MTH 102 University of Nevada Las Vegas Linear Algebra Questions

Chapter 3: ● 3.2 – p.193 # 2 – 10 even, 12, 13, 29-33 odd, 36, 40, 41, 65, 66

Chapter 4: ● 4.1 – p.258 # 2 – 16 even, 23, 25, 47, 50, 52

● 4.2 – p.270 # 2 – 8 even, 10, 12-20 even, 22, 27, 31

● 4.3 – p. 283 #5, 6, 7, 10, 22, 23, 24, 25, 33, 35

Skills Check 3.2
In Exercises 1-10, use factoring to solve the equations
1. x2 -3.2 – 10 = 0
2.×2 – 92 +18 = 0
3.×2 – 11x + 24 = 0
4. x2 + 3x – 10 = 0
5. 2×2 + 2x – 12 = 0
6. 282 +8-6=0
7.0=2+2 -11t+12
8. 622-132 +0
9. 6×2 + 10x = 4
10. 10×2 +11x = 6
In Exercises 11-16, find the x-intercepts algebraically.
=
=
=
4.22
11. f(x)
3:22 5.2 – 2
12. f(x)
5.22 70 + 2
13. f(x)
9
14. f (x) = 4x² + 20x + 25
15. f (x) = 3×2 + 4.0 – 4
16. f (x) = 9×2 – 1
In Exercises 29-34, use the square root method to solve the quadratic equations.
29. 4c? – 9=0
30. 22-20 = 0
31. 2? – 32 = 0
32.522 – 25 = 0
33. (x – 5)2 = 9
34. (2x + 1)2 = 20
In Exercises 35-38, complete the square to solve the quadratic equations.
35. x2 – 42 -9=0
36. x2 – 6x +1=0
37. x2 – 3x +2 = 0
38. 2×2 – 9x + 8 = 0
In Exercises 39–42, use the quadratic formula to solve the equations.
40. 3.22
39. x2 – 52 +2 = 0
6x – 12 = 0
41. 5x + 3×2 = 8
30x – 180 = 0
42. 3.42
Exercises 3.2
In Exercises 63–74, solve analytically and then check graphically.
63. Flight of a Ball If a ball is thrown upward at 96 feet per second from the top of a building that is 100 feet high, the height of the ball can be modeled by S(t) = 100 +961 – 16t2 feet,
where t is the number of seconds after the ball is thrown. How long after the ball is thrown is the height 228 feet?
64. Falling Object A tennis ball is thrown into a swimming pool from the top of a tall hotel. The height of the ball above the pool is modeled by D (t) =-16t2 – 4t + 200 feet, where t is the
time, in seconds, after the ball is thrown. How long after the ball is thrown is it 44 feet above the pool?
65. Break-Even The profit for an electronic reader is given by P(x) = -12×2 + 13202 – 21, 600, where x is the number of readers produced and sold. How many readers give break-even
(that is, give zero profit) for this product?
66. Break-Even The profit for Coffee Exchange coffee beans is given by P (2) = -15×2 + 180.0 – 405 thousand dollars, where x is the number of tons of coffee beans produced and sold.
How many tons give break-even (that is, give zero profit) for this product?
Skills Check 4.1
In Exercises 1-16, (a) sketch the graph of each pair of functions using a standard window, and (b) describe the transformations used to obtain the graph of the second function from the first
function
1. y=x”, y=x + 5
2. y=x, y=2? + 3
3.y= x,y= V2-4
4. y=x?, y=(x+2)
5.y=ya,y= 3x + 2-1
6. Y=r?, y=(x – 5)3 – 3
7. Y= x,y= 2 – 2+1
8. y = x, y = 12 +3 – 4
9. y= , y=-22 +5
10. y= vy=-x-2
11. y=y=1-3
12. y = 1, y = 3
13. f(x) = x2, g(I) = 322
14. f(x) = r°, 9(2) = 0.4×3
15. f(x) = x1, g(x) = 3x
16. f(x) = 12,9(x) = 4/5
23. Suppose the graph of f(x) = x is shifted up 3 units and to the left 6 units. What is the equation of the new graph? Verify the result graphically.
24. Suppose the graph of f(x) = x2 is shifted up 2 units and to the right 5 units. What is the equation of the new graph? Verify the result graphically.
