Given the graph of f(x), determine the range of f−1(x).
Rational function with one piece decreasing from the left in quadrant 2 asymptotic to the line y equals 3 and passing through the point 0 comma 2 and asymptotic to the line x equals 2 and another piece decreasing from the left in quadrant 1 asymptotic to the line x equals 2 passing through the point 4 comma 4 asymptotic to the line y equals 3.
ℝ
(2, ∞)
(−∞, 2) ∪ (2, ∞)
(−∞, 3) ∪ (3, ∞)
Given the graph of f(x), the range of f⁻¹(x) is; (−∞, 2) ∪ (2, ∞)
How to find the range of the function?The range of a function is defined as the set that is composed by all the output values on a function. Thus, on the graph, the range of a function is composed by the values of y of the function.
For the inverse function, the input and the output are exchanged, and as such given the graph of a function, we can say that the range of the inverse function is given by the domain of the initial function, which is, the values of x of the graph of the original function.
From the description of the rational function, we are told that the asymptote is given as: x = 2
Thus, the domain of the graphed function, would be the same as the range of the inverse function, will be given by the interval:
(−∞, 2) ∪ (2, ∞).
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can I get help with this practice I'm having a lot of trouble with this can I get some help
The formula to find the slope is m = (y₂ - y₁)/(x₂ - x₁), hence option B is correct and the slope of the line will be -1/2.
What is slope?It is possible to determine a line's direction and steepness by looking at its slope. Finding the slope between lines inside a coordinate plane can aid in anticipating if the lines are perpendicular, parallel, or none at all without physically using a compass.
As per the data mentioned in the question,
1.
The formula to find the slope of the line between two points is,
m = (y₂ - y₁)/(x₂ - x₁)
2.
The given points are,
(-4, 3) and (12, -5)
So, the slope of the line that connects given points,
m = (-5 - 3)/(12+4)
m = -8/16
m = -1/2
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For the equation f(x) = 1.1 * (0.44) ^ x state the initial value C, the growth or decay factor a, and percent change R for each unit increase in x
c = (Type an integer or a decimal)
a = (Type an integer or a decimal)
R = % (Simplify your answer. Type an integer or a decimal)
The parameters of the exponential function in this problem are given as follows:
c = 1.1.a = 0.56.R = -56%.What is an exponential function?The standard format of an exponential function is given as follows:
y = c(1 - a)^t.
This is the case for a decaying exponential function, and the meaning of each parameter is given as follows:
c is the initial value, value assumed by y when t = 0.a is the decay rate.In this problem, the function is given as follows:
f(x) = 1.1(0.44)^x.
Hence the values for the parameters are given as follows:
c = 1.1, which is the initial value.a = 0.56, as 1 - a = 0.44 -> a = 0.56.Then the percent of change is of -56%, as it is a decaying exponential function with a = 0.56.
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What values of v and w make Δ P Q R ≅ Δ K J I ?
v=_____
w=_____
The values of v and w that will make ΔPQR congruent to ΔKJI are:
v = 12; w = 20.
What are Congruent Triangles?Based on the CPCTC, every corresponding sides and corresponding angles of two congruent triangles are equal to each other.
Therefore, given that ΔPQR ≅ ΔKJI
PR = KI, and QR = JI
Substitute the values:
5v = 6v - 12
5v - 6v = -12
-v = -12
v = 12
w - 1 = 3w - 41
w - 3w = 1 - 41
-2w = -40
w = -40/-2
w = 20
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when conducting a study comparing more than two groups of parametric (normally distributed) data, which of the following statistical tests should be used? g
Answer:
.
Step-by-step explanation:
sorry i hit my daily limit and i need more answers
An investor decides to invest some cash in an account paying12%
annual interest, and to put the rest in a stock fund that ends up earning 8%over the course of a year. The investor puts $1500more in the first account than in the stock fund, and at the end of the year finds the total interest from the two investments was $1880. How much money was invested at each of the two rates? Round to the nearest integer.
Let A be the amount of money invested at a 12% annual interest rate and B be the amount of money invested at a 8% annual interest rate.
We know that A = B + 1500, and the total interest earned by the two investments is $1880.
