(a) Find the maximum rate of change of the function f(x, y, z) -xy t yz- xz at the point Po (3, -1,4) (b) Find the unit vector direction in which the greatest rate of change occurs. (Your instructors prefer angle bracket notation < > for vectors.)

Answer:

- 3, - 2, 0, 5

Step-by-step explanation:

1.4 (d - 2) - 0.2d ≤ 3.2 ← distribute parenthesis and simplify left side

1.4d - 2.8 - 0.2d ≤ 3.2

1.2d - 2.8 ≤ 3.2 ( add 2.8 to both sides )

1.2d ≤ 6 ( divide both sides by 1.2 )

d ≤ 5

the only value less than or equal to 5 are

- 3, - 2, 0 ,5

The surface area of the sphere of radius of 7cm is 616 square centimeters.

We know that the surface area of a sphere of radius r is given by the formula:

S = 4*(22/7)*r²

Here we want to find the surface area of a sphere whose radius is r = 7 cm.

Replacing it in the formula above, we will get:

S = 4*(22/7)*7²

S = 4*22*7

S = 616

And the units are square centimeters, so the correct option is C.

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Since the limit is a finite value, the integral is convergent. Furthermore, the value of the convergent integral is 21e^(-1), which is approximately 7.713.

whether the integral is convergent or divergent.

First, let's rewrite the integral using proper notation:

∫(1 to ∞) 21e^(-x) dx

Now, to determine if the integral is convergent or divergent, we'll perform the following steps:

1. Apply the limit as the upper bound approaches infinity:

lim(b→∞) ∫(1 to b) 21e^(-x) dx

2. Evaluate the improper integral using the antiderivative:

F(x) = -21e^(-x)

Now, we need to find the limit as b approaches infinity:

lim(b→∞) (F(b) - F(1))

3. Calculate the limit:

lim(b→∞) (-21e^(-b) - (-21e^(-1)))

As b approaches infinity, e^(-b) approaches 0. Therefore, the limit is:

-(-21e^(-1)) = 21e^(-1)

Since the limit is a finite value, the integral is convergent. Furthermore, the value of the convergent integral is 21e^(-1), which is approximately 7.713.

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The missing statement in the given argument is a premise.

Premise: Carrie Underwood has won several Grammy awards.

Conclusion: Carrie Underwood is a talented singer.

Revised argument:

Premise: Winning several Grammy awards is an indication of talent.

Premise: Carrie Underwood has won several Grammy awards.

Conclusion: Therefore, Carrie Underwood is a talented singer.

The given statement is an example of an enthymeme, which is an argument with an implied premise or conclusion. In this case, the implied premise is that winning several Grammy awards is an indication of talent.

The argument is based on the assumption that the audience agrees with this premise, and therefore, the conclusion that Carrie Underwood is a talented singer follows logically.

However, it is important to note that the relationship between winning Grammy awards and talent is not necessarily causative, as other factors such as marketing, popularity, and the preferences of the voting committee can also influence the outcome.

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A formula that can be used to find the value of x MU² - UM * MV - MV * TV = x² * (MU - UM). The value of x is x ≈ ±3.55.

Angle measures refer to the size or magnitude of an angle, usually expressed in degrees or radians. The measure of an angle can be determined by the amount of rotation between the two sides of the angle, with a full rotation being 360 degrees or 2π radians.

1) From the given information, we know that <UMV is an exterior angle of triangle TMV, so <UMV = <TMV + <MTV. Substituting the given angle measures, we get:

m<UMV = x² + 31

Also, by the intersecting secants theorem, we have:

MU * MW = MV * MT

Substituting the given segment lengths, we get:

(MU + UW) * (MU - UW) = MV * TV

Simplifying this equation, we get:

MU² - UW² = MV * TV - UW * MU

Substituting the given angle measure and simplifying further, we get:

MU² - UW² = MV * TV - UW * MU

MU² - MW² - UW² = -UW * MU

(MU - MW) * (MU + MW) - UW² = -UW * MU

(MU + MW) = (UW² - MU * UW) / (MU - UW)

Substituting the given angle measure, we get:

tan(145) = UW / UM

Simplifying this equation, we get:

UW = UM * tan(145)

Substituting this expression for UW, we get:

MU + UM * tan(145) = (UM² - MU * UM) / (MU - UM)

Simplifying further, we get:

MU² - UM * MV - MV * TV = x² * (MU - UM)

