) A(x) = 400x− x2

(b) **A(x) = 200x− x2

(c) A(x) = 200− 2x

(d) A(x) = 400x2

(e) None of the above

28. (2013-final) What is f ′(x), if f(x) =∫ x3

0

√tdt?

(a) 2x√x

3

(b) **3x3√x

(c)√x

(d) x√x

(e) x4√x

29. (2013-final) Suppose f is a differentiable function with derivative f ′(x) = x(x− 1)2(x− 3). Wheredoes f(x) have local extrema?

(a) ** Local maximum at x = 0; local minimum at x = 3

(b) Local maximum at x = 0 and x = 3; local minimum at x = 1

(c) Local maximum at x = 3; local minimum at x = 0

(d) Local maximum at x = 1; local minimum at x = 0 and x = 3

(e) None of the above

9

30. (2013-final) Evaluate∫ 10 x√x2 + 1 dx (hint: substition u = x2 + 1)

(a) 2√2

3

(b) **2√2−12

(c) 2√2−23

(d) 13

(e) 2√2−13

31. (2013-final) If the sides of a square are growing at a constant rate of 4 cm/min, how fast is the areaincreasing when the sides are 25 cm long?

(a) 100 cm2/min

(b) **200 cm2/min

(c) 300 cm2/min

(d) 400 cm2/min

(e) 500 cm2/min

32. (2013-final) Evaluate∫

x2√1−2x3

dx (hint: substitution u = 1− 2x3)

(a) **−√1−2x3

3 + C

(b) −√x3 + C

(c) 2x3 − 1 + C

(d) −√1−2x3

2 + C

(e) − 6√1−2x3

+ C

10

33. (2013-final) Evaluate∫ 10 x2(3 + x) dx:

(a) −15

(b) 1

(c) 15

(d) 0

(e) **54

34. (2013-final) Compute∫ 41

1√x(1+

√x)2

dx (hint: substition u = 1 +√x)

(a) 6

(b) −16

(c) 34

(d) −13

(e) **13

35. (2013-final) If f(x) =(x− 1

x

)2, find f ′(2).

(a) 3

(b) 94

(c) 158

(d) 52

(e) **154

11

36. (2013-final) The height of a triangle is increasing at a rate of 4 cm/min. The base is decreasing ata rate of 5 cm/min. At what rate is the area changing when the height measures 10 cm, and thebase measures 15 cm?

(a) 1500 cm2/min

(b) **5 cm2/min

(c) -1 cm2/min

(d) -10 cm2/min

(e) -20 cm2/min

37. (2013-final) Find the equation of the tangent line to the curve y2 +4xy+x2 = 13 at the point (2, 1).

(a) y = x− 11

(b) y = x + 13

(c) y = −x + 15

(d) y = −54x + 7

2

(e) **y = −45x + 13

5

38. (2013-final) A box with a square base and no top is to have a volume of 32 cm3. Find the leastamount of material (surface area) needed to construct such a box.

(a) 64 cm2/min

(b) **48 cm2/min

(c) 32 cm2/min

(d) 16 cm2/min

(e) 128 cm2/min

39. (2013-midterm) A company estimates that it will need $100,000 in 8 years to replace a computer.If it establishes a sinking fund by making fixed monthly payments into an account paying 7.5%compounded monthly, how much should each payment be?

(a) $1,388.39 (b) $720.26 (c) $755.87 (d) ** $763.39 (e) $1,203.56

12

40. (2013-midterm) American Express’s online banking division offered a money market account withan APY of 4.86%. If interest is compounded monthly, what is the equivalent annual nominal rate?

(a) ** 4.755% (b) 4.932% (c) 4.723% (d) 4.631% (e) 4.86%

41. (2013-midterm) A newborn child receives a $20,000 gift toward college from her grandparents. Howmuch will the $20,000 be worth in 17 years if it is invested at 7% compounded quarterly?

(a) $63,176.3 (b) ** $65,068.4 (c) $43,800.0 (d) $57,692.5 (e) $67829.4

42. (2013-midterm) A note will pay $20,000 at maturity 10 years from now. How much should you bewilling to pay for the note now if money is worth 5.2% compounded continuously?

(a) $33,640.6 (b) $13,257.8 (c) $12,305.7 (d) **$11,890.4 (e) $32,650.2

43. (2013-midterm) Which of the following matrices is in reduced form?