The interest earned by the investment at a 12% annual interest rate is 0.12 * A = 0.12A
The interest earned by the investment at a 8% annual interest rate is 0.08 * B = 0.08B
Therefore, we can write the following equation to represent the situation:
0.12A + 0.08B = 1880
Since A = B + 1500, we can substitute this into the equation to get:
0.12(B + 1500) + 0.08B = 1880
Solving for B, we get:
0.12B + 180 + 0.08B = 1880
Combining like terms, we get:
0.2B + 180 = 1880
Subtracting 180 from both sides, we get:
0.2B = 1700
Dividing both sides by 0.2, we get:
B = 8500
Since A = B + 1500, we can substitute this value into the equation to find the value of A:
A = 8500 + 1500 = 10000
Therefore, the investor invested $8,500 at an 8% annual interest rate and $10,000 at a 12% annual interest rate.
Answer:
Account A (12%)= $10,000
Account B (8% stock fund) = $8,500
Step-by-step explanation:
Annual Interest Formula
[tex]\large \text{$ \sf I=P\left(1+r\right)^{t} -P$}[/tex]
where:
I = Interest.P = Principal amount.r = Interest rate (in decimal form).t = Time (in years).Account A:
P = x + 1500r = 12% = 0.12t = 1 year[tex]\implies \sf Interest=(x+1500) (1+0.12)^1-(x+1500)[/tex]
[tex]\implies \sf Interest=(x+1500) (1+0.12)-(x+1500)[/tex]
[tex]\implies \sf Interest=(x+1500)+0.12(x+1500)-(x+1500)[/tex]
[tex]\implies \sf Interest=0.12(x+1500)[/tex]
[tex]\implies \sf Interest=0.12x+180[/tex]
Account B (stock fund):
P = xr = 8% = 0.08t = 1 year[tex]\implies \sf Interest=x (1+0.08)^1-x[/tex]
[tex]\implies \sf Interest=x (1+0.08)-x[/tex]
[tex]\implies \sf Interest=x+0.08x-x[/tex]
[tex]\implies \sf Interest=0.08x[/tex]
If the total interest from the two investments was $1880:
[tex]\implies \sf 0.12x+180+0.08x=1880[/tex]
[tex]\implies \sf 0.2x+180=1880[/tex]
[tex]\implies \sf 0.2x=1700[/tex]
[tex]\implies \sf x=8500[/tex]
Therefore, the money invested in each of the two accounts is:
Account A = $8,500 + $1,500 = $10,000Account B = $8,50010+2a<4 PLSS HELPPPP SOLVE FOR A ⚠️⚠️⚠️
[tex]10+2a < 4[/tex]
Simplify:
[tex]2a+10 < 4[/tex]
Subtract 10 from both sides:
[tex]2a+10-10 < 4-10[/tex]
[tex]2a < -6[/tex]
Divide both sides by 2:
[tex]\dfrac{2a}{2} < \dfrac{-6}{2}[/tex]
[tex]a < -3[/tex]
[tex]\fbox{Second Option}[/tex]
Write an equation for the absolute value function below:
The absolute value equation represented in the graph is
|x + 3|
What is absolute value?Without taking direction into account, absolute value describes how far away from zero a certain number is on the number line.
A number can never have a negative absolute value.
How to write the absolute value equationThe equation is written in the form below
|x|
The a transformation which involves a translation of 3 units to the left took place. the transformation rule for 3 units to the left is addition of 3, hence we have the equation
|x + 3|
We can therefore conclude that the absolute value equation of the function is |x + 3|
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The y-intercept can be described all except for one of them. Identify the statement that does not always represent the y-intercept.
Write an expression equivalent to (2/5)^4 using a positive exponent.
According to the given statement the positive exponent is [tex]\frac{16}{625}[/tex].
What do an exponent and an example mean?Exponents are a method of expressing enormous magnitudes in terms of their respective powers. The amount of times some number has already been multiplied in itself is the exponent, so to speak. For instance, the result of multiplying the number 6 on it's own four times is 6 6 6 6. This may be expressed as 64. Here, the exponent and base are 4 and 6, respectively.
The solution to powers and exponents.The exponent is the exact quantity of times that base would be compounded on its own. As a result, if two powers have the same foundation, they can be multiplied. Whenever two powers are multiplied, exponents are added. If necessary, we can also divide the abilities.
Briefing:= [tex]( \frac{2}{5} )^{4}[/tex]
= [tex]\frac{2}{5} \times\frac{2}{5} \times\frac{2}{5} \times\frac{2}{5} \\\\\frac{16}{625}[/tex]
Hence, the required exponent is [tex]\frac{16}{625}[/tex].