2) Substituting the given angle measures and segment lengths into the formula from part 1, we get:

MU² - UM * MV - MV * TV = x² * (MU - UM)

MU² - 2 * MU * MV * sin(31) - MV * sin(x²) = x² * (MU - UM)

Substituting the expression for UW from part 1, we get:

MU + UM * tan(145) = (UM² - MU * UM) / (MU - UM)

MU² - MU * UM - UM * tan(145) = -MU * (MU - UM)

MU * (MU - UM + UM * tan(145)) = MU² - UM * tan(145)

MU = (UM * tan(145)) / (1 - tan(145))

Substituting this expression for MU, we get:

(UM * tan(145))² / (1 - tan(145)) + UM * MV * sin(31) - MV * sin(x²) = x² * ((UM * tan(145)) / (1 - tan(145)) - UM)

Simplifying this equation and solving for x, we get:

x ≈ ±3.55

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A formula that can be used to find the value of x MU² - UM * MV - MV * TV = x² * (MU - UM). The value of x is x ≈ ±3.55.

Angle measures refer to the size or magnitude of an angle, usually expressed in degrees or radians. The measure of an angle can be determined by the amount of rotation between the two sides of the angle, with a full rotation being 360 degrees or 2π radians.

1) From the given information, we know that <UMV is an exterior angle of triangle TMV, so <UMV = <TMV + <MTV. Substituting the given angle measures, we get:

m<UMV = x² + 31

Also, by the intersecting secants theorem, we have:

MU * MW = MV * MT

Substituting the given segment lengths, we get:

(MU + UW) * (MU - UW) = MV * TV

Simplifying this equation, we get:

MU² - UW² = MV * TV - UW * MU

Substituting the given angle measure and simplifying further, we get:

MU² - UW² = MV * TV - UW * MU

MU² - MW² - UW² = -UW * MU

(MU - MW) * (MU + MW) - UW² = -UW * MU

(MU + MW) = (UW² - MU * UW) / (MU - UW)

Substituting the given angle measure, we get:

tan(145) = UW / UM

Simplifying this equation, we get:

UW = UM * tan(145)

Substituting this expression for UW, we get:

MU + UM * tan(145) = (UM² - MU * UM) / (MU - UM)

Simplifying further, we get:

MU² - UM * MV - MV * TV = x² * (MU - UM)

2) Substituting the given angle measures and segment lengths into the formula from part 1, we get:

MU² - UM * MV - MV * TV = x² * (MU - UM)

MU² - 2 * MU * MV * sin(31) - MV * sin(x²) = x² * (MU - UM)

Substituting the expression for UW from part 1, we get:

MU + UM * tan(145) = (UM² - MU * UM) / (MU - UM)

MU² - MU * UM - UM * tan(145) = -MU * (MU - UM)

MU * (MU - UM + UM * tan(145)) = MU² - UM * tan(145)

MU = (UM * tan(145)) / (1 - tan(145))

Substituting this expression for MU, we get:

(UM * tan(145))² / (1 - tan(145)) + UM * MV * sin(31) - MV * sin(x²) = x² * ((UM * tan(145)) / (1 - tan(145)) - UM)

Simplifying this equation and solving for x, we get:

x ≈ ±3.55

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The probability of at least one male getting an earring is approximately 1 - 0.0173 ≈ 0.9827 or 98.27%.

To find the probability of at least one male getting an earring, we'll use the complementary probability.

There are 12 crackers with 4 containing earrings, so the probability of a cracker not having earrings is 2/3.

With 6 males at the party, the probability of all males not getting earrings is (2/3)^6 ≈ 0.0173.

Therefore, the probability of at least one male getting an earring is approximately 1 - 0.0173 ≈ 0.9827 or 98.27%.

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The covariance of the random variables X and Y is 1/120.

Exercise 3.49 on page 106 states:

"Suppose that the joint probability density function of X and Y is given by f(x,y) = 3x, 0 ≤ y ≤ x ≤ 1, 0 elsewhere. Find E[X], E[Y], and cov(X,Y)."

