If W is a subspace, then
1. Geometric description of W is a line passing through the origin with a slope of 2.
2. W is not a subspace of R ^ 2.
3. Here, W is the union of two lines passing through the origin with slopes of 1 and -1.
4. In this case, W is not a subspace of R ^ 2.
5. Geometrically, W is a line passing through the origin with a slope of 0.
6. Here, Geometrically, W is a single point at the origin.
1. W = \{x / x_{1} = 2x_{2}\} is a subspace of R^2 because it satisfies the three conditions of being a subspace: it contains the zero vector (0,0), it is closed under addition, and it is closed under scalar multiplication. Geometrically, W is a line passing through the origin with a slope of 2.
2. W = \{x / x_{1} - x_{2} = 2\} is not a subspace of R^2 because it does not contain the zero vector (0,0). Geometrically, W is a line that does not pass through the origin.
3. W=\ x / x_{1} = x_{2} or x_{1} = - x_{2} is a subspace of R^2 because it satisfies the three conditions of being a subspace. Geometrically, W is the union of two lines passing through the origin with slopes of 1 and -1.
4. W = {x: x₁ and x_{2} are rational numbers} is not a subspace of R^2 because it is not closed under scalar multiplication. For example, if we multiply the vector (1/2, 1/3) by the scalar √2, we get the vector (√2/2, √2/3), which is not in W. Geometrically, W is the set of all points with rational coordinates.
5. W = \{x / x_{1} = 0\} is a subspace of R^2 because it satisfies the three conditions of being a subspace. Geometrically, W is a line passing through the origin with a slope of 0 (the x-axis).
6. W = \{x / |x_{1}| + |x_{2}| = 0\} is a subspace of R^2 because it contains only the zero vector (0,0), which satisfies the three conditions of being a subspace. Geometrically, W is a single point at the origin.
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3x+4y=36 find y and x
Answer:
y = 36
x = 12
Hope it helps!
Read and interpret the following conditions imposed on the variables \( a, b, c, d \), and \( x \). Determine and state whether the statements in Exercises 1 - 12 are true or false. If they are false,
The given conditions imposed on the variables \( a, b, c, d \) and \( x \) are:
\( a+b = c \) \( d = a^2 + b^2 \) \( x = a^3 + b^3 \)
To determine if the statements in Exercises 1-12 are true or false, use the given conditions to evaluate the expressions in the statement. If the statement matches the conditions, it is true; if it does not, it is false. For example, if the statement is: " \( a + b = d \) ", then this is false, as \( a + b \neq d \).
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McKenzie and Lindy work on a landscaping crew. They can complete the landscaping job in 4 hours if they work together. McKenzie generally takes 6 hours less Lindy. How long would it take McKenzie to complete the landscaping job if Lindy calls in sick?
McKenzie would take 6 hours to complete the landscaping job alone if Lindy calls in sick.
If McKenzie and Lindy can complete the landscaping job in 4 hours working together, it means their combined work rate is 1/4 of the job per hour. Let x be the number of hours it takes Lindy to complete the job alone, then McKenzie can complete the job in x-6 hours.
Using the formula for their individual work rates, we have:
[tex]1/x + 1/(x-6) = 1/4[/tex]
Multiplying both sides by [tex]4x(x-6)[/tex], we get:
[tex]4(x-6) + 4x = x(x-6)[/tex]
Expanding and simplifying, we get:
[tex]2x^2 - 12x - 48 = 0[/tex]
Dividing both sides by 2 and using the quadratic formula, we get:
[tex]x = (12 ± \sqrt{12^2 + 4248}) / (2*2)[/tex]
x = (12 ± 18) / 4
x = 7.5 or -1.5
Since we cannot have a negative time, the answer is that it would take Lindy 7.5 hours to complete the job alone, and McKenzie would take 1.5 hours less, or 6 hours, to complete the job alone if Lindy calls in sick.
Therefore, McKenzie would take 6 hours to complete the landscaping job alone if Lindy calls in sick.
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1. Show that the seriesn=1∑[infinity](−1)n−1(3n+5)3ln(n+2)+31is convergent, and determine how many terms of the series we need to add to find the sum within0.05. Is this series absolutely convergent? Justify your answer. 2. Determine whether the seriesn=1∑[infinity](−1)n−13n+4ln(5n+2)is absolutely convergent, conditionally convergent or divergent.