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Sheryl creates a scatter plot to analyze how an increase in the outside temperature above 52°F affects the sale of hot chocolate at her shop. The line shown on the graph is the line of best fit for the data. Which statements are true about the graph? Select all that apply.
1. The y-intercept represents the orders for hot chocolate on a day when the outside temperature was 52°F.
2. Sheryl gets 160 orders for hot chocolate on a day when the outside temperature is 52°F.
3. Sheryl gets 2 fewer orders for each degree increase above 52°F in the outside temperature.
4. Sheryl gets 16 fewer orders for each degree increase above 52°F in the outside temperature.
The true statements about the graph are 1, 2, and 3.
What is a line of fit?A line of the best fit is a straight line that reduces the distance between it and some data. The line of best fit is used to represent a relationship in a scatter plot with numerous data points.
Given:
The temperature above 52° F affects the sale of hot chocolate at her shop,
As you can see from the graph, the y-axis shows the orders for hot chocolate on a day when the outside temperature was 52° F.
From the graph,
at x = 0, y = 160
Thus, Sheryl gets 160 orders for hot chocolate on a day when the outside temperature is 52° F
At x = 0, y = 160 and at x = 16, y = 128
So the difference is 160-128 / 16 - 0 = 2
Thus, Sheryl gets 2 fewer orders for each degree increase above 52° F in the outside temperature.
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2. The width, w, of a rectangular garden is x-2. The area of the garden is ³-2x-4. What is
an expression for the length of the garden?
Ox²-2x-2
Ox²+2x-2
Ox²-2x+2
Ox²+2x+2
Consequently, the rectangular garden's length is [tex]x^{2} +2x+2[/tex]
option D is correct
what is area?Area is the entire amount of space occupied by a flat (2-D) surface or an object's shape. Surface area refers to an open surface or the perimeter of a three-dimensional object, whereas plane region or plane area refers to a shape or planar lamina.
given
the width(w) of the rectangular garden= [tex]x[/tex][tex]-2[/tex]
area of the rectangular garden = [tex]x^{3} -2x-4[/tex]
The area of a rectangle is,
area = length × width(w) ,
If l is the rectangle's length,
w is the width of the rectangle,
By substituting values,
[tex]x^{3}-2x-4[/tex] = ([tex]x-2[/tex]) * length ( l )
[tex]l=\frac{x^{3}-2x-4 }{x-2}[/tex] = [tex]x^{2} +2x+2[/tex] (by long division method)
Consequently, the rectangular garden's length is [tex]x^{2} +2x+2[/tex]
option D is correct
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*What two numbers have a sum of 352 and a difference of 104?
Answer:
124 and 228
Step-by-step explanation:
Two numbers will be presented as x and y
x + y = 352
x - y = 104
x + y + x - y = 352 + 104
2x = 456
Divide both side by 2
x = 228
x + y = 352
y = 352 - x
y = 352 - 228
y = 124
Select the correct answer.
Naomi is building a circuit board. The final microchip should have a surface area of 864 square millimeters. The height of the microchip can be a
maximum of 4 millimeters. What are the maximum dimensions of the microchip she can use?
The maximum dimension of a rectangular microchip she can use is 12 mm and 24 mm.
What is the maximum dimension of a rectangular microchip?A rectangular microchip, commonly known as an integrated circuit or chip, resembles a flat rectangle approximately small in size with a slew of wires designated by pins protruding from it.
The maximum dimension of a rectangular microchip can be determined by finding its length and width.
From the information given:
The surface area of the microchip = 864 mm²The height of the microchip = 4 mmThe surface area of the rectangle can be expressed by using the formula:
S = 2(lw + lh + wh)
where;
l = 2xw = xh = 4The surface area then becomes:
864 = 2(2x² + 8x + 4x)
Divide both sides by 2, and we have:
432 = 2x² + 12x
We can now have a quadratic expression as:
2x² + 12x - 432 = 0
x² + 6x - 216 = 0
using the factorization method;
(x + 18) (x - 12) = 0
x = - 18 or x = 12;
Since x cannot be negative, then x = 12. Therefore, width = 12 mm, and the length = 2(12) = 24 mm. Therefore, we can conclude that the maximum dimension of the microchip she can use is 12 mm and 24 mm.
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Answer:
I went with B. 9 mm and 18 mm PLATO
Step-by-step explanation: mad maths skills
Solve the proportion using cross products. Round to the nearest hundredth if necessary.