A =

1 2 0 0 10 1 0 0 10 0 0 1 1

, B =

1 2 0 0 00 0 1 0 00 0 0 1 00 0 0 0 0

, C =

1 0 0 00 1 0 00 0 2 0

.

(a) A,B and C are all in reduced form.

(b) Only A and B are in reduced form.

(c) **Only B is in reduced form.

(d) Both B and C are in reduced form.

(e) None of them is in reduced form.

44. (2013-midterm) Suzanne imports tea and oranges from China. The total weight is 100 crates. Shepaid import tax of $5 for each crate of tea, and $3 for each crate of oranges. If she paid $360 intoal, how many crates of tea did she import?

(a) 70 (b) 48 (c) 60 (d) 25 (e) **30

13

45. (2013-midterm) For what values of h is the matrix A NOT invertible (that is, A−1 does not exist)?Here A is given as

A =

1 0 −2−3 1 42 −3 h

.

(a) h = 1

(b) **h = 2

(c) h = 3

(d) h = −2

(e) A is invertible for any h

46. (2013-midterm) Find the entries in the second row of AB where A =

2 −5 0−1 3 −46 −8 −7−3 0 9

, B =

4 −67 13 2

.

(a) **[5, 1].

(b) [7, 2]

(c) [15, 36]

(d) [−27,−17]

(e) [2, 3]

47. (2013-midterm) Find the equation of the line tangent to the graph of f(x) = ex − e(1x + 2 lnx

)at

x = 1.

(a) y = 3(x− 1)

(b) **y = 0

(c) y + 2e = x− 1

(d) y = e(x− 1)

(e) y + 2 = 0

14

48. (2013-midterm) Find f ′(1) for the given function

f(x) =

√x− 3x

x2.

(a) −12

(b) 0

(c) 12

(d) **32

(e) None of the above

49. (2011-midterm) Janelle makes monthly deposits of 120 dollars into an account that pays 5.3%interest rate compounded monthly. How much will she have in the account immediately after the25th deposit, rounded to the nearest dollar?

(a) ** $3,165 (b) $2,834 (c) $6,975 (d) $3,349 (e) None of the previous

50. (2011-midterm) Allan borrows 2,250 dollars from his uncle. Two years later, he borrows another1,230 dollars. If his uncle charges him 7.5% interest rate compounded annually, how much doesAllan owe 6 years after the first loan, rounded to the nearest dollar?

(a) $4,862 (b) $4,647 (c) $5,370 (d) **$5,115 (e) None of the previous

51. (2011-midterm) John has started a saving plan in which a fixed amount of money will be depositedinto an account at the end of every month in the coming 36 months. The annual interest rate onthe account is 6%, compounded monthly. If John wants to have a total sum of $ 36,000 at theend of the 36 month period, what is the amount of the monthly deposit John would need to make,rounded to the nearest dollar?

(a) $1,030 (b) $1,051 (c) **$915 (d) $951 (e) $917

15

52. (2011-midterm) Let A =

(3 −1−2 −1

). Then A−1 =

(a)

(1 −1−2 3

)(b)

(−3 −2−1 1

)(c)

(−0.2 0.20.4 0.6

)

(d) **

(0.2 −0.2−0.4 −0.6

)(e) None of the previous

53. (2011-midterm) Find limx→3+

f(x) from its graph shown below.

(a) **7 (b) 6 (c) 4 (d) 0 (e) does not exist

54. (2011-midterm) Find the limit limx→ 1

2

−4x2 + 2x

4x2 − 8x + 3.

(a) **12 (b) 0 (c) −1

2 (d) Does not exist (e) None of the previous

16

55. (2011-midterm) Let f(x) =x2 − 2x− 8

(x + 2)2(x− 3)(x− 4). Which of the following is true about the graph

of y = f(x)?

(a) x = 3 is the only vertical asymptote of the graph of y = f(x).

(b) **y = 0 is a horizontal asymptote of the graph of y = f(x).

(c) x = −2 is not a vertical asymptote of the graph of y = f(x).

(d) The graph of y = f(x) has three vertical asymptotes.

(e) y = 3 is a horizontal asymtptote of the graph of y = f(x).

56. (2011-midterm) Given the following augmented matrix

1 3 0 −1 20 0 1 0 50 0 0 1 20 0 0 0 0

.

Which of the following statements about the above augmented matrix is true?

(a) The augmented matrix is in reduced form.

(b) The corresponding system of linear equations has unique solution.

(c) **The corresponding system of linear equations is consistent.