The series converges.
1. This is an alternating series, and therefore, the series converges by the Alternating Series Test. This is because the sequence of the absolute value of the terms, {|3n+5|}^(1/3), is a monotonically decreasing sequence, and it is bounded. To determine how many terms are necessary to get within 0.05 of the sum, we use the formula:
Sn≈s[infinity]+a1r1/1−r
where a1 is the first term, and r is the common ratio. In this case, we have a1 = (3+5)^(1/3) = 2, and r = (-1)^(1/3) = -1. Thus, Sn ≈ 2 + (2)(-1)/1 - (-1) = 4.
Since the series is alternating, it is absolutely convergent.
2. This series is also an alternating series, so it is absolutely convergent by the Alternating Series Test. The sequence of the absolute value of the terms, {|3n+4|}^(1/ln(5n+2)), is a monotonically decreasing sequence, and it is bounded. Therefore, the series converges.
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A circle has 36 pi square meters what is the circumference of this circle
Answer: 18cm
Step-by-step explanation:
The length of the radius of the circle with center O has a circumference of 36π is 18cm.
A toymaker is creating a toy that has different gears. He needs to determine how far the center of each gear is from the knob, so he can connect wire to those points inside the toy. Here's what he knows: The distance between the center of a metal gear and a plastic gear is 14 inches. The distance between the center of the metal gear and a wooden gear is 10 inches. All radii are even whole numbers, and all are greater than 2 inches. No two gears are the same size. He has labeled distances between the knob & points of tangency to the gears in his diagram. You can help him decide which gear is made of which material. ● ● ● 1. Now help him find the distance from the knob to the center of each gear! Show all of your work. Use the page below for workspace as needed. Distance from knob to... Metal gear: Plastic gear: Wooden gear:
The use of gears enables the exchange of torque for angular velocity (i.e., if the gears are of unequal size, the larger gear will experience higher torque but lower angular velocity than the smaller gear).
What is system?Any organized assembly of resources and procedures united and regulated by interaction or interdependence to accomplish a set of specific functions.
here, we have,
Two gears or pulleys must rotate at the same angular velocity if they are attached to the same axle.
Since the angle doesn't change the unit in any way, the work unit is also in ft-lbs. The rotating equivalent of a formula you might remember from physics is "Work = Force * Distance."
Energy is preserved in a gear train if friction-related heat losses are disregarded.
A gear ratio is the comparison of the number of rotations made by the driving gear system and the driven gear.
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solve pls
3x+5y=15
x+y=3
Answer:
Solve for the first variable in one of the equations, then substitute the result into the other equation.
Point Form:
(
0
,
−
3
)
Equation Form:
x
=
0
,
y
=
−
3
Step-by-step explanation:
Answer:
x=0 y=3
Step-by-step explanation:
3(0) + 5(3) = 15
0 + 15 = 15
15=15
HELLOOOO PLEASE HELP ME 15 POINTS!!!!!
Answer: C
Step-by-step explanation: Just replace x=1,2,3,4 to see which answers satisfy the given conditions
y = f(x) if f”(x) = 6x-2. f’(0) =3, f(0) 7
a) f(x) = x^3 -x^2 + 3x +7
b) f(x) = x^3 - x^2
c) f(x) = 0
d) f(x) = x^3 - x^2 + 3x
a) f(x) = x^3 -x^2 + 3x +7 is the correct answer
What is the slope of the line shown below? slope =
sloperise/run
Answer:
m= 3/4
Step-by-step explanation:
hope this helps
Simplify each expression by performing the indici (a) z+3z; (b) z*3z; (c) -z-3z; (d) (-z)(-3z) (a) z+3z=1
The simplified expressions are: (a) z = 1/4, (b) 3z^2, (c) -4z, and (d) 3z^2.
To simplify each expression by performing the indici, we need to follow the order of operations and combine like terms. Here are the steps for each expression:
(a) z + 3z = 1
First, we need to combine the like terms on the left side of the equation. Since both terms have the variable z, we can add them together:
4z = 1
Next, we need to solve for z by isolating the variable on one side of the equation. We can do this by dividing both sides of the equation by 4:
z = 1/4
(b) z * 3z
To simplify this expression, we just need to multiply the two terms together:
3z^2
(c) -z - 3z
To simplify this expression, we need to combine the like terms. Since both terms have the variable z, we can add them together:
-4z
(d) (-z)(-3z)
To simplify this expression, we just need to multiply the two terms together. Remember that a negative times a negative is a positive:
3z^2
So the simplified expressions are: (a) z = 1/4, (b) 3z^2, (c) -4z, and (d) 3z^2.