25. Suppose the graph of f(x) = 1 is stretched vertically by a factor of 4 and then shifted down 3 units. What is the equation of the new graph? Verify the result graphically.
In Exercises 47-52, determine whether the function is even, odd, or neither.
47. f(x) = 3 – 5
48. f(x) = 12 – 21
49. g(x) = Vr+3
50. f(x) = {x8 – 1
51. g(x) = 3
52. g(x) = 4x + 2?
Skills Check 4.2
In Exercises 1-8, find the following: (a) (f +9)(), (b) (8 – 9)(Ⓡ), (c) (f – 9) (2) (a) (1) (2), and (c) the domain of
f.
1. f(x) = 3x – 5; 9(2) = 4 – 2
2. f(x) = 2x – 3; g(x) = 5 – 2
3. f(x) = x2 – 2x; g(x) =1+2
4. f(x) = 2×2 – 2; g(x) = 2x +1
5. f(x) = }; 9(x) = 21
6. f(x) = 772; g(x) = 1
7. f(x) = x; g(x)=1-22
8. f(x) = xº; g(x) = x+3
9. If f(x) = x2 – 5x and g(x) = 6 – 2°, evaluate
a. (f +9)(2)
b. (9- )(-1)
c. (f.g)(-2)
( ) (3)
d.
10. If f(x) = 4 – 22 and g(x) = 23 + 2, evaluate
a (f+9) (1)
b. (f-9)(-2)
c. (f.9)(-3)
( ) (2)
d.
In Exercises 11-20, find (a) (fog)(x) and (b) (gof)(x).
11. f(x) = 2x – 6; g(x) = 3x – 1
12. f(x) = 3x – 2; g(x) = 2x – 2
13. f(x) = x2; 9(2) =
14. f(0) = 2; g(x) =
15. f(x) = x – 1; g(x) = 2x – 7
16. f(x) = 3 – r; g(x) = x – 5
17. f(x) = |2 – 31; 9(2) = 4.0
18. f(x) = [4 – x\; g(x) = 2x +1
19. f(x) = 35+; g(x) = 2771
20. f(x) = 3x +1; g(x) = 23 +1
In Exercises 21 and 22, use f(x) and g(x) to evaluate each expression.
21. f(x) = 2×2; g(x) = 1
a (fog)(2)
b. (gof)(-2)
22. f() = (x – 1)?; g(x) = 3x – 1
a (fog)(2)
b. (gof)(-2)
27. Revenue and Cost The total revenue function for LED TVs is given by R= 1050x dollars, and the total cost function for the TVs is C = 10,000+ 30x + x² dollars, where x is the
number of TVs that are produced and sold.
a. Which function is quadratic, and which is linear?
b. Form the profit function for the TVs from these two functions.
c. Is the profit function a linear function, a quadratic function, or neither of these?
28. Revenue and Cost The total monthly revenue function for Easy-Ride golf carts is given by R=26, 600x dollars, and the total monthly cost function for the carts is
C = 200,000 + 4600x + 2×2 dollars, where x is the number of golf carts that are produced and sold.
a. Which function is quadratic, and which is linear?
b. Form the profit function for the golf carts from these two functions.
c. Is the profit function a linear function, a quadratic function, or neither of these?
29. Revenue and Cost The total weekly revenue function for a certain digital camera is given by R=550x dollars, and the total weekly cost function for the cameras is
C = 10,000+ 30x + x2 dollars, where x is the number of cameras that are produced and sold.
a. Find the profit function
b. Find the number of cameras that gives maximum profit.
c. Find the maximum possible profit.
30. Revenue and Cost The total monthly revenue function for camcorders is given by R 6600x dollars, and the total monthly cost function for the camcorders is
C = 2000 + 4800x + 2×2 dollars, where x is the number of camcorders that are produced and sold.
a. Find the profit function.
b. Find the number of camcorders that gives maximum profit.
c. Find the maximum possible profit.
