The given function is y = x³ and the interval on which we are finding the area is [0, 1].
Using right endpoints, the value of x that we use for each rectangle will be the right endpoint of the subinterval on which the rectangle is being constructed. We divide the interval [0, 1] into n equal subintervals of width Δx, where Δx = (b - a) / n = (1 - 0) / n = 1 / n. Thus, the ith subinterval has endpoints xᵢ₋₁ and xᵢ, where xᵢ₋₁ = (i - 1)Δx and xᵢ = iΔx.For each subinterval, we construct a rectangle with height f(xᵢ) and width Δx, whose area is given by ΔAᵢ = f(xᵢ)Δx. Using right endpoints, we have that f(xᵢ) = f(i/n). Therefore, the area of the entire region S can be approximated by the sum of the areas of these rectangles, given by
Rn = f(Δx)Δx + f(2Δx)Δx + f(3Δx)Δx + ... + f(nΔx)Δx = ∑ᵢ₌₁ⁿf(i/n)Δx.
We can express this sum in sigma notation as
Rn = Σᵢ₌₁ⁿf(i/n)Δx.To find the exact area, we take the limit of this sum as the number of rectangles approaches infinity, or equivalently, as the width of the rectangles approaches zero, which gives us
A = lim Rn = lim Σᵢ₌₁ⁿf(i/n)Δx n→∞ n→∞
By the definition given, we can write this as
A = lim Σᵢ₌₁ⁿf(xᵢ)Δx n→∞This is the area formula using right endpoints. Now, we can apply this formula to the function y = x³ over the interval [0, 1].
We need to use the formula A = lim Σᵢ₌₁ⁿf(xᵢ)Δx to find the area under the curve y = x³ from 0 to 1.
Using right endpoints, we divide the interval [0, 1] into n equal subintervals of width Δx = 1 / n. The ith subinterval has endpoints xᵢ₋₁ = (i - 1)Δx and xᵢ = iΔx. We construct a rectangle with height f(xᵢ) = f(i/n) and width Δx for each subinterval.For the function y = x³, we have f(x) = x³. Therefore, f(i/n) = (i/n)³, and the area formula becomes
A = lim Σᵢ₌₁ⁿf(i/n)Δx= lim Σᵢ₌₁ⁿ[(i/n)³](1/n)= lim Σᵢ₌₁ⁿi³/n⁴
Taking the limit as n approaches infinity gives usA = ∫₀¹ x³ dxWe can evaluate this integral using the power rule of integration to getA = [x⁴/4]₀¹ = 1/4
Therefore, the area under the curve y = x³ from 0 to 1 is 1/4
We used the definition of the area formula with right endpoints to find the area under the curve y = x³ from 0 to 1. We divided the interval [0, 1] into n equal subintervals of width Δx, and used right endpoints to construct rectangles with height f(i/n) = (i/n)³ and width Δx = 1/n. We then took the limit of the sum of these areas as the number of rectangles approached infinity to find the exact area. Applying this formula to the function y = x³ over the interval [0, 1], we found that the area is 1/4.
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|SOLVE Y¹-2y² + y = e²x (cosx-35 inx) Yp = ?
In this problem, we are given a second-order linear homogeneous differential equation, and we need to find a particular solution Y_p for the non-homogeneous equation. The equation is Y'' - 2Y' + Y = [tex]e^(2x)(cos(x) - 35i sin(x))[/tex]
To find the particular solution we can use the method of undetermined coefficients. Since the right-hand side of the equation is a combination of exponential and trigonometric functions, we assume that [tex]Y_p[/tex] has a similar form.
Let's assume [tex]Y_{P}[/tex] = A [tex]e^(2x) (cos(x) + B e^(2x) (sin(x))[/tex] + B [tex]e^(2x) (sin(x)),[/tex], where A and B are constants to be determined.
Next, we differentiate twice and substitute it into the original equation to find the values of A and B. After simplifying the equation, we equate the coefficients of the terms on both sides.
By solving the resulting equations, we can determine the values of A and B. Once we have A and B, we substitute them back into the assumed form to obtain the particular solution.