12Miles/27hours=4miles/Xhours
A. 28
B. 12
C. 9
D. 18
Answer:
C) x = 9
Step-by-step explanation:
[tex]\frac{12}{27}[/tex] = [tex]\frac{4}{x}[/tex]
[tex]\frac{12}{27}[/tex] ÷ [tex]\frac{3}{3}[/tex] = [tex]\frac{4}{9}[/tex] This is the fraction simplified
[tex]\frac{4}{9}[/tex] = [tex]\frac{4}{x}[/tex] Simplified, I think that it is easier to see that x = 9
In angle DEF, BD = 87 and WE = 38. Find BW, CW, and CE.
Answer:
Step-by-step explanation:
The value of y is unclear given the information provided. However, if BC intersects AD at E, then it stands to reason that y would be equal to the value of x. This is because, if BC intersects AD at E, then the two lines are parallel and therefore have the same value for y. Calculating the internal angles of triangle DEF, we can find the length of BW, CW and CE. Since BD=87 and WE=38, the remaining angle must be 55°. Therefore, using trigonometry and the known side lengths, we can calculate BW = 87 sin 55° = 77.5 , CW = 87cos55° = 48.5 and CE = 38sin55° = 33.2 . As you can see then, these are all easily calculable from the given data. In angle DEF, BD = 87 and WE = 38. Therefore, BW = 87 - 38 = 49, CW = 49 - 38 = 11, and CE = 11 - 1 = 10.
Please answer this question.
By applying vertical compression by a factor of 3, then vertical reflection and finally horizontal translation by 1 unit to the right we can transform into required graph.
What if transformation?A transformation is a mathematical operation that changes the position, size, or shape of a geometric object. In other words, it is a way to manipulate the object by moving, scaling, rotating, or reflecting it.
What is reflection?Reflection is a transformation that flips an object over a mirror line, or line of reflection. It creates a mirror image of the object on the other side of the line.
In mathematics, reflection is often represented by a matrix called the reflection matrix. This matrix encodes the rules for reflecting points in a particular direction.
To transform the graph of y = x^2 into y = -3(x+1)^2, we can apply the following transformations:
Vertical stretch or compression by a factor of 3: This transformation stretches or compresses the graph vertically by a factor of 3. To apply this transformation, we need to multiply the y-values of the points on the graph by 3.
Vertical reflection: This transformation reflects the graph across the x-axis. To apply this transformation, we need to multiply the y-values of the points on the graph by -1.
Horizontal translation by 1 unit to the right: This transformation shifts the graph 1 unit to the right. To apply this transformation, we need to add 1 to the x-values of the points on the graph.
Applying these transformations to the graph of y = x^2, we get the graph of y = -3(x+1)^2.
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The Federal Helium Reserve held about 16 billion
cubic feet of helium in 2010 and is being depleted by
about 2.1 billion cubic feet each year.
a. Give a linear equation for the remaining federal
helium reserves, R, in terms of t, the number of
years since 2010.
b. In 2015, what will the helium reserves be?
c. If the rate of depletion doesn’t change, in what year
will the Federal Helium Reserve be depleted
a) The linear equation for R(t) is given as follows: R(t) = -2.1t + 16.
b) The helium reserves in 2015 will be of: 5.5 billion.
c) The reserve will be depleted in the year of: 2018.
How to define the linear function?The linear function in this problem has the definition in slope-intercept format presented as follows:
R(t) = mt + b.
In which:
The slope m represents the yearly rate of change of the amount of the reserves.The intercept b represents the initial amount of the reserves.Considering the measures in billions, the parameters are given as follows:
m = -2.1, b = 16.
Then the equation is defined as follows:
H(t) = -2.1t + 16.
2015 is five years after 2010, hence the estimate is calculated as follows:
H(5) = -2.1(5) + 16 = 5.5 billion.
The reserve will be depleted in the year t + 2010 + 1, when R(t) = 0, hence:
-2.1t + 16 = 0
2.1t = 16
t = 16/2.1
t = 7.61 years.
Hence the reserves will be depleted during the year of 2018.
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NO LINKS!!