31. Average Cost If the monthly total cost of producing 27-inch television sets is given by C(x) = 50,000+ 105x, where x is the number of sets produced per month, then the average cost
per set is given by
50,000 + 1053
T(2) =
2
a. Explain how C(x) and another function can be combined to obtain the average cost function.
b. What is the average cost per set if 3000 sets are produced?
Skills Check 4.3
In Exercises 1 and 2, determine if the function f defined by the arrow diagram has an inverse. If it does, create an arrow diagram that defines the inverse. If it does not explain why not.
f
Domain
Range
2
3
1
3
7
8
Domain
Range
2.
4
8
13
18
6
L8
In Exercises 3 and 4, determine whether the function f defined by the set of ordered pairs has an inverse. If it does, find the inverse.
3. {(5, 2), (4, 1), (3, 7), (6, 2)}
4. {(2, 8), (3,9), (4, 10), (5, 11)}
5. If f (x) = 3x and g(x) = ý,
a. What are f (g(x)) and g (f (2)) ?
b. Are f(x) and g(x) inverse functions?
6. If f (x) = 4.1 – 1. and g(x) = ??,
a. What are f(g(x)) and g(f(x))
b. Are f(x) and g(x) inverse functions?
7. If f(x) = x3 +1 and g(1) = 5x – 1 are f(x) and g(x) inverse functions?
8. If f(x) = (x – 2) and g(x) = 3x + 2 are f(x) and g(x) inverse functions?
9. For the function f defined by f(x) = 3x – 4 complete the tables below for fandf
10. For the function g defined by g(x) = 2×3 – 1, complete the tables below for g and g-1
X
g(x)
-2
-17
-1
0
1
2
X
g 1(0)
-17
-2
-3
-1
1
15
21. If function h has an inverse and h-(-2) = 3, find h(3).
22. Find the inverse of f (x) =
23. Find the inverse of g(x) = 4x +1.
24. Find the inverse of f (x) = 4×2 for x > 0.
25. Find the inverse of g (2) = r2 – 3 for 2 > 0.
26. Graph g(x) = va and its inverse g-|(2) for x > 0 on the same axes.
27. Graph g(x) = ya and its inverse g-() on the same axes.
28. f (x) = (x – 2)2 and g(x) = ve +2 are inverse functions for what values of X?
29. Is the function f (x) = 2×3 +1 a one-to-one function? Does it have an inverse?
30. Sketch the graph of y=f(x) on the axes with the graph of y = f(2) shown below.
2
0
2
6
33. Shoe Sizes If x is the size of a man’s shoe in the United States, then t(x) = x + 34.5 is its Continental size.
a. Find a function that will convert Continental shoe size to U.S. shoe size.
b. Use the inverse function to find the U.S. size if the Continental size of a shoe is 43.
(Source: Kuru International Exchange Association)
34. Investments if x dollars are invested at 10% for 6 years, the future value of the investment is given by S(x) = x +0.62.
a. Find the inverse of this function.
b. What do the outputs of the inverse function represent?
c. Use this function to find the amount of money that must be invested for 6 years at 10% to have a future value of $24,000.
35. Currency Conversion Suppose the function that converts from Canadian dollars to U.S. dollars is f (x) = 1.0136x, where x is the number of Canadian dollars and f(x) is the number of
U.S. dollars.
a. Find the inverse function for f and interpret its meaning
b. Use fand f1 to determine the money you will have if you take 500 U.S. dollars to Canada, convert them to Canadian dollars, don’t spend any, and then convert them back to U.S.
dollars. (Assume that there is no fee for conversion and the conversion rate remains the same.)
(Source: Expedia.com)