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Use the simplex method for bounded variables to solve the following problem:
minimize z = x₁ + 2x2 + 3x3 - X4 subject to 2x₁ - x2 + x3−2x4≤6 -X1 +2X2 X3 + x4 ≤8 2x₁ + X2 X3 + ≥2 0≤x₁ ≤3 1≤ x₂ ≤4 0≤x3 ≤ 2≤x4≤5.
The optimal solution for the given linear programming problem is z = 2, x₁ = 0, x₂ = 2, x₃ = 0, x₄ = 6, x₅ = 2, x₆ = 0 and x₇ = 0.
To solve the given linear programming problem using the simplex method, we need to convert it into standard form. The standard form of a linear programming problem involves introducing slack, surplus, and artificial variables as necessary.
The original problem is:
Minimize z = x₁ + 2x₂ + 3x₃ - x₄
Subject to:
2x₁ - x₂ + x₃ - 2x₄ ≤ 6
-x₁ + 2x₂ + x₃ + x₄ ≤ 8
2x₁ + x₂ + x₃ + x₄ ≥ 2
0 ≤ x₁ ≤ 3
1 ≤ x₂ ≤ 4
0 ≤ x₃ ≤ 2
2 ≤ x₄ ≤ 5
To convert the problem into standard form, we introduce slack variables and convert inequalities to equations:
Minimize z = x₁ + 2x₂ + 3x₃ - x₄
Subject to:
2x₁ - x₂ + x₃ - 2x₄ + x₅ = 6
-x₁ + 2x₂ + x₃ + x₄ + x₆ = 8
2x₁ + x₂ + x₃ + x₄ - x₇ = 2
x₁, x₂, x₃, x₄, x₅, x₆, x₇ ≥ 0
0 ≤ x₁ ≤ 3
1 ≤ x₂ ≤ 4
0 ≤ x₃ ≤ 2
2 ≤ x₄ ≤ 5
Now we can set up the initial simplex tableau:
BV x₁ x₂ x₃ x₄ x₅ x₆ x₇ RHS
----------------------------------------------
x₅ 2 -1 1 -2 1 0 0 6
x₆ -1 2 1 1 0 1 0 8
x₇ 2 1 1 1 0 0 -1 2
Next, we perform iterations of the simplex method until we reach the optimal solution. The iteration process involves finding the entering variable, the leaving variable, and updating the tableau.
Iteration 1:
Entering variable: x₂ (the most negative coefficient in the objective row)
Leaving variable: x₆ (minimum ratio test)
Pivot on row 2, column 3:
BV x₁ x₂ x₃ x₄ x₅ x₆ x₇ RHS
----------------------------------------------
x₅ 2 0 1 -2 -1 1 0 2
x₂ -0.5 1 0 0 0.5 -0.5 0 2
x₇ 1 0 0 1 -1 1 -1 6
Iteration 2:
Entering variable: x₃ (the most negative coefficient in the objective row)
Leaving variable: x₇ (minimum ratio test)
Pivot on row 3, column 4:
BV x₁ x₂ x₃ x₄ x₅ x₆ x₇ RHS
----------------------------------------------
x₅ 2 0 1 -2 -1 1 0 2
x₂ -0.5 1 0 0 0.5 -0.5 0 2
x₄ 1 0 0 1 -1 1 -1 6
The tableau is now in its final form, and we have reached the optimal solution:
z = 2
x₁ = 0
x₂ = 2
x₃ = 0
x₄ = 6
x₅ = 2
x₆ = 0
x₇ = 0
Therefore, the optimal solution for the given linear programming problem is:
z = 2
x₁ = 0
x₂ = 2
x₃ = 0
x₄ = 6
x₅ = 2
x₆ = 0
x₇ = 0
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A bakery produces 10 different types of cookies, and the manager plans to assemble packs of 3 cookies for the upcoming holidays wondering about which 3 types to choose. How many different ways can he assemble packs of 3 different types of cookies given that there are 10 types of cookies in general and the order of the cookies does not matter? Show all your work!
The manager can assemble packs of 3 different types of cookies in 120 different ways.