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PLEASE HURRY IM FAILING
Answer:
A. 3/2
Step-by-step explanation:
what is a dilation? a dilation can make objects bigger or smaller.
10 times [tex]\frac{3}{2}[/tex] is 15
4 times [tex]\frac{3}{2}[/tex] is 6
8 times [tex]\frac{3}{2}[/tex] is 12
therefore the answer is A.. [tex]\frac{3}{2}[/tex]
Answer:
A. 3/2
Step-by-step explanation:
What is dilation?
The original image is DEFG. It is called the pre-image
The dilated image is D'E'F'G' and is called image
A transformed image has the coordinates labeled with a '
In this case the transformed image is a dilation of the original pre-image. A dilated image is a cop.y of the pre-image but with either an expansion or a compression
If the image is larger than the pre-image it is an expansion, if smaller than the pre-image then it is a compression
With dilation, comes a scale factor which indicates how much a pre-image has been expanded or compressed
Solution
If you look at the two images, take any side in the pre-image and compare its length with the corresponding side you can find the scale factor
Here if you take a side, say DE = 4 and look at the corresponding side D'E' = 6you see that the DE side has been expanded with a scale factor of 6/4 = 3/2
This scale factor applies to all sides, so you need to take only one side to find the scale factor
Answer: Scale Factor of dilation = 3/2
This is option A
Jamal works as an electrician’s apprentice. He rewired 4 electrical outlets in 1 and one-halfhours. If he works at the same pace for 7.5 hours, how many outlets will he rewire?
Answer:
Below
Step-by-step explanation:
4 outlets/ 1.5 hr * 7.5 hr = 20 outlets ( see how 'hr' cancels out and you are left with 'outlets ?)
Answer: 20
Step-by-step explanation:
In 1.5 hours, the no. of electrical outlets rewired = 4
Then, in 1 hour the no. of electrical outlets rewired = 4/1.5
So, in 7.5 hours the no. of electrical outlets rewired =( 4/1.5 ) × 7.5 = 20.
Write the slope-intercept equation of the function f whose graph satisifies the given conditions.
The graph of f passes through (-9,-3) and is perpendicular to the line whose equation is x=-3
This is a horizontal line passing through the point (-9, -3).
What is Slope?Slope is a measure of how steep a line is. It is defined as the ratio of the change in the y-coordinates of two points on a line to the change in the x-coordinates of those same points. In other words, slope is the amount by which the y-coordinate changes when the x-coordinate changes by 1 unit.
Since the given line is vertical, its slope is undefined. The slope of a line perpendicular to it is zero. Therefore, the slope of the graph of f is zero.
We can use the point-slope form of a linear equation to write the equation of the line passing through the point (-9, -3) with a slope of zero:
y - (-3) = 0(x - (-9))
Simplifying, we get:
y + 3 = 0
Subtracting 3 from both sides, we get:
y = -3
So the equation of the function f is:
f(x) = -3
Therefore, this is a horizontal line passing through the point (-9, -3).
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Find the equation of the line through the point (3,-7) that is parallel to the line 4x + 7y - 10 = 0. Write the answer in the point-slope form y - y1 = m(x – x1).
The equation of the line through the point (3, -7) that is parallel to the line 4x + 7y - 10 = 0 is y = (-4/7)x - (37/7) in the point-slope form
To find the equation of the line through the point (3, -7) that is parallel to the line 4x + 7y - 10 = 0, we first need to find the slope of the given line. We can do this by rearranging the equation to the slope-intercept form, y = mx + b, where m is the slope.
4x + 7y - 10 = 0
7y = -4x + 10
y = (-4/7)x + (10/7)
The slope of the given line is -4/7. Since we want a line that is parallel to this one, the slope of our new line will also be -4/7.
Now we can use the point-slope form of a line, y - y1 = m(x - x1), to write the equation of the new line. We plug in the given point (3, -7) for (x1, y1) and the slope -4/7 for m.
y - (-7) = (-4/7)(x - 3)
y + 7 = (-4/7)x + (12/7)
y = (-4/7)x + (12/7) - 7
y = (-4/7)x - (37/7)
So the equation of the line through the point (3, -7) that is parallel to the line 4x + 7y - 10 = 0 is y = (-4/7)x - (37/7) in the point-slope form.