Find the average rate of change of the function from x1 to x2.
function f(x) = -9x + 4
x-values x1 = -5, x2 = 0
Answer:
-9
Step-by-step explanation:
Given function:
[tex]f(x)=-9x+4[/tex]
Given x-values:
x₁ = -5x₂ = 0Calculate the value of the function for the two given values of x:
[tex]\begin{aligned}\implies f(x_1)&=-9(-5)+4\\&=45+4\\&=49\end{aligned}[/tex]
[tex]\begin{aligned}\implies f(x_2)&=-9(0)+4\\&=0+4\\&=4\end{aligned}[/tex]
[tex]\boxed{\begin{minipage}{6.3 cm}\underline{Average rate of change of function $f(x)$}\\\\$\dfrac{f(b)-f(a)}{b-a}$\\\\over the interval $a \leq x \leq b$\\\end{minipage}}[/tex]
As -5 < 0:
a = x₁ = -5b = x₂ = 0Therefore:
[tex]\begin{aligned} \implies \textsf{Average rate of change}&=\dfrac{f(x_2)-f(x_1)}{x_2-x_1}\\\\&=\dfrac{4-49}{0-(-5)}\\\\&=\dfrac{-45}{5}\\\\&=-9\end{aligned}[/tex]
The velocity of a car was read from its speedometer at 10-second intervals and recorded in the table. Use the Midpoint Rule to estimate the distance traveled by the car. (Use the Midpoint Rule with 5 subintervals. (Round your answer to one decimal place.)
_________ mi
t(s) v(mi/h) t(s) v(mi/h)
0 0 60 56
10 34 70 55
20 52 80 50
30 54 90 49
40 55 100 45
50 51
The estimated distance traveled by the car is 73.5 mi.
To use the Midpoint Rule, we need to divide the time interval into subintervals and evaluate the average velocity over each subinterval.
The time interval is from 0 seconds to 100 seconds, so we will divide this interval into 5 subintervals of length 20 seconds each. The midpoint of each subinterval is the average time, which we can use to estimate the average velocity over that subinterval.
The midpoints of the subintervals are:
Subinterval 1: (0 + 20)/2 = 10 secondsSubinterval 2: (20 + 40)/2 = 30 secondsSubinterval 3: (40 + 60)/2 = 50 secondsSubinterval 4: (60 + 80)/2 = 70 secondsSubinterval 5: (80 + 100)/2 = 90 secondsUsing the data from the table, we can calculate the average velocity over each subinterval:
Subinterval 1: (0 + 34)/2 = 17 mi/hSubinterval 2: (52 + 54)/2 = 53 mi/hSubinterval 3: (55 + 51)/2 = 53 mi/hSubinterval 4: (50 + 49)/2 = 49.5 mi/hSubinterval 5: (49 + 45)/2 = 47 mi/hWe can now use the Midpoint Rule to estimate the distance traveled by the car:
distance = (20/6) * (17 + 53 + 53 + 49.5 + 47)
= (20/6) * 220.5
= <<20/6*220.5=73.5>>73.5 mi
So, the estimated distance traveled by the car is 73.5 mi.
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Jen drew a scale drawing of a summer camp
Answer:
This is a statement. Not a question.
Step-by-step explanation:
to test whether or not there is a difference between treatments a, b, and c, a sample of 12 observations has been randomly assigned to the 3 treatments. you are given the results below.
The required null hypothesis of the given observation is μ1 = μ2 = μ3.
What is the null hypothesis?The null hypothesis is a typical mathematical theory that asserts that there is no statistical relationship and significance between two sets of observed data and measured phenomena for each set of specified, single observable variables.
The null hypothesis can be evaluated to determine whether or not there is a relationship between two measured phenomena, which makes it valuable.
It can let the user know if the outcomes are the product of random chance or deliberate manipulation of a phenomenon.
The null hypothesis, often known as "H-nought," "H-null," or "H-zero," is written as H0 to distinguish it from other hypotheses.
The null hypothesis of the given observation:
μ1 = μ2 = μ3
Therefore, the required null hypothesis of the given observation is μ1 = μ2 = μ3.
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Complete question:
To test whether or not there is a difference between treatments A, B, and C, a sample of 12 observations has been randomly assigned to the 3 treatments. You are given the results below.
Treatment
Observations
A
20
30
25
33
B
22
26
20
28
C
40
30
28
22
The null hypothesis for this ANOVA problem is?
-16(d + 1) = -21 solve for d
Step-by-step explanation:
To solve the equation -16(d + 1) = -21 for d, we can first use the distributive property to expand the expression -16(d + 1):
-16(d + 1) = -16d - 16
Then, we can set the two sides of the equation equal to each other and solve for d:
-16d - 16 = -21
-16d = -5
d = 5/16
Therefore, the value of d that satisfies the equation -16(d + 1) = -21 is d = 5/16.
