Several Calculus Questions

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I have several calculus questions that I need help with. You MUST show your work and therefore must have a thorough understanding of calculus concepts and theories. You must NOT simply get answers from other online sites such as Coursehero etc. All work will be checked for accuracy so do not just copy answers you find online.

ANSWERS MUST BE TYPED!

Pre-Calculus Quarter 4 Exam Name: _________________________ Score: ______ / ______ 1. Find the indicated sum. Show your work. 2. Locate the foci of the ellipse. Show your work. π‘₯2 𝑦2 + =1 36 11 1 Pre-Calculus Quarter 4 Exam 3. Solve the system by the substitution method. Show your work. 2y - x = 5 x2 + y2 - 25 = 0 4. Graph the function. Then use your graph to find the indicated limit. You do not have to provide the graph f(x) = 5x - 3, 5. f(x) Use Gaussian elimination to find the complete solution to the system of equations, or state that none exists. Show your work. 4x - y + 3z = 12 x + 4y + 6z = -32 5x + 3y + 9z = 20 2 Pre-Calculus Quarter 4 Exam 6. Solve the system of equations using matrices. Use Gaussian elimination with backsubstitution. x+y+z = x - y + 3z = 4x + y + z = -5 -1 -2 7. A woman works out by running and swimming. When she runs, she burns 7 calories per minute. When she swims, she burns 8 calories per minute. She wants to burn at least 336 calories in her workout. Write an inequality that describes the situation. Let x represent the number of minutes running and y the number of minutes swimming. Because x and y must be positive, limit the boarders to quadrant I only. Short Answer Questions: Type your answer below each question. Show your work. 8. A statement Sn about the positive integers is given. Write statements S1, S2, and S3, and show that each of these statements is true. Show your work. Sn: 12 + 42 + 72 + . . . + (3n - 2)2 = 3 𝑛(6𝑛2 βˆ’3π‘›βˆ’1) 2 Pre-Calculus Quarter 4 Exam 9. A statement Sn about the positive integers is given. Write statements Sk and Sk+1, simplifying Sk+1 completely. Show your work. Sn: 1 βˆ™ 2 + 2 βˆ™ 3 + 3 βˆ™ 4 + . . . + n(n + 1) = [n(n + 1)(n + 2)]/3 10. Joely's Tea Shop, a store that specializes in tea blends, has available 45 pounds of A grade tea and 70 pounds of B grade tea. These will be blended into 1 pound packages as follows: A breakfast blend that contains one third of a pound of A grade tea and two thirds of a pound of B grade tea and an afternoon tea that contains one half pound of A grade tea and one half pound of B grade tea. If Joely makes a profit of $1.50 on each pound of the breakfast blend and $2.00 profit on each pound of the afternoon blend, how many pounds of each blend should she make to maximize profits? What is the maximum profit? 11 Your computer supply store sells two types of laser printers. The first type, A, has a cost of $86 and you make a $45 profit on each one. The second type, B, has a cost of $130 and you make a $35 profit on each one. You expect to sell at least 100 laser printers this month and you need to make at least $3850 profit on them. How many of what type of printer should you order if you want to minimize your cost? 4 Pre-Calculus Quarter 4 Exam 12 A statement Sn about the positive integers is given. Write statements S1, S2, and S3, and show that each of these statements is true. Show your work. Sn: 2 + 5 + 8 + . . . + ( 3n - 1) = n(1 + 3n)/2 13 Use mathematical induction to prove that the statement is true for every positive integer n. Show your work. 2 is a factor of n2 - n + 2 14 A statement Sn about the positive integers is given. Write statements S1, S2, and S3, and show that each of these statements is true. Show your work. Sn: 2 is a factor of n2 + 7n 5 Pre-Calculus Quarter 4 Exam 15 (i.) f(x) (ii.) f(x) (iii.) What can you conclude about f(x)? How is this shown by the graph? (iv.) What aspect of costs of renting a car causes the graph to jump vertically by the same amount at its discontinuities? 16 Use mathematical induction to prove that the statement is true for every positive integer n. 8 + 16 + 24 + . . . + 8n = 4n(n + 1) 6 Pre-Calculus Quarter 4 Exam 17 The following piecewise function gives the tax owed, T(x), by a single taxpayer on a taxable income of x dollars. T(x) = (i) Determine whether T is continuous at 6061. (ii) Determine whether T is continuous at 32,473. (iii) If T had discontinuities, use one of these discontinuities to describe a situation where it might be advantageous to earn less money in taxable income. 18 A statement Sn about the positive integers is given. Write statements Sk and Sk+1, simplifying Sk+1 completely. Sn: 1 + 4 + 7 + . . . + (3n - 2) = n(3n - 1)/2 7 Pre-Calculus Quarter 4 Exam 19 An artist is creating a mosaic that cannot be larger than the space allotted which is 4 feet tall and 6 feet wide. The mosaic must be at least 3 feet tall and 5 feet wide. The tiles in the mosaic have words written on them and the artist wants the words to all be horizontal in the final mosaic. The word tiles come in two sizes: The smaller tiles are 4 inches tall and 4 inches wide, while the large tiles are 6 inches tall and 12 inches wide. If the small tiles cost $3.50 each and the larger tiles cost $4.50 each, how many of each should be used to minimize the cost? What is the minimum cost? 20 The Fiedler family has up to $130,000 to invest. They decide that they want to have at least $40,000 invested in stable bonds yielding 5.5% and that no more than $60,000 should be invested in more volatile bonds yielding 11%. How much should they invest in each type of bond to maximize income if the amount in the stable bond should not exceed the amount in the more volatile bond? What is the maximum income? 8 Pre-Calculus Quarter 4 Exam 9

Tutor Answer

Borys_S
School: New York University

The solutions are ready. Please read them and ask if something is unclear, or if you want more explanations. etc.