(d) The corresponding system of linear equations has no solution.

(e) The corresponding system of linear equations has 3 free variables.

57. (2011-midterm) Solve the inequalityx− 3

(2x− 5)2(x− 4)≤ 0.

(a) x ≤ 3 or x ≥ 4

(b) 3 ≤ x < 4

(c) x < 2.5 or 2.5 < x ≤ 3 or x ≥ 4

(d) x < 2.5 or 3 ≤ x ≤ 4

(e) **3 ≤ x ≤ 4

17

58. (2011-midterm) Let f(x) =√x2 + 1. Which of the following statements is true?

(a) f ′(1) = limh→0

√2 + h−

√2

h

(b) f ′(1) = limh→0

√h2 + 2−

√2

h

(c) f ′(1) = limh→1

√h2 + 2−

√2

h

(d) f ′(1) = limh→1

√(1 + h)2 + 1−

√2

h

(e) **f ′(1) = limh→0

√h2 + 2h + 2−

√2

h

59. If the sides of a square are growing at a constant rate of 5 cm/min, how fast is the area increasingwhen the area is 400 cm2? (Answer is b)

(a) 100 cm2/min (b) 200 cm2/min (c) 300 cm2/min

(d) 400 cm2/min (e) 500 cm2/min

60. A point is moving on the graph of 4x2 + 9y2ey = 36. When the point is at (3, 0), its y coordinate isdecreasing by 2 units per second. How fast is its x coordinate changing at that moment? (Answeris c)

(a) -1 unit per second (b) 3 unit per second (c) 0 unit per second

(d) -2 units per second (e) None of the above

61. Find limx→3+x|3−x|x−3 (Answer is b)

(a) -3 (b) 3 (c) 0

(d) 1 (e) The limit does not exist

18

62. Given the function

f(x) =

{2 if x is an integer1 if x is not an integer

Find limx→2 f(x). (Answer is a)

(a) 1 (b) 2 (c) 3

(d) 0 (e) The limit does not exist

63. Find the derivative of f(x) at x = 1 where

f(x) =2x5 − 4x3 + 2x

x3.

(Answer is a)

(a) 0 (b) 1/2 (c) 1/4

(d) 3 (e) None of the above

64. Find d yd t at t = 1 where y = (1 + et) ln t. (Answer is a)

(a) 2e + 1 (b) e + 1 (c) 2 + e2

(d) 0 (e) None of the above

65. If y = ln(tet), find d yd t at t = 1. (Answer is b)

(a) 2e + 1 (b) 2 (c) 1 + e

(d) e−1 (e) None of the above

66. If f(x) = 3√x + 1− ex/3, find f ′′(0) ((Answer is e))

(a) 1/3 (b) -2/9 (c) -1/9

(d) 0 (e) -1/3

19

67. What is the slope of the tangent line to the curve y = ex2+1 at the point (x, y) = (0, e)? ((Answer

is d))

(a) 3e (b) 2e (c) e

(d) 0 (e) −e

20

Part II: Answer each of the following 4 long questions. Unless otherwise specified, numer-ical answers should be either exact or correct to 2 decimal places. Write all steps.

68. [25 pts] (2014-midterm-1) A person purchased a $250,000 home 10 years ago by paying 20% downand signing a 30-year mortgage at 12% compounded monthly.

(a) Find the monthly payment for the mortgage.

(b) Find the unpaid balance of the mortgage now (or how much does he still owe to the bank).

(c) How much interest has he paid to the bank so far?

21

(d) Now the owner got a deal from a bank: he paid down 20% of the unpaid balance, and signeda new 20-year mortgage at 8% compounded monthly. What is the monthly payment for thenew mortgage?

(e) With refinancing, what is the total interest the person will pay during the next 20-year period?

22

69. [25 pts] (2014-midterm-1) An economy is based on two sectors, energy (E) and water (W). Toproduce one dollar’s worth of E requires 0.6 dollar’s worth of E and 0.1 dollar’s worth of W, andto produce one dollar’s worth of W requires 0.2 dollar’s worth of E and 0.7 dollar’s worth of W.

(a) Find the technology matrix M for the economy.

(b) Find the total output for each sector that is needed to satisfy a final demand of $40 billion forenergy and $30 billion for water.

(c) Find the final demand for each sector if the total output of energy is $60 billion and the totaloutput of water is $70 billion.