Find an equation for (-8,1) parallel to x-4y=4
Answer:
y = 1/4x + 1
Step-by-step explanation:
Help for Pre Calc please
The correct statement regarding the inverse function of f(x) = 4x^4 is given as follows:
[tex]f^{-1}(x) = \pm \left(\frac{x}{4}\right)^{\frac{1}{4}}[/tex]; f^(-1)(x) is not a function.
How to obtain the inverse function?The function in this problem is defined as follows:
f(x) = 4x^4.
To obtain the inverse of a function y = f(x), first the variables y and x are exchanged, as follows:
x = 4y^4.
Isolating the variable y, we have that:
y^4 = (x/4).
The inverse operation of the fourth power is the fourth root, hence:
[tex]y = \pm \sqrt[4]{\frac{x}{4}}[/tex]
[tex]f^{-1}(x) = \pm \sqrt[4]{\frac{x}{4}}[/tex]
[tex]f^{-1}(x) = \pm \left(\frac{x}{4}\right)^{\frac{1}{4}}[/tex]
The plus/minus symbol means that for each input of x, the inverse function gives two outputs, meaning that there are multiple outputs mapped to each input, and thus the inverse is not a function.
This means that the second statement is correct.
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A dance teacher divides 5 dance classes into 6 equal groups. Each dance class has 18 students. How many dance students are in each group
First, multiply [tex]18*5 = 90[/tex]
Then, divide [tex]90 / 6 = 15[/tex]
Therefore, each group has 15 students.
Convergence or Divergence? Prove using a test.
Emma wrote 8.4x+6.3x+12.6 as an equivalent expression for 4.2(2x+1.5x+3). She said that her equivalent expression is simplified. Do you agree? Explain.
Yes, I agree that this equivalent expression has been simplified.
The solutions to these issues are as follows
Reorder and gather like terms: (8.4x +6.3 x) +12.6
Collect coefficients for the like terms: (8.4 +6.3) × x +12.6
Calculate the sum or difference: 14.7 x +12.6
Answer : 14.7 x + 12.6
What exactly is equivalent expression, and how do we recognize it?
Equivalent expressions are expressions that work the same even though they look different. If two algebraic expressions are equivalent, then the two expressions have the same value when we plug in the same value(s) for the variable(s).
Distribute any coefficients.Combine any like terms on each side of the equation.Arrange the terms in the same order, usually x-term before constants.If all of the terms in the two expressions are identical, then the two expressions are equivalent.To learn more about coefficients refer to:
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if s is the part of the sphere that lies above the cone find the following: 1. s can be parametrized by a vector equation
If surface , S is the part of the sphere, x² + y²+ z² =1 ,lies above the cone, x² + y² = z
a)| ⃗rᵩ × ⃗rθ| = ⟨sin²φ cosθ,sin²φsinθ, sinφcosφ⟩
b)∫∫ z² ds = ₀∫ˣ₀∫ʸcos²φ dφdθ,
S
where x = 2π and y = π/4
What is Parametrizing Surfaces?
A surface in space given in Cartesian coordinates as f(x,y,z) = 0, can be parametrized as a vector function with two parameters,
r(u,v)= ⟨r₁(u,v) , r₂(u,v) , r₃(u,v)⟩ , (u,v)∈R²
We have, S is a part of Sphere , x² + y²+ z² =1 , lies above the cone, x² + y² = z and surface S parametrized by following vector equation ,
r(θ, φ) = ⟨sinφ cosθ , sinφsinθ,cosφ⟩ --(1)
⃗rᵩ = ∂r/∂φ =⟨cos φ cosθ,cos φ sinθ,-sin φ⟩
⃗rθ= ∂r/∂θ=⟨- sinφ sinθ , sinφ cosθ ,0⟩
| ⃗rᵩ × ⃗rθ| =| i j k |
|cosφcosθ cosφ sinθ -sinφ |
|- sinφsinθ sinφ cosθ 0 |
= i( 0 + sinφ cosθsinφ) -j(0- (- sinφsinθ)(-sinφ)) + k(sinφ cosθ cosφcosθ - (-sinφ sinθ ) cosφ sinθ)
= (sin²φcosθ )i + (sin²φsinθ)j + (sinφcosφ(sin²θ+cos²θ))k
= sin²φcosθ )i + (sin²φsinθ)j +(sinφcosφ)k
| ⃗rᵩ× ⃗rθ|=⟨sin²φcosθ ,sin²φsinθ,sinφcosφ⟩
b) from part (a) we get,
x = sinφ cosθ
y = sinφsinθ
z = cosφ
are spherical coordinates of S where, 0≤φ≤π/4 and 0≤ θ≤ 2π.