To find the covariance of X and Y, we first need to find the expected values of X and Y:

E[X] = ∫∫ x f(x,y) dy dx = ∫0¹ ∫y¹ 3[tex]x^2[/tex] dy dx = ∫0¹ [tex]x^3[/tex] dx = 1/4

E[Y] = ∫∫ y f(x,y) dy dx = ∫0¹ ∫y¹ 3xy dy dx = ∫0¹ [tex]x^2[/tex]/2 dx = 1/6

Next, we need to use the formula for covariance:

cov(X,Y) = E[XY] - E[X]E[Y]

To find E[XY], we integrate the joint probability density function multiplied by XY:

E[XY] = ∫∫ xy f(x,y) dy dx = ∫0¹ ∫y¹ 3x^2y dy dx = ∫0¹ [tex]x^4[/tex]/2 dx = 1/10

Putting it all together, we have:

cov(X,Y) = E[XY] - E[X]E[Y] = 1/10 - (1/4)(1/6) = 1/120

Therefore, the covariance of the random variables X and Y is 1/120.

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The differential of f(x,y)= √(x³ + y²) at the point (1,2) is (3/2)dx + (2/√5)dy.

To find the differential of f(x,y)= √(x³ + y²) at the point (1,2), we first need to find the partial derivatives of f with respect to x and y:

∂f/∂x = (3x² / (2 √(x³ + y²))
∂f/∂y = (y / √(x³ + y²))

Then, we can evaluate these partial derivatives at the point (1,2):

∂f/∂x (1,2) = (3(1)²) / (2 √(1³ + 2²)) = 3/2
∂f/∂y (1,2) = (2) / √(1³ + 2²) = 2/√5

Finally, we can use the formula for the differential of f:

df = (∂f/∂x)dx + (∂f/∂y)dy

Substituting the values we found, we get:

df = (3/2)dx + (2/√5)dy

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Five  5 critical values for the daily temperatures in Tucson, Arizona, in December: are- 49, 56.5, 58, 59.5 and 67.

The graphical tool used to illustrate the data is the box and whisker plot. For the data to be plotted, some summary statistics are required. The first quartile, median, third quartile, and maximum are those values. It is applied to determine if an outlier exists in the data.

Given data for the Average daily temperatures in Tucson, Arizona.

58, 60, 59, 50, 67, 53, 57, 62, 58, 57, 56, 63, 57, 53, 58, 58, 59, 49, 64, 58

Arrange is the ascending order;

49, 50, 53, 53, 56, 57, 57, 57, 58, 58, 58, 58, 58, 59, 59, 60, 62, 63, 64, 67,

n = 20

n/2 = 10 th term - 58

(n + 1)/2 = 11th term - 58

The median Q2 - (n/2 + (n+1)/2) /2

(58+58) / 2 = 58

Now, consider the middle numbers before the median for lower quartile :Q1 - (5th + 6th)/2

(56 + 57) / 2 = 56.5

Consider middle numbers after the median for upper quartile:

Q3 - (15th +16th)/2

(59 + 60) / 2 = 59.5

Five  5 critical values are-

49, 56.5, 58, 59.5 and 67.

Thus, the  Box and whisker plot for the all four estimated quratiles are formed.

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Rounding this to the nearest hundredth gives an expected count of 120 people with profile B.

What is an expected count?

Expected count is a term used in statistical analysis, particularly in the context of contingency tables and hypothesis testing. It refers to the number of observations that would be expected in a particular category of a contingency table if there was no association between the variables being examined.

Expected counts are calculated by multiplying the marginal totals of a contingency table to obtain the total number of observations that would be expected under the null hypothesis. Expected counts are then compared to the observed counts in the contingency table to assess whether there is a significant association between the variables being examined.

To find the expected count of people with profile B, we need to multiply the total sample size (300) by the expected percentage of people with profile B (40% or 0.4):

Expected count of B = 0.4 x 300 = 120

Rounding this to the nearest hundredth gives an expected count of 120 people with profile B.

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we have shown that if f(n) = f(n2), then n = n2, which means that f is one-to-one.

The question asks us to define a function f from the set of integers to itself, where f(n) = 2 - 3n for each integer n. We then need to prove that this function is one-to-one.

To prove that f is one-to-one, we need to show that for any two integers n and n2, if f(n) = f(n2), then n = n2. Here's how we can do that:

Proof:

1. Suppose n and n2 are any integers such that f(n) = f(n2).

2. Substituting from the definition of f gives us:

2 - 3n = 2 - 3n2

3. Simplifying this equation, we get:

-3n = -3n2

4. Dividing both sides by -3, we get:

n = n2

5. Therefore, we have shown that if f(n) = f(n2), then n = n2, which means that f is one-to-one.

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The given series is convergent, and its sum is approximately 0.1524.