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Using the integer root theorem, list out all possibl (e)/(c)andidate integer roots of f(x)=-x^(5)+22x^(4)-8x^(3)-8x^(2)+5x+100. Use commas to separate.
The Integer Root Theorem states that the possible integer roots of a polynomial are the divisors of the constant term (100). Therefore, the candidate integer roots for the polynomial [tex]f(x) = -x^5 + 22x^4 - 8x^3 - 8x^2 + 5x + 100[/tex] are: 1, -1, 2, -2, 4, -4, 5, -5, 10, -10, 20, -20, 25, -25, 50, and -50.
According to the integer root theorem, the possible integer roots of a polynomial f(x) are the factors of the constant term divided by the factors of the leading coefficient. In the case of f(x)=-x^(5)+22x^(4)-8x^(3)-8x^(2)+5x+100, the constant term is 100 and the leading coefficient is -1.
The factors of 100 are: 1, 2, 4, 5, 10, 20, 25, 50, and 100. The factors of -1 are: 1 and -1.
Therefore, the possible integer roots of f(x) are: ±1, ±2, ±4, ±5, ±10, ±20, ±25, ±50, and ±100.
So, the candidate integer roots of [tex]f(x) = -x^5 + 22x^4 - 8x^3 - 8x^2 + 5x + 100[/tex] are: 1, -1, 2, -2, 4, -4, 5, -5, 10, -10, 20, -20, 25, -25, 50, -50, 100, and -100.
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PLEASE HELP :((( show BOTH distribution and FOIL to find the product of (3x - 2)and(2z + 6).
According to the given information product of (3x - 2) and (2z + 6) is 6xz + 18x - 4z - 12.
What is expression ?In mathematics, expressions are also combinations of constants, variables, operators, and function calls that represent mathematical operations or relationships.
According to given conditions:Let's start with distributing the first term of the first expression to both terms of the second expression, then distributing the second term of the first expression to both terms of the second expression:
(3x - 2)(2z + 6)
= 3x(2z + 6) - 2(2z + 6)
= 6xz + 18x - 4z - 12
Now, let's use the FOIL method to find the same product:
(3x - 2)(2z + 6)
= 3x(2z) + 3x(6) - 2(2z) - 2(6)
= 6xz + 18x - 4z - 12
As you can see, both methods result in the same product: 6xz + 18x - 4z - 12.
Therefore, according to the given information product of (3x - 2) and (2z + 6) is 6xz + 18x - 4z - 12.
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There are 5 yellow, 4 green and 3 black balls in a bag. All the 12 balls are drawn one by one and arranged in
a row. Find out the number of different arrangements possible.
There are 27720 different arrangements possible.
The number of different arrangements possible can be found using the formula for permutations with repetition:
n! / ([tex]p{1}[/tex]! * [tex]p{2}[/tex]! * ... *[tex]p_{k}[/tex]!)
where n is the total number of items, and [tex]p{1}[/tex], p2, ..., pk are the number of times each item is repeated.
In this case, n = 12 (the total number of balls), [tex]p{1}[/tex] = 5 (the number of yellow balls), [tex]p{2}[/tex] = 4 (the number of green balls), and [tex]p{3}[/tex] = 3 (the number of black balls).
So the number of different arrangements possible is:
12! / (5! * 4! * 3!)
= 479001600 / (120 * 24 * 6)
= 27720
Therefore, number of different arrangements possibleis 27720.
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FIrst Question: What proportion of U.S. residents receive a jury summons each year? A polling organization plans to survey a random sample of 500 U.S. residents to find out. Let P^
be the proportion of residents in the sample who received a jury summons in the previous 12 months. According to the National Center for State Courts, 15% of U.S. residents receive a jury summons each year. Suppose that this claim is true.
What sample size would be required to reduce the standard deviation of the sampling distribution to one-half the original value?
Second Question: A USA Today poll asked a random sample of 1012 U.S. adults what they do with the milk in their cereal bowl after they have eaten. Let p^
be the proportion of people in the sample who drink the cereal milk. A spokesman for the dairy industry claims that 70% of all U.S. adults drink the cereal milk. Suppose this claim is true.