so, ∫∫ z² ds = ₀∫ˣ₀∫ʸcos²φ dφ dθ ,
S
where x = 2π and y = π/4
So,Surface integeral is equals to ₀∫ˣ₀∫ʸcos²φ dφ dθ.
To learn more about Parametrizing Surfaces, refer:
https://brainly.com/question/29772771
#SPJ4
Complete question:
If Sis the part of the sphere x2 + y2 + x2 = 1 that lies above the cone z= x2 + y2, find the following: a)S can be parametrized by a vector equation
r(θ, φ) = ⟨sinφ cosθ , sinφsinθ , cosφ ⟩ then a)| ⃗rᵩ × ⃗rθ| = ?
b) ∫∫ z² ds = ?
S
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Use the properties to expand the expression as a sum, difference, and/or constant multiple of logarithms. (Assume the variable is positive)
1. ln((xy)^5)
2. ln (seventh root(t))
3. ln (√x²/y^5)
Properties of log:
[tex]log\ ab=log\ a + log\ b[/tex][tex]log\ a/b =log\ a-log\ b[/tex][tex]log\ a^b=b\ log\ a[/tex]Evaluate given using the properties above:
Q1
[tex]ln((xy)^5) = 5ln(xy) = 5(ln x + ln y) = 5\ ln\ x + 5\ ln\ y[/tex]Q2
[tex]ln(\sqrt[7]{t} ) = ln(t^{1/7})=1/7\ ln\ t[/tex]Q3
[tex]ln(\sqrt{x^2}/y^5)=ln(x/y^5)= ln\ x - ln\ y^5 = ln\ x - 5\ ln\ y[/tex]Answer:
[tex]\textsf{1.} \quad 5 \ln x + 5 \ln y[/tex]
[tex]\textsf{2.} \quad \dfrac{1}{7} \ln t[/tex]
[tex]\textsf{3.} \quad \ln x - \dfrac{5}{2}\ln y \;\; \;\;\textsf{or} \;\;\;\; \ln x - 5\ln y[/tex]
Step-by-step explanation:
[tex]\boxed{\begin{minipage}{6 cm}\underline{Natural log laws}\\\\Product law: \;$\ln xy=\ln x + \ln y$\\\\Quotient law: $\ln \left(\dfrac{x}{y}\right) = \ln x - \ln y$\\\\Power law: \;\;\;\;$\ln x^n=n \ln x$\\\end{minipage}}[/tex]
Question 1Apply the power law followed by the product law:
[tex]\begin{aligned}\ln (xy)^5 & = 5 \ln (xy)\\ & = 5\left( \ln x + \ln y \right) \\ & = 5 \ln x + 5 \ln y\end{aligned}[/tex]
Question 2[tex]\textsf{Apply the exponent rule} \quad \sqrt[n]{a}=a^{\frac{1}{n}}[/tex]
then apply the power law:
[tex]\begin{aligned}\ln \sqrt[7]{t} & = \ln t^{\frac{1}{7}} \\ & = \dfrac{1}{7} \ln t\end{aligned}[/tex]
Question 3It is not completely clear where the square root sign begins and ends, so I have provided answers for both permutations:
[tex]\begin{aligned}\ln \left(\sqrt{\dfrac{x^2}{y^5}}\right) & = \ln \left(\dfrac{\sqrt{x^2}}{\sqrt{y^5}}\right) \\\\& = \ln \left(\dfrac{x}{y^{\frac{5}{2}}}\right) \\\\&=\ln x - \ln y^{\frac{5}{2}}\\\\&=\ln x - \dfrac{5}{2}\ln y\end{aligned}[/tex]
[tex]\begin{aligned}\ln \left(\dfrac{\sqrt{x^2}}{y^5}\right)& = \ln \left(\dfrac{x}{y^5}\right) \\\\&=\ln x - \ln y^5\\\\&=\ln x - 5\ln y\end{aligned}[/tex]