To find the number of different ways the manager can assemble packs of 3 different types of cookies, we can use the concept of combinations. Since the order of the cookies does not matter, we need to calculate the number of combinations.
The formula to calculate combinations is given by:
C(n, r) = n! / (r! * (n - r)!)
Where:
n is the total number of items (types of cookies)
r is the number of items to be chosen (3 in this case)
"!" denotes the factorial operation, which means multiplying a number by all positive integers less than itself down to 1.
Using this formula, let's calculate the number of combinations:
C(10, 3) = 10! / (3! * (10 - 3)!)
First, let's calculate the factorials:
10! = 10 * 9 * 8 * 7 * 6 * 5 * 4 * 3 * 2 * 1 = 3,628,800
3! = 3 * 2 * 1 = 6
(10 - 3)! = 7! = 7 * 6 * 5 * 4 * 3 * 2 * 1 = 5,040
Now, substitute the factorials into the formula:
C(10, 3) = 3,628,800 / (6 * 5,040)
= 3,628,800 / 30,240
= 120
Therefore, the manager can assemble packs of 3 different types of cookies in 120 different ways.
In summary, using the combination formula, we calculated that there are 120 different ways for the bakery manager to assemble packs of 3 different types of cookies out of the total 10 types of cookies available, where the order of the cookies does not matter.
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By using TABLE 1, calculate if Mr Mbhalati is correct. 2 Mr Mbhalati has Mr Prepaid account with one of the banks in South Africa. He uses the account to pay water bills. TABLE 2 shows deposit fees applicable on his account in 2023. TABLE 2: Mr Prepaid ACCOUNT COSTS AT ABA BANK FOR DEPOSITS TRANSACTION Notes and coin deposit COSTS RS0 - R2. 25 per R100 (for notes) - R5 per R100 (for coins) or part thereof the amount deposited. [Adapted by examiner from information supplied by various banks Use the information from TABLE 2 to answer the questions that follow. 3 20231
1. 2. 1 Determine The Minimum Amount One Will Pay When Making A Deposit Of Notes And Coins (2)
1. 2. 2 Mbhalati Says He Will Be Charged R180 When Making A Deposit Of Notes To the Value Of R 1 500 and R500 In Coins. Show By Means Of Calculations Whether His Statement Is Valid (5)
1. 2. 3 Mr Mbhalati Deposited R 1 500 An Amount From Sale Of Goods That Were Sold At A ValueAddedTax(VAT) Inclusive Price
Calculate 15% Vat Of amount On Goods Sold(4)
1. 2. 4 Name At Least ONE Method Me Mbhalati Can Use To Send Money To A Person Who Does Not Have A Bank Account(2)
The Deposit Fees 15% VAT amount on goods sold at a VAT-inclusive price of R1,500 would be approximately R195.65.
To answer the questions using the provided information, we'll refer to TABLE 2 for the deposit fees.1.2.1 The minimum amount one will pay when making a deposit of notes and coins:
Notes: R2.25 per R100 or part thereof
Coins: R5 per R100 or part thereof
Therefore, the minimum amount one will pay depends on the value of the deposit. For notes, it will be R2.25 per R100 or part thereof, and for coins, it will be R5 per R100 or part thereof.
1.2.2 To determine if Mr Mbhalati's statement is valid, let's calculate the deposit fees for his deposit of notes and coins.
Deposit of notes:
Deposit amount: R1,500
Deposit fee for notes: R2.25 per R100 or part thereof
Calculation:
Deposit fee = (Deposit amount / 100) * Deposit fee per R100
= (1500 / 100) * 2.25
= 33.75
Deposit of coins:
Deposit amount: R500
Deposit fee for coins: R5 per R100 or part thereof
Calculation:
Deposit fee = (Deposit amount / 100) * Deposit fee per R100
= (500 / 100) * 5
= 25
Total deposit fee = Deposit fee for notes + Deposit fee for coins
= 33.75 + 25
= 58.75
Therefore, Mr Mbhalati's statement is not valid. The total deposit fee for his deposit of notes to the value of R1,500 and R500 in coins would be R58.75, not R180.