To determine whether the series ∑n=1∞ 1/(9+[tex]e^{(-n)}[/tex]) is convergent or divergent, we can use the comparison test with the series 1/n.

Since for all n, [tex]e^{(-n)}[/tex] > 0, we have [tex]9 + e^{(-n)}[/tex] > 9, and so [tex]1/(9+e^{(-n)})[/tex] < 1/9.

Now, we can compare the given series with the series ∑n=1∞ 1/9, which is a convergent p-series with p=1.

By the comparison test, since the terms of the given series are smaller than those of the convergent series 1/9, the given series must also converge.

To find the sum of the series, we can use the formula for the sum of an infinite geometric series:

S = a/(1-r)

where a is the first term and r is the common ratio. In this case, the first term is 1/10 (since [tex]e^{(-1)}[/tex] is very small compared to 9.

We can approximate [tex]9+e^{(-n)}[/tex] as 9 for large n), and the common ratio is [tex]e^{(-1)} < 1[/tex]. Therefore, the sum of the series is:

S = (1/10)/(1 - [tex]e^{(-1)}[/tex]) = (1/10)/(1 - 0.3679) ≈ 0.1524

Therefore, the given series is convergent, and its sum is approximately 0.1524.

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Answer:

3.2 miles

Step-by-step explanation:

[tex]\frac{5684.106yds}{1}[/tex] · [tex]\frac{3ft}{1yd}[/tex] · [tex]\frac{1mile}{5280ft}[/tex] You can cross cancel words just like numbers.  Cross cancel the words: yards and feet.  That will leave you with just miles

[tex]\frac{5684.106}{1 }[/tex] ·[tex]\frac{3}{1}[/tex] · [tex]\frac{1mile}{5280}[/tex]

[tex]\frac{17052.318}{5280}[/tex]

3.22960568182

This rounded to the nearest tenth would be: 3.2

Helping in the name of Jesus.

The vector Xp=(1 1)e^t+(1 -1)te^t is a particular solution of the given system.

To verify that Xp=(1 1)e^t+(1 -1)te^t is a particular solution of the given system, we need to substitute it into the given system and check if it satisfies the equations.

The given system is:

X'=(2 1 3 4)X-(1 7)e^t

Substituting Xp=(1 1)e^t+(1 -1)te^t into the above system, we get:

Xp'=(2 1 3 4)Xp-(1 7)e^t

Differentiating Xp with respect to t, we get:

Xp'=(1 1)e^t+(1 -1)e^t+(1 -1)te^t

Substituting the above expression into the system, we get:

(1 1)e^t+(1 -1)e^t+(1 -1)te^t=(2 1 3 4)((1 1)e^t+(1 -1)te^t)-(1 7)e^t

Simplifying, we get:

(1 1)e^t+(1 -1)e^t+(1 -1)te^t=(2e^t+2te^t+3e^t-3te^t)-(1 7)e^t

Combining like terms, we get:

(1 1)e^t+(1 -1)e^t+(1 -1)te^t=(2e^t+2te^t+3e^t-3te^t)-(1 7)e^t

(1 1)e^t+(1 -1)e^t+(1 -1)te^t=(2e^t+2te^t+3e^t-3te^t-1e^t-7e^t)

(1 1)e^t+(1 -1)e^t+(1 -1)te^t=(4e^t-3te^t)

Comparing the left-hand side and the right-hand side, we can see that they are equal, which means Xp=(1 1)e^t+(1 -1)te^t satisfies the given system of equations. Therefore, Xp=(1 1)e^t+(1 -1)te^t is a particular solution of the given system.

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p(f | e) = p(e | f) * p(f) / p(e) = (1/5) / (1/3) = 3/5

Therefore, p(f | e) = 3/5.

We can use Bayes' theorem to find p(f | e):

p(f | e) = p(e | f) * p(f) / p(e)

We know that p(e) = 1/3 and p(f) = 1/2. To find p(e | f), we can use the conditional probability formula:

p(e | f) = p(e ∩ f) / p(f)

We are given that p(e | f) = 2/5, so we can rearrange the formula to get:

p(e ∩ f) = p(e | f) * p(f) = (2/5) * (1/2) = 1/5

Now we have all the information we need to apply Bayes' theorem:

p(f | e) = p(e | f) * p(f) / p(e) = (1/5) / (1/3) = 3/5

Therefore, p(f | e) = 3/5.