What sample size would be required to reduce the standard deviation of the sampling distribution to one-half the original value?
a sample size of approximately 246 U.S. adults would be required to reduce the standard deviation of the sampling distribution to one-half the original value.
what is the standard deviation?The standard deviation is a metric that reveals how much variance from the mean there is, including spread, dispersion, and spread. A "typical" variation from the mean is shown by the standard deviation. Because it uses the data set's original units of measurement, it is a well-liked measure of variability.
from the question:
Assuming a 95% confidence interval and an error rate equal to half the initial standard deviation, or E = 0.5, the following results are obtained:
[tex]n = [(1.96 / 0.5σ) / 0.70(1 - 0.70)]^2[/tex]
The population's standard deviation must be known in order to solve for n; however, this information is not provided in the problem description. Based on prior research or experience, we may substitute a standard deviation of 0.05 into the following formula:
[tex]n = [(1.96 / 0.5(0.05)) / 0.70(1 - 0.70)]^2 ≈ 246[/tex]
Hence, to cut the standard deviation of the sampling distribution to half its initial value, a sample size of about 246 American adults would be needed.
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Linier ALgebra : Let W E R^nxn, s E R, and λ an eigenvalue of W. Prove that λ – s is an eigenvalue of W – sλ. (As usual, I = Inxn denotes the n x n identity matrix.)
$\lambda - s$ is an eigenvalue of W – s$\lambda$ and $\mathbf{v}$ is the corresponding eigenvector.
Linear Algebra: Let W $\in$ $\mathbb{R}^{n \times n}$, s $\in$ $\mathbb{R}$, and $\lambda$ an eigenvalue of W. To prove that $\lambda$ – s is an eigenvalue of W – s$\lambda$, we will use the definition of eigenvalues and eigenvectors:
An eigenvalue $\lambda$ of a square matrix A $\in$ $\mathbb{R}^{n \times n}$ is a scalar such that there exists a nonzero vector $\mathbf{v} \in \mathbb{R}^n$ for which the following equation holds:
A$\mathbf{v}$ = $\lambda \mathbf{v}$
Therefore, we can rearrange the equation to show that W – s$\lambda$ $\mathbf{v}$ = $(\lambda -s) \mathbf{v}$, which implies that $\lambda - s$ is an eigenvalue of W – s$\lambda$ and $\mathbf{v}$ is the corresponding eigenvector.
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Find the area of the trapezoid.
Answer:
area = (1/2) · (p + q) · h
Step-by-step explanation:
PLEASE HELP!!!
Tanner is spray painting an arrow on the side of a building to point to the entrance of his store. The can of gold spray paint he wants to use covers up to 12 square feet. Does Tanner have enough spray paint for his arrow?
Yes, Tanner has enough spray paint for his arrow.
What is an Area?
The amount of space occupied by a flat (2-D) surface or an object's shape is known as its area. A planar figure's area is the area that its perimeter encloses. The quantity of unit squares that completely encircle the surface of a closed figure is its area. Square measurements for area include cm2 and m2.
Given : paint available in can = 12 ft²
We know that the arrow is comprised of a triangle and a rectangle.
So, the area of given arrow = area of rectangle + area of triangle
Now, area of triangle = 1/2 ×base × height
= 1/2 × 3 × (6 - 5 1/3)
= 3/2 × ( 6 - 16/3)
= 3/2 × ( 18-16)/3
= 3/2 × 2/3
= 1 ft²
Similarly, area of rectangle = length × breadth
= 5 1/3 × 2
= 16/3 × 2
= 32/3 ft²
Hence, area of arrow = area of triangle +area of rectangle
= 1 + 32/3
= 35/3 ft²
= 11.67 ft²
So, he has sufficient paint to cover the arrow.
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The volume of a cylinder is 180 pi cubic inches and the radius of the cylinder is 3 inches. what is the height of the cylinder 
Using the formula of the cylinder we know that the height of the given cylinder is 6.36 inches.
What is a cylinder?One of the most fundamental curvilinear geometric shapes, a cylinder has historically been a three-dimensional solid.
It is regarded as a prism with a circle as its base in basic geometry.
In several contemporary fields of geometry and topology, a cylinder can alternatively be characterized as an infinitely curved surface.