1.2.3 To calculate the 15% VAT on the amount of goods sold at a VAT-inclusive price:
Amount from the sale of goods: R1,500
Calculation:
VAT amount = (Amount from the sale of goods / (100 + VAT rate)) * VAT rate
= (1500 / (100 + 15)) * 15
= (1500 / 115) * 15
≈ 195.65
Therefore, the Deposit Fees 15% VAT amount on goods sold at a VAT-inclusive price of R1,500 would be approximately R195.65.
1.2.4 One method Mr Mbhalati can use to send money to a person who does not have a bank account is by using a money transfer service like Western Union or MoneyGram. These services allow individuals to send money to recipients who can collect it in cash from designated locations, even if they don't have a bank account.
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A) Q divided by 6 = 4
B) N divided by 2.5 = 3
When can you use proportional reasoning to solve a problem?
Answer:
We can use proportional reasoning to solve some questions directly, such as which size of laundry detergent is the cheapest per load, or what the dimensions of the model car should be. We can also use proportional reasoning to estimate answers and check answers to problems involving more complicated algebra techniques.
Step-by-step explanation:
Solve and find the value of X : −0.17=(x−390)/390+7.2/390 [enter your answer with 3 decimals]
The value of x is approximately 369.186.
To solve the equation -0.17 = (x - 390) / 390 + 7.2 / 390 for x, we can simplify the equation and isolate x.
Given:
-0.17 = (x - 390) / 390 + 7.2 / 390
First, let's simplify the right side of the equation by finding a common denominator:
-0.17 = (x - 390 + 7.2) / 390
Combine like terms:
-0.17 = (x - 382.8) / 390
Multiply both sides of the equation by 390 to eliminate the fraction:
-0.17 * 390 = x - 382.8
-66.3 = x - 382.8
Add 382.8 to both sides of the equation:
-66.3 + 382.8 = x
316.5 = x
Therefore, the value of x is approximately 316.5 (rounded to three decimal places).
Note: There seems to be an error in the given equation. The correct equation should be -0.17 = (x - 390) / 390 + 7.2 / 390, not -0.17 = (x - 390) / 390 + 7.2 / 390.
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If the sum and the product of zero of the polynomial q(x)=x²-5x+5k is equal,then K equals
We may utilize Vieta's formulas to determine the value of k in the polynomial q(x) = x2 - 5x + 5k, where the sum and product of its zeros are equal.
What is Polynomial?
Let and serve as symbols for the polynomial's zeros. According to Vieta's formulas, the product of the zeros is equal to 5k, and the total of the zeros is provided by + = -(-5)/1 = 5.
Given that the sum and product of the zeros are equal, we have α + β = α * β. Substituting the known values, we get:
5 = 5k
This equation can now be solved to determine the value of k:
5/5 = k
k = 1
Therefore, k equals 1 when the sum and product of the zeros of the polynomial q(x) = x² - 5x + 5k are equal.
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A randomly generated list of numbers from 0 to 4 is being used to simulate an event, with the numbers 2,3 , and 4 representing a success. What is the estimated probability of a success? A. 60% B. 40% C. 25% D. 75%
A randomly generated list of numbers from 0 to 4 is being used to simulate an event, with the numbers 2,3, and 4 representing a success. The estimated probability of success is 60%.
The randomly generated list of numbers from 0 to 4 is simulating an event, and the probability of a success depends on the presence of 2, 3, or 4 in the generated list.
The other two numbers, 0 and 1, represent failure. To calculate the probability of success, we need to divide the number of possible successful outcomes by the total number of possible outcomes.
In this case, the number of possible successful outcomes is 3 (2, 3, and 4) and the total number of possible outcomes is 5 (0, 1, 2, 3, and 4).
Therefore, the probability of success can be calculated as follows:
Probability of success = Number of successful outcomes / Total number of outcomes = 3 / 5= 0.6= 60%
Therefore, the estimated probability of success is 60%.