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Answer: 7.35 inches

Step-by-step explanation:

The formula for the surface area of a cylinder is 2πrh + 2πr^2, where r is the radius and h is the height of the cylinder.

Given that the diameter of the tub is 10 inches, the radius (r) is half of that, which is 5 inches.

So, the equation for the surface area of the cylinder can be written as:

2π(5)(h) + 2π(5)^2 = 387.79

Simplifying the equation gives:

10πh + 50π = 387.79

Dividing both sides by 10π gives:

h + 5 = 12.34

Subtracting 5 from both sides gives:

h = 7.34

Therefore, the height of the tub is 7.35 inches (rounded to two decimal places).

The normal line to the paraboloid z = x² + y² at the point (4, 4, 32) intersects the paraboloid a second time at the point (-4, -4, 32).

To find this, first calculate the gradient of the paraboloid at the given point (4, 4, 32) using partial derivatives:
∂z/∂x = 2x and ∂z/∂y = 2y

At the point (4, 4, 32), the gradient is (8, 8). Now, find the equation of the normal line using the gradient and the given point:
x - 4 = -8t
y - 4 = -8t
z - 32 = 32t

Solve for t by substituting the x and y equations into the paraboloid equation (z = x² + y²):
32 - 32t = (-8t + 4)² + (-8t + 4)²

Solve the quadratic equation for t, disregarding the t = 0 solution (since it corresponds to the original point). The other solution gives the second intersection point (-4, -4, 32).

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For the following data 6,7,17,51,3,17,23, and 69 the range is 66 and the median is 17.

We need to find the range and median of the given dataset: 3, 6, 7, 17, 17, 23, 51, 69.

Range: The range is the difference between the largest and smallest values in the dataset. To find it, first identify the largest and smallest numbers:

Largest number: 69
Smallest number: 3

Next, subtract the smallest number from the largest number:

Range = 69 - 3 = 66

Median: The median is the middle value in an ordered dataset. Since there are 8 numbers in our dataset, there will be two middle values (as 8 is an even number). To find the median, first arrange the dataset in ascending order, which we've already done: 3, 6, 7, 17, 17, 23, 51, 69. Now, identify the two middle values:

Middle values: 17 and 17

To find the median, calculate the average of these two middle values:

Median = (17 + 17) / 2 = 34 / 2 = 17

So, for the given dataset, the range is 66 and the median is 17. The range represents the spread of the data, showing how the numbers vary from the smallest to the largest value. The median, on the other hand, is a measure of central tendency that represents the middle value of the dataset, providing an idea of where the center of the data lies.

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A particular solution of the given differential equation y'' + 3y' + 4y = 8x + 2 is: y_p = 2x - 1 (option c).

The particular solution of the given differential equation can be found by using the method of undetermined coefficients. We assume that the particular solution has the same form as the right-hand side of the equation, i.e., y_p = Ax + B, where A and B are constants. We then substitute this into the differential equation and solve for A and B.

y" + 3y' + 4y = 8x + 2

y_p = Ax + B
y'_p = A
y"_p = 0

Substituting these into the equation, we get:

0 + 3A + 4Ax + 4B = 8x + 2

Comparing the coefficients of x and the constant term, we get:

4A = 8  =>  A = 2
4B = 2  =>  B = 1/2

Therefore, the particular solution is y_p = 2x + 1, which is option a.

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The coefficient of x^9 in the expansion of (x+1)^14 + x^3(x+2)^15 is 320322.

To find the coefficient of x^9, we need to look at the terms in the expansion that have x^9.

For (x+1)^14, the term that includes x^9 is:

C(14,9) * x^9 * 1^5

where C(14,9) is the binomial coefficient or combination of 14 things taken 9 at a time. We can calculate this coefficient using the formula:

C(14,9) = 14! / (9! * 5!) = 2002

So the term that includes x^9 in (x+1)^14 is:

2002 * x^9 * 1^5 = 2002x^9

For x^3(x+2)^15, the term that includes x^9 is:

C(15,6) * x^3 * 2^6

where C(15,6) is the binomial coefficient or combination of 15 things taken 6 at a time. We can calculate this coefficient using the formula:

C(15,6) = 15! / (6! * 9!) = 5005

So the term that includes x^3(x+2)^15 is:

5005 * x^3 * 2^6 * x^6 = 5005 * 64x^9

Adding the coefficients of x^9 from both terms, we get:

2002 + 5005 * 64 = 320322

Therefore, the coefficient of x^9 in the expansion of (x+1)^14 + x^3(x+2)^15 is 320322.