So, the cylinder formula for the volume is:
V = πr²h
Now, substitute the values as follows:
V = πr²h
180 = 3.14*3²h
180 = 28.26h
180/28.26 = h
6.36 = h
Therefore, using the formula of the cylinder we know that the height of the given cylinder is 6.36 inches.
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Luis wants to buy a skateboard that usually sells for $79.28. All merchandise is discounted by 12%. What is the total cost of the skateboard If Luis has to pay a state sales tax of 8.25%. Round your intermediate calculations and answer to the nearest cent.
Answer:
The discount on the skateboard is 12% of its original price, so the discounted price is:
Discounted price = $79.28 - 0.12($79.28) = $69.78
Now we need to calculate the sales tax on the discounted price. The sales tax rate is 8.25%, so the amount of sales tax is:
Sales tax = 0.0825($69.78) = $5.76
Adding the discounted price and the sales tax, we get the total cost of the skateboard:
Total cost = $69.78 + $5.76 = $75.54
Therefore, the total cost of the skateboard, including the discount and sales tax, is $75.54.
Sarah was explaining her first step factoring, but made an error in her explanation.
Factor: 5x^2-4x-9
Sarah said, “I know that a=5, b=-4, and c=-9. Now, I need to find factors of -9 that add to -4.”
Explain the error in Sarah’s thought process.
The expression cannot be factored over integers since the roots are not integers.
The error in Sarah's thought process is that she is assuming that the factoring of a quadratic equation can always be done by finding two numbers that add up to the coefficient of the middle term (in this case, -4) and multiply to the coefficient of the quadratic term (in this case, 5). However, this method only works when the quadratic expression is factorable over the integers, which is not always the case.
In fact, the expression [tex]5x^2-4x-9[/tex] cannot be factored over the integers, which means that there are no two integers whose product is [tex]5*(-9)=-45[/tex] and whose sum is -4. To confirm this, we can use the quadratic formula (-b ± \sqrt{(b^2-4ac)} )/2a, which gives the roots of the quadratic equation [tex]ax^2+bx+c=0[/tex]. For this expression, [tex]a=5, b=-4[/tex], and c=-9, so the roots are:
[tex]x=[-(-4)±sqrt{(-4)^2-45(-9)}]/(2*5\leq)[/tex]
[tex]x=(4±sqrt(196))/10\\x=(4±14)/10\\x=1 or x=-9/5[/tex]
Since the roots are not integers, the expression cannot be factored over the integers.
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Line Handout #1f: Find the equation of the line parallel to 3x +y=-3 through the point (3,2)
The straight line that is parallel to 3x + y = -3 and passes through point (3,2) is y = -3x + 11.
We find the slope of the given line: 3x + y= -3 can be rearranged to y = -3x - 3, which is in the slope-intercept form y = mx + b. The slope of this line is -3.
Since parallel lines have the same slope, the slope of the line we are looking for is also -3.
We use the point-slope form of an equation, y - y₁ = m(x - x₁), where m is the slope and (x₁, y₁) is the point the line passes through.
Plug in the values we have: y - 2 = -3(x - 3)
We simplify the equation: y - 2 = -3x + 9 => y = -3x + 11
Therefore, the equation of the line parallel to 3x +y=-3 through the point (3,2) is y = -3x + 11.
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Use the suggested substitution to write the expression as a
trigonometric expression. Simplify your answer as much as possible.
Assume 0≤θ≤π2.
√1−x^2, x=cos(θ)
The expression √1−x^2, with the substitution x=cos(θ), can be simplified to sin(θ).
To write the expression as a trigonometric expression using the suggested substitution, we will replace x with cos(θ) in the expression. This gives us:
√1−cos^2(θ)
Now, we can use the Pythagorean identity, sin^2(θ) + cos^2(θ) = 1, to simplify the expression further. Rearranging the identity, we get:
sin^2(θ) = 1 - cos^2(θ)
Substituting this back into our original expression, we get:
√sin^2(θ)
Taking the square root of both sides, we get:
sin(θ)
Therefore, the expression √1−x^2, with the substitution x=cos(θ), can be simplified to sin(θ).
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Graph Y=2x on this chart thanks
Answer:
Step-by-step explanation:
The slope is 2 so rise / run = 2 / 1, or up two right one.