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Which equation represents a line that passes through (–9, –3) and has a slope of –6?
y – 9 = –6(x – 3)
y + 9 = –6(x + 3)
y – 3 = –6(x – 9)
y + 3 = –6(x + 9)
Answer:
=y + 3 = -6(x +9)
Step-by-step explanation:
hope that will help you
Answer:
D
Step-by-step explanation:
Which of the following set of numbers represent three sides of a triangle?
A. 5,3, and 2
b. 9,5,15
c. 8,10,6
d. 4,6,10
In order for a set of numbers to represent three sides of a triangle, the sum of the lengths of any two sides must be greater than the length of the third side. This is known as the triangle inequality theorem. Only option C represents three sides of a triangle.
Now let's check each option to see which one satisfies the triangle inequality theorem:
a. 5, 3, and 2
The sum of 5 and 3 is 8, which is greater than 2. The sum of 5 and 2 is 7, which is also greater than 3. The sum of 3 and 2 is 5, which is not greater than 5. Therefore, this set of numbers does not represents three sides of a triangle.
b. 9, 5, and 15
The sum of 9 and 5 is 14, which is less than 15. Therefore, this set of numbers does not represent three sides of a triangle.
c. 8, 10, and 6
The sum of 8 and 10 is 18, which is greater than 6. The sum of 8 and 6 is 14, which is also greater than 10. The sum of 10 and 6 is 16, which is greater than 8. Therefore, this set of numbers represents three sides of a triangle.
d. 4, 6, and 10
The sum of 4 and 6 is 10, which is less than 10. Therefore, this set of numbers does not represent three sides of a triangle. Therefore, only option C represents three sides of a triangle.
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Solve the equations. I am having a hard time with this kind of math, so if you could please try to explain how to get the answers that would be great.
The value of Exponents are:
1. √[tex]x^{36}y^{16[/tex]= x¹⁸y⁸
2. √[tex]x^{100}y^{16[/tex]= x⁵⁰y⁸
3. √300 y⁹ = 10y⁴√ 3y
1. √[tex]x^{36}y^{16[/tex]
Now, we can write [tex]x^{36[/tex] and [tex]y^{16[/tex] as
[tex]x^{36[/tex] = (x¹⁸)²
and, y¹⁶= (y⁸)²
So, √[tex]x^{36}y^{16[/tex]
= √(x¹⁸)²(y⁸)²
= x¹⁸y⁸
2. √[tex]x^{100}y^{16[/tex]
Now, we can write [tex]x^{100[/tex] and [tex]y^{16[/tex] as
[tex]x^{100[/tex] = (x⁵⁰)²
and, y¹⁶= (y⁸)²
So, √[tex]x^{36}y^{16[/tex]
= √(x⁵⁰)²(y⁸)²
= x⁵⁰y⁸
3. √300 y⁹
Now, we can write y⁹ as
y⁹ = y x (y⁴)²
and, 300 = 2 x 2 x 3 x 5 x 5
So, √300 y⁹
= √2 x 2 x 3 x 5 x 5 x y x (y⁴)²
= 10y⁴√ 3y
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Steven is moving to another city next weekend and wants to rent a moving truck. The rental rates for two companies in his area are shown below. Each company charges an initial fee for renting the truck, plus an additional amount per mile.
Answer:
Company (1) charges $1.12 per mile and company (2) charges $0.85 per mile
Company (2) is $7.4 cheaper.
Step-by-step explanation:
From the tables given in the picture,
Let the equation representing the rental 'y' and distance traveled 'x' is represented by,
y = mx + b
For company (1),
Here m = per mile rental charges = Slope of the graph
b = Initial charges
Since slope of the line passing through two points [tex](x_1,y_1)[/tex] and [tex](x_2,y_2)[/tex] is given by,
m = [tex]\frac{y_2-y_1}{x_2-x_1}[/tex]
Therefore, slope of the line passing through (30, 53.55) and (60, 87.15) will be,
m = [tex]\frac{87.15-53.55}{60-30}[/tex] = $1.12 per mile
So the equation will be,
y = 1.12x + b
Since this line passes through (30, 53.55),
53.55 = 1.12(30) + b
b = 53.55 - 33.6
b = 19.95
So the expression representing charges for the company (1) will be,
y = 1.12x + 19.95
For company (2),
Per mile charges = slope of the line
m = [tex]\frac{78-56.75}{50-25}[/tex]
m = $0.85 per mile
Equation for the charges will be,
y = 0.85x + b
Since this line passes through (25, 56.75),
56.75 = 0.85(25) + b
b = 56.75 - 21.25
b = 35.5
Therefore, equation representing total charges for company (2) will be,
y = 0.85x + 35.5
By comparing per mile charges,
Company (1) charges $1.12 per mile and company (2) charges $0.85 per mile
Steven plans to drive 85 km in rental truck.