23

70. [25 pts] (2014-midterm-2) A box with an open top is to be constructed from a square piece ofcardboard, 30 cm on each side, by cutting out a square of length x cm from each of the four cornersand bending up the sides.

(a) (5 pts) Find the volume of the box V (x) in terms of x.

(b) (5 pts) State the possible range of values of x.

(c) (15 pts) Find the dimensions of the box (the length of each side and height) to maximize thevolume. (Use either the 1st or the 2nd derivative test.)

24

71. [25 pts] (2014-midterm-2) Consider the graph of y = f(x) = x3 − x2 − 2x.

(a) (10 pts) Find the x-intercepts of the graph y = f(x) (that is the points of the graph on thex-axis).

(b) (15 pts) Find the area of the finite region bounded by the graph y = f(x) and the x-axis.

25

72. [15 pts] (2013-final) Find the area between the curves y = x and y = x2 for 0 ≤ x ≤ 2.

26

73. [15 pts] (2013-final) A 200-room hotel in Reno is filled to capacity every night at a rate of $40 perroom. For each $1 increase in the nightly rate, 4 fewer rooms are rented. If each rented room costs$8 a day to service, how much should the management charge per room in order to maximize grossprofit? What is the maximum gross profit? Use either the 1st or the 2nd derivative test to checkthat this is indeed the absolute maximum.

27

74. [15 pts] (2013-final) A corporation has a taxable income of $7,650,000. At this income level, thefederal income tax rate is 50%, the state tax rate is 20%, and the local tax rate is 10%. The wholetax liability is lower than 50% + 20% + 10% = 80%, however, because it is customary to deducttaxes paid to one agency before computing taxes for the other agencies. Assume that the federaltaxes are based on the income that remains after the state and local taxes are deducted (say forexample, if it is known in advance that the state tax is $500,000 and the local tax is $500,000, thenthe federal income tax should be 0.50 × (7, 650, 000 − 500, 000 − 500, 000) = 3, 325, 000), and thatstate and local taxes are computed in a similar manner.

(1). Let x, y and z (dollars) be the federal, state and local taxes paid by the company respectively.What are the equations that you could establish from the above assumption? (Hint: threelinear equations.)

(2). Solve the above equations for x, y and z. Provide all details. You may write on the back.

(3). What is the tax liability of the corporation (the total taxes paid as a percentage of taxableincome $7,650,000)?

28

75. [15 pts] (2013-final) Find

(1).∫x ln 2x dx.

(2).∫xe1+x2

dx.

(3).∫

(lnx)2 dx.

29

76. [10 pts] (2013-midterm)

(1). If f(x) = x lnx, find f ′(x).

(2). If f(x) = 1−ex1+ex , find f ′(x).

(3). If f(x) = x lnxex , find f ′(x).

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77. [10 pts] (2013-midterm) An economy is based on two sectors, energy (E) and water (W). To produceone dollar’s worth of E requires 0.4 dollar’s worth of E and 0.2 dollar’s worth of W, and to produceone dollar’s worth of W requires 0.1 dollar’s worth of E and 0.3 dollar’s worth of W.

(a) Find the technology matrix M for the economy.

(b) Find the total output for each sector that is needed to satisfy a final demand of $40 billion forenergy and $30 billion for water.

(c) Suppose the final demand for energy and water in (a) is increased by 12% and 25% respectively.Find the percentage increase in the corresponding total outputs for each sector.

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78. [10 pts] (2013-midterm) A person purchased a $250,000 home 20 years ago by paying 20% downand signing a 30-year mortgage at 12% compounded monthly.

(a) Find the monthly payment for the mortgage.

(b) Find the unpaid balance of the mortgage now.

(c) Interest rates have dropped and the owner wants to refinance the unpaid balance by signinga new 10-year mortgage at 8% compounded monthly. What is the monthly payment for thenew mortgage?

(d) With refinancing, what is the total interest the person will pay during the 30-year period?

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79. [10 pts] (2013-midterm) For the system of linear equationsx1 + x2 + 4x3 + x4 = 2x1 + x3 = 12x1 + x3 + 3x4 = 52x1 + 5x2 + 11x3 + 3x4 = 5

(a) Write down the augmented matrix

(b) Use Gauss-Jordan elimination to get a reduced form for the augmented matrix (you may writeon the back if there is not enough space)

(c) Find the solution set of the above system of linear equations.

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80. [8 pts] (2011-midterm) Let f(x) = 2− 1√x

.