1. βˆ‘4π‘˜=1(βˆ’1)π‘˜ (π‘˜ + 11) = βˆ’12 + 13 βˆ’ 14 + 15 = 𝟐.

2.

π‘₯2

𝑦2

+ 11 = 1. 36 > 11 so it is a horizontal ellipse. The center is at the origin.
36

π‘Ž = √36 = 6, 𝑏 = √11, 𝑐 = βˆšπ‘Ž2 βˆ’ 𝑏 2 = 5, thus the foci are (βˆ’πŸ“, 𝟎), (πŸ“, 𝟎).

2𝑦 βˆ’ π‘₯ = 5,
From the first equation π‘₯ = 2𝑦 βˆ’ 5, substitute it to the second:
π‘₯ + 𝑦 2 βˆ’ 25 = 0.
(2𝑦 βˆ’ 5)2 + 𝑦 2 βˆ’ 25 = 0, 5𝑦 2 βˆ’ 20𝑦 = 5𝑦(𝑦 βˆ’ 4) = 0, 𝑦 = 0 π‘œπ‘Ÿ 𝑦 = 4.
So π‘₯ = βˆ’5 π‘œπ‘Ÿ π‘₯ = 3 and the answers are (βˆ’πŸ“, 𝟎), (πŸ‘, πŸ’).
3. {

2

4. lim(5π‘₯ βˆ’ 3) = 5 lim π‘₯ βˆ’ 3 = 5 βˆ™ 5 βˆ’ 3 = 𝟐𝟐.
π‘₯β†’5

π‘₯β†’5

4 βˆ’1 3 12
0 βˆ’17 βˆ’21 140
5. (1 4 6 βˆ’32) β†’ |𝑅1 = 𝑅1 βˆ’ 4𝑅2 , 𝑅3 = 𝑅3 βˆ’ 5𝑅2 | β†’ (1
4
6
βˆ’32).
5 3 9 20
0 βˆ’17 βˆ’21 180
We see that the first and the second rows make a contradiction. So no solutions.

1 1
6. (1 βˆ’1
4 1

1 βˆ’5
1 1
1 βˆ’5
3 βˆ’1) β†’ |𝑅2 = 𝑅2 βˆ’ 𝑅1 , 𝑅3 = 𝑅3 βˆ’ 4𝑅1 | β†’ (0 βˆ’2 2
4 )β†’
0 βˆ’3 βˆ’3 18
1 βˆ’2
1 1 1 βˆ’5
1 1 1 βˆ’5
1
1
β†’ |𝑅2 = βˆ’ 𝑅2 , 𝑅3 = βˆ’ 𝑅3 | β†’ (0 1 βˆ’1 βˆ’2) β†’ |𝑅3 = 𝑅3 βˆ’ 𝑅1 | β†’ (0 1 βˆ’1 βˆ’2) β†’
2
3
0 1 1 βˆ’6
0 0 2 βˆ’4
1
1
1
βˆ’5
1
1
1
βˆ’5
1
β†’ |𝑅3 = 𝑅3 | β†’ (0 1 βˆ’1 βˆ’2) β†’ |𝑅2 = 𝑅2 + 𝑅3 | β†’ (0 1 0 βˆ’4) β†’
2
0 0 1 βˆ’2
0 0 1 βˆ’2
1 0 0 1
β†’ |𝑅1 = 𝑅1 βˆ’ 𝑅2 βˆ’ 𝑅3 | β†’ (0 1 0 βˆ’4), which means 𝒙 = 𝟏, π’š = βˆ’πŸ’, 𝒛 = βˆ’πŸ.
0 0 1 βˆ’2

7. First, we know π‘₯ β‰₯ 0, 𝑦 β‰₯ 0. At least 336 calories means πŸ•π’™ + πŸ–π’š β‰₯ πŸ‘πŸ‘πŸ”.
[these inequalities give the first quadrant except the triangle (0,42), (0,0), (48,0)]

8. 𝑆1 : 12 =

1βˆ™2
2

𝑆2 : 12 + 42 =

, 1 = 1, which is true.
2βˆ™(24βˆ’6βˆ’1)
2

𝑆3 : 12 + 42 + 72 =

, 17 = 17, which is true.

3βˆ™(6βˆ™9βˆ’9βˆ’1)
2

, 66 =

3βˆ™44
2

= 66, which is true.

9. π‘†π‘˜ : βˆ‘π‘˜π‘—=1 𝑗(𝑗 + 1) =

π‘˜(π‘˜+1)(π‘˜+2)

(π‘˜+1)(π‘˜+2)(π‘˜+3)

3

3

, so π‘†π‘˜+1 : βˆ‘π‘˜+1
𝑗=1 𝑗(𝑗 + 1) =

.

[I am not sure what is "simplify π‘†π‘˜+1 "]

10. Denote π‘₯ the number of pounds of breakfast tea and 𝑦 of afternoon tea.
Then the constraints are:
1
1
2
1
π‘₯ β‰₯ 0, 𝑦 β‰₯ 0,
π‘₯ + 𝑦 ≀ 45,
π‘₯ + 𝑦 ≀ 70.
3
2
3
2
The profit functions ...

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