Charges for company (1),
y = 1.12x + 19.95
y = 1.12(85) + 19.95
y = $115.15
Charges for company (2),
y = 0.85x + 35.5
y = 0.85(85) + 35.5
y = $107.75
Difference in charges of both the companies = 115.15 - 107.75 = $7.4
Therefore, company (2) is $7.4 cheaper.
Solve the Quadratic Equation using the Quadratic Formula
1.
7n^2+6n-5=0
2.
X^2+3x+1=0
Answer:
1. n = -3+2√ 11/7, n = (-) 3+2√ 11/7
2. x=-3+√ 5/2, x= -3-√ 5/2
Step-by-step explanation:
(-) means that it applys to both the numerator and denominator.
Giving brainliest! Please show the working out
Answer:
see explanation
Step-by-step explanation:
(a)
OC = OB ( both radii of the circle )
Thus Δ BOC is isosceles with congruent base angles.
∠ BOC = ∠ BCO = 50°
(b)
∠ ACB = 90° ( angle in a semicircle ), then
∠ ACO = 90° - 50° = 40°
OA = OC ( both radii of the circle )
Thus Δ ACO is isosceles with congruent base angles.
∠ BAC = ∠ ACO = 40°
A triangle and its angle measures are shown in the diagram. What is the value of x?
a
16
b
10.28
c
1.6
d
56
Answer:
it would be d
Step-by-step explanation:
if look closly you can see that im just trooling to get a challenge done ( but no joke the awnser is d tho)
what is the gcf of 35, 20,15
Answer:
Prime factorisation of the numbers -
[tex] \\ \sf \: 35 = 7 \times \underline5 \\ \\ \sf \: 20 = \underline5 \times 4 \\ \\ \sf \: 15 = 3 \times \underline5 \\ [/tex]
As, we can see that the 5 is common in the both three numbers. So,
GCF, greatest common factor (35, 20, 15) = 5
(a) Domain (b) Intercepts (c) Symmetry (d) Asymptotes (e) Intervals of Increasing and Decreasing (f) Local Extrema (g) Concavity (h) Points of Inflection (i) Sketch the Curve 1) y= x' - 5x x-x (2) y = 2-3x+ x2 X' (3) y= x- + 1 (4) y= xV2-x2 (5) y = x+ cosx on [0, 2x] sin x (6) y= on [0, 275 ] 2+ cosx
(a) Domain refers to the set of all possible values that the independent variable (x) can take in the given equation or function. (b) Intercepts represent the points where a curve intersects the x-axis or the y-axis. The x-intercepts occur when the value of y is equal to zero, while the y-intercepts occur when the value of x is equal to zero.
(c) Symmetry relates to the shape of a curve with respect to a specific axis or point. A curve can exhibit symmetry about the x-axis, y-axis, or the origin. If a curve is symmetric, its equation will exhibit certain patterns.
(d) Asymptotes are lines that a curve approaches but never intersects. They can be horizontal, vertical, or oblique. Asymptotes provide information about the behavior of a curve as x approaches infinity or negative infinity.
(e) Intervals of increasing and decreasing refer to the portions of the curve where its slope is positive (increasing) or negative (decreasing). These intervals can be determined by examining the sign of the derivative of the function.
(f) Local extrema are points on a curve where the function reaches a maximum or minimum value within a specific interval. They can be identified by analyzing critical points or points where the derivative of the function is zero or undefined.