(a) Find the range of values of x such that f(x) is defined.

(b) State the definition of f ′(x) and use it to find f ′(x).

(c) Let g(x) = (f(x))2. Find g′(x) by the power rule.

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81. [10 pts] (2011-midterm) Mr. Smith purchased a house 5 years ago for $2, 500, 000. The house wasfinanced by paying 20% down and signing a 25-year mortgage at 3.75% compounded monthly. Thecurrent market value of the house is $3, 000, 000.

(a) Find the monthly payment for the mortgage.

(b) Find the unpaid loan balance after making 60 monthly payments.

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(c) After making 60 payments, Mr. Smith wants to sell the house at its current market valueand buy a bigger house for $4, 500, 000. After selling the old house and repaying the unpaidloan balance, he puts all the remaining money as down payment for buying the new house.Then Mr. Smith takes out a new 20-year mortgage for the loan at 3.5% compounded monthly.Comparing to the monthly payment in the previous mortgage, how much more does he needto pay every month now?

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82. [12 pts] (2011-midterm) A proposed network of irrigation canals is described in the figure below,where the flows at interchanges A, B, C and D are shown (in cubic meter per minute).

(a) Using the fact that the amount of flow into an interchange is equal to the amount of flow outfrom it per minute (e.g. At interchange A, 55 = f1 + f4), set up a system of linear equationsfor f1, f2, f3 and f4.

(b) Solve the system in (a) and find the maximum possible amount of flow per minute in canalBC.

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(c) If canal BC is closed, what range of flow on AD must be maintained so that no canal carriesa flow of more than 30 cubic meter per minute?

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83. [10 pts] (2011-midterm) The technology matrix for an economy based on energy (E) and water (W)is

E W

M =EW

(0.4 0.10.3 0.2

)(a) Find the total output for each sector that is needed to satisfy a final demand of $45 billion for

energy and $36 billion for water.

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(b) Suppose the total outputs for energy and water in (a) are increased by 10% and 20% respec-tively. Find the percentage increase in the corresponding final demand for each sector.

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84. [10 pts] (2012-final) An economy is based on two sectors, coal (C) and steel (S). To produce onedollar’s worth of C requires 0.4 dollar’s worth of C and 0.5 dollar’s worth of S, and to produce onedollar’s worth of S requires 0.3 dollar’s worth of C and 0.6 dollar’s worth of S.

(a) Find the technology matrix M for the economy.

(b) Find the total output for each sector that is needed to satisfy a final demand of $60 billion forcoal and $40 billion for steel.

(c) Suppose the final demand for coal and steel in (b) is increased by 15% and 30% respectively.Find the percentage increase in the corresponding total outputs for each sector.

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85. [10 pts] (2012-mid) Bob and Mary Rodgers want to purchase a new house and feel that they canafford a mortgage payment of $4,800 a month. They are able to obtain a 30-year 7.6% mortgage(compounded monthly), but must put down 25% of the cost of the house.

(a) Assuming that they have enough savings for the down payment, how expensive a house canthey afford?

(b) Suppose Bob and Mary eventually purchase a $750,000 house. How much do they need to payevery month for the mortgage after paying 25% down?

(c) How much interest will they pay in the first 15 years?

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86. [10 pts] (2012-mid) A system of linear equations is given as follows:x − y − 2z = 0x − 3y − 5z = 0

y + 2z = 3(1)

(a) Use the augmented matrix method and the Gauss-Jordan elimination to solve the system forx, y, z.

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(b) We can write the system (1) of linear equations above as a matrix equation

MX = b

where X =

xyz

, b =

003

. What is the matrix M? Write down M explicitly. Then find M−1

and compute M−1b.

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87. [10 pts] (2012-mid) A person is ordered by a doctor to take 10 units of vitamin A, 9 units of vitaminD, and 19 units of vitamin E each day. The person can choose from three brands of vitamin pills.Brand X contains 2 units of vitamin A, 3 units of vitamin D, and 5 units of vitamin E; brand Yhas 1, 3 and 4 units, respectively; and brand Z has 1 unit of vitamin A, none of vitamin D, and1 of vitamin E. (a) Find all possible combinations of pills that will provide exactly the required

amounts of vitamins.

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(b) If brand X costs 1 cent a pill, brand Y 6 cents, and brand Z 3 cents, are there any combinationsin part (a) costing exactly 15 cents a day?

(c) What is the least expensive combination in part (a)? the most expensive?

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