1. Let the point \( P \) be \( (-1,3) \) and the point \( Q \) be \( (3,7) \). Find the following. a. \( \mathbf{v}=\overrightarrow{P Q} \) b. \( \|\mathbf{v}\| \) c. \( \overrightarrow{P Q}+\overrigh
The answers are:
a. \( \mathbf{v}=\overrightarrow{P Q} = (4, 4) \)
b. \( \|\mathbf{v}\| = 4\sqrt{2} \)
c. \( \overrightarrow{P Q}+\overrightarrow{Q P} = (0, 0) \)
The given points are point \( P \) be \( (-1,3) \) and point \( Q \) be \( (3,7) \).
a. To find \( \mathbf{v}=\overrightarrow{P Q} \), we subtract the coordinates of point \( P \) from the coordinates of point \( Q \):
\( \mathbf{v}=\overrightarrow{P Q} = (3-(-1), 7-3) = (4, 4) \)
b. To find \( \|\mathbf{v}\| \), we use the distance formula:
\( \|\mathbf{v}\| = \sqrt{(4-0)^2 + (4-0)^2} = \sqrt{16 + 16} = \sqrt{32} = 4\sqrt{2} \)
c. To find \( \overrightarrow{P Q}+\overrightarrow{Q P} \), we add the coordinates of \( \overrightarrow{P Q} \) and \( \overrightarrow{Q P} \):
\( \overrightarrow{P Q}+\overrightarrow{Q P} = (4, 4) + (-4, -4) = (0, 0) \)
Therefore, the answers are:
a. \( \mathbf{v}=\overrightarrow{P Q} = (4, 4) \)
b. \( \|\mathbf{v}\| = 4\sqrt{2} \)
c. \( \overrightarrow{P Q}+\overrightarrow{Q P} = (0, 0) \)
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Determine where functions are continuous: (a) f(x)=(9x^(2)-4)/(3x-2) (b) f(x)=x2^(sinx) (c) f(x)=sin(1)/(2x) (d) f(x)={(x^(2)-1,x>3),(8,n=3),(2^(x),x<3):}
The functions are continuous at all points except where the denominator of a fraction is equal to zero. This is because division by zero is undefined and causes a discontinuity in the function.
(a) f(x)=(9x^(2)-4)/(3x-2): This function is continuous everywhere except where 3x-2=0, which is when x=2/3. Therefore, the function is continuous at all points except x=2/3.
(b) f(x)=x2^(sinx): This function is continuous everywhere because there are no denominators that could equal zero.
(c) f(x)=sin(1)/(2x): This function is continuous everywhere except where 2x=0, which is when x=0. Therefore, the function is continuous at all points except x=0.
(d) f(x)={(x^(2)-1,x>3),(8,n=3),(2^(x),x<3):} This function is continuous for x>3 and x<3, but there is a discontinuity at x=3 because the function is not defined for x=3. Therefore, the function is continuous at all points except x=3.
In conclusion, the functions are continuous at all points except where the denominator of a fraction is equal to zero, causing a discontinuity. The functions are continuous at all other points.
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The continuity of a function can be determined by examining the values of the function at different points in its domain. If the function is continuous at a point, it means that the limit of the function as it approaches that point from both the left and the right is equal to the value of the function at that point. A function is continuous over an interval if it is continuous at every point in that interval.
(a) The function f(x)=(9x^(2)-4)/(3x-2) is continuous everywhere except at x = 2/3, where the denominator is equal to zero and the function is undefined.
(b) The function f(x)=x2^(sinx) is continuous everywhere. The exponential function 2^(sinx) is continuous for all values of x, and the product of two continuous functions is also continuous.
(c) The function f(x)=sin(1)/(2x) is continuous everywhere except at x = 0, where the denominator is equal to zero and the function is undefined.
(d) The function f(x)={(x^(2)-1,x>3),(8,n=3),(2^(x),x<3):} is continuous for x > 3 and x < 3, but it is not continuous at x = 3, where there is a jump discontinuity from 8 to 2^(3).
In conclusion, the functions are continuous at the following points:
(a) f(x)=(9x^(2)-4)/(3x-2): continuous everywhere except at x = 2/3
(b) f(x)=x2^(sinx): continuous everywhere
(c) f(x)=sin(1)/(2x): continuous everywhere except at x = 0
(d) f(x)={(x^(2)-1,x>3),(8,n=3),(2^(x),x<3):}: continuous for x > 3 and x < 3, but not continuous at x = 3
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