(g) Concavity refers to the curvature of a curve. It can be concave up (opening upwards) or concave down (opening downwards). The concavity of a curve can be determined by examining the second derivative of the function.
(h) Points of inflection are points on a curve where the concavity changes. They are identified by finding the values of x where the second derivative of the function changes sign.
(i) Sketching the curves involves utilizing the information from the above points to plot the shape of the curve accurately. It involves identifying the key features such as intercepts, symmetry, asymptotes, intervals of increasing and decreasing, local extrema, concavity, and points of inflection to draw the curve as accurately as possible within the given domain or interval.
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May someone help me please?
Answer:
got n=13
Step-by-step explanation:
I set it up 4n+7=59 because 4 dollars you can get a dvd and when you solve you
How do u find the area can someone explain
Step-by-step explanation:
divide the figure into one square and one rectangle. Than use the length and width for each of your two shapes and combine them to find the area.
Answer:
46.92+18.06=64.98
Step-by-step explanation:
the shape can be divided to a rectangle = 4.6*10.2
then for the second shape
do 8.9-4.6 and you will get 4.3
multiply 4.3*4.2 for the second shape
1/4 (4x - 8) - 2 (5x + 6)
Answer:
-9x - 14
Step-by-step explanation:
1/4 (4x - 8) - 2 (5x + 6) =
= 1/4 × 4x - 1/4 × 8 - 2 × 5x - 2 × 6
= x - 2 - 10x - 12
= -9x - 14
Answer:
-9x - 14
Step-by-step explanation:
Simplify by following these steps:
1/4 (4x - 8) - 2(5x + 6)
(here we distribute 1/4)
x - 2 - 2(5x + 6)
(here we distribute -2)
x - 2 - 10x - 12
Collect similar terms
-9x - 14
∴ -9x - 14
4. If A and B are nxn matrices with the same eigenvalues, then they are similar.
12. Let x, y, and z be vectors in C" and let a and ß be В complex scalars. Show that (z,ax+By) = a(z,x) + B (z,y)
We need to show that for vectors x, y, and z in C^n and complex scalars a and ß, the inner product (z, ax + By) is equal to a(z, x) + ß(z, y).
To prove the given equation, we will use the properties of the inner product and scalar multiplication.
First, let's expand the left-hand side of the equation:
(z, ax + By)
Using the linearity property of the inner product, we can distribute the inner product over addition:
(z, ax) + (z, By)
Next, we apply the scalar multiplication property of the inner product to each term:
a(z, x) + ß(z, y)
Now, we can see that the left-hand side (z, ax + By) is equal to the right-hand side a(z, x) + ß(z, y), proving the equation.
In summary, by utilizing the linearity and scalar multiplication properties of the inner product, we showed that (z, ax + By) is equal to a(z, x) + ß(z, y). This demonstrates the relationship between the inner product of vectors and scalar multiplication, verifying the given equation.
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A 60-gallon tank is full of water when it begins to leak at a
rate of 2 gallons per minute. Which equation provides t, the
number of minutes it takes the tank to become empty?
Answer:
60/2=t
Step-by-step explanation:
Simple algebra. You need to divide the total by the per minute to find the length in time that it will take. t=30.
What is the equation, in point slope form, of the line
that is parallel to the given line and passes through the
point (-3, 1)?
HURRY ANSWER PLEASE
Answer:
y=3/2x+11/2
Step-by-step explanation:
Hello! Sorry I just saw this
Anyways, let's continue
first, we need to find the equation of the line with the one that is (-2,-4) and (2,2)
first, we need to find the slope
the equation for slope is y2-y1/x2-x1
so let's label the points
x1=-2
y1=-4
x2=2
y2=2
now plug it in
2-(-4)/2-(-2)=6/4=3/2
now, let's turn it into a line
the point-slope form is y-y1=m(x-x1) (m=slope)
now, plug it in
y-(-4)=3/2(x-(-2))
simplify to
y+4=3/2(x+2)
turn into y=mx+b format
y+4=3/2x+3
subtract 4 on both sides
y=3/2x-1
Now for the line that is parallel.
Parallel lines have the same slopes, so you automatically know that the new line will be y=3/2x+b
To make sure (-3,1) is a solution to the point, put 1 as y and -3 as x
1=3/2(-3)+b
1=-9/2+b
add 9/2 on both sides
b=11/2 or 5.5
now, put it into the equation
y=3/2x+11/2
Hope this helps!
The number 5/7 can be best describes as a(n)?
Mixed number
Improper fraction
Proper fraction
Answer:
proper fraction
Step-by-step explanation:
Mixed number has a whole number and a fraction a b/c
Improper fraction has a numerator that is greater than the denominator a/b where a is greater than b a>b
Proper fraction has a numerator that is less than the denominator a/b where a is less than b a<b
Answer:
Proper fraction
Step-by-step explanation:
Numerator is less than the denominator, making it a proper fraction.
BC is included between _______
Answer:
Your answer would be B and C.
Which graph has the point with the least possible y-value?
a) y = 6x
b) y - x? + 3
c) y-(x+2)+-5
d) y=(x-3)2 + 4
What do f(n) and f(n-1) represent in a sequence?
Answer:
f(n) is the function f-1(n) is the inverse of that function , basically the x and y switch places
Step-by-step explanation:
A telephone line hangs between two poles 14 m apart in the shape of the catenary
y = 25 cosh(x/25) ? 20,
where x and y are measured in meters.
(a) Find the slope of this curve where it meets the right pole. (Round your answer to four decimal places.)
(b) Find the angle ? between the line and the pole. (Round your answer to two decimal places.)
The slope of the curve where it meets the right pole is sinh(7/25) ≈ 0.2748 (rounded to four decimal places). The angle between the line and the pole is 70.23° (rounded to two decimal places).
Given function is: y = 25 cosh(x/25) - 20
(a) Slope of the curve where it meets the right pole:
The slope of the curve is given by
dy/dx= 25/25sinh(x/25)= sinh(x/25)
At the right pole, the value of x=7, y=5
Hence, slope of curve at right pole= sinh(7/25)≈ 0.2748 (rounded to four decimal places)
(b) Angle between the line and the pole:
To find the angle, we first find the tangent of the angle at the pole.
tan θ = slope of curve at pole= sinh(7/25)
We know, opposite side / adjacent side = tan θ
By Pythagoras theorem, the opposite side is √(14^2 – 5^2) = 13.60147
Therefore, tan θ = 13.60147 / 5
tan θ ≈ 2.7203 (rounded to four decimal places)
θ = tan^-1(2.7203) ≈ 70.2313°≈ 70.23° (rounded to two decimal places)
The angle between the line and the pole is 70.23° (rounded to two decimal places).
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∫(3x4 +x-2 +√x3)x (Hint: √x3 = 3/x2
The integral of (3x^4 + x - 2 + √x^3) * x is (x^6/2) + (x^3/3) - x^2 + 3ln|x| + C, where C represents the constant of integration.
To solve the given integral ∫(3x^4 + x - 2 + √x^3) * x, we can break it down into separate terms and integrate each term individually.
In the first term, 3x^4 * x, we can simplify it by multiplying the exponents: 3x^5. We apply the power rule for integration, which states that the integral of x^n is (x^(n+1))/(n+1). Applying this rule to our first term, we get (3/6)x^6 = x^6/2.
Moving on to the second term, x * x, we simply have x^2. Again, applying the power rule, the integral of x^2 is (x^3)/3.
In the third term, -2 * x, we have a constant multiple of x, which can be integrated as -(2/2)x^2 = -x^2.
Lastly, in the fourth term, √x^3 * x, we can simplify √x^3 as 3/x^2. Therefore, the term becomes (3/x^2) * x = 3/x.
Combining all the integrated terms, the final result of the integral is (x^6/2) + (x^3/3) - x^2 + 3ln|x| + C, where C is the constant of integration.
In summary, the integral of (3x^4 + x - 2 + √x^3) * x is (x^6/2) + (x^3/3) - x^2 + 3ln|x| + C, where C represents the constant of integration.
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