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Answer:

45

Step-by-step explanation:

Trapeziod Area - 1/2(a + b)×h

1/2(6 + 12)×5

1/2(18)×5

(9) × 5

Area= 45 cm sq.

Answer:

2 5/8

Step-by-step explanation:

1.5*1.75=2.625=2 5/8

a. The finite population correction factor is not necessary. The expected value of the sample proportion is 0.34 and the standard deviation of the sample proportion is 0.0508.

b. The probability that the sample proportion is greater than 0.37 is approximately 0.2776.

a. To determine if the finite population correction factor is necessary, we need to check if the sample size is large enough in relation to the population size. If the sample size is less than 5% of the population size, then the correction factor is not necessary. In this case, n = 100 is less than 5% of N = 3,000, so we don't need to apply the finite population correction factor.

The expected value of the sample proportion is equal to the population proportion, so E(p) = p = 0.34.

The formula for the standard deviation of the sample proportion is

σ(p) = sqrt[p(1-p)/n]

Substituting in the values, we get:

σ(p) = sqrt[(0.34)(1-0.34)/100] = 0.0508

Therefore, the expected value of the sample proportion is 0.34 and the standard deviation of the sample proportion is 0.0508.

b. We want to find the probability that the sample proportion is greater than 0.37. We can use the z-score formula and standard normal distribution to find this probability.

The z-score formula is:

z = (P - p) / σ(P)

Substituting in the values, we getp

z = (0.37 - 0.34) / 0.0508 = 0.591

Using a standard normal distribution table or calculator, we can find that the probability of z being greater than 0.591 is approximately 0.2776.

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The partial derivative of w=3zexyz with respect to z is obtained by differentiating exyz with respect to z, treating x and y as constants. This gives ∂w/∂z = 3exyz.

To find the partial derivatives of the given function f(x,y,z), we need to differentiate the function with respect to each variable, treating the other variables as constants.

We have the function:

f(x, y, z) = 6x sin(y − z) w=3zexyz

Let's find the first partial derivative of f with respect to x, y, and z.

Partial derivative of f with respect to x:

f_x = ∂f/∂x

f_x = 6 sin(y - z)

Partial derivative of f with respect to y:

f_y = ∂f/∂y

f_y = 6x cos(y - z)

Partial derivative of f with respect to z:

f_z = ∂f/∂z

f_z = -6x cos(y - z) + 3exyz

The partial derivative of w=3zexyz with respect to z is obtained by differentiating exyz with respect to z, treating x and y as constants. This gives ∂w/∂z = 3exyz.

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The perimeter and the area of the triangle are given as follows:

First we obtain the area, as we have the two parameters, as follows:

The area is half the multiplication of the base and the height, hence it is given as follows:

[tex]A = 0.5 \times 16 \times 8\sqrt{3}[/tex]

[tex]A = 64\sqrt{3}[/tex] cm².

For the perimeter, we must obtain the lateral segments, considering the bisection and the Pythagorean Theorem, as follows:

[tex]l^2 = 8^2 + (8\sqrt{3})^2[/tex]

l² = 64 + 192

l² = 256

l = 16.

Hence the perimeter is given as follows:

P = 3 x 16

P = 48 cm.

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Therefore, there are 81,920 different pizzas that this parlor offers.

Since there are five different sizes of pizza, a customer can choose any one of the five sizes. For each size, the customer can choose to have any combination of the 14 toppings, or no toppings at all. This means that for each size of pizza, there are $2^{14}$ different possible topping combinations, including the option of having no toppings. So the total number of different pizzas that the parlor offers is:

=5*2¹⁴

=5*16,384

=81,920

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Answer:Complex

Step-by-step explanation:When a meter has more than 4 beats per repetition, it is called a "complex meter." Examples of complex meters include 5/4, 7/8, and 11/8, among others. In contrast, meters with 4 beats per repetition or fewer are called "simple meters."

We are given the differential equation [tex]y" + 2y = -18x^2e^2x[/tex]n:. To find the complementary function, we first solve the homogeneous equation:[tex]y" + 2y = 0[/tex]. The answer is the particular solution is:[tex]y_p(x) = -3/2*x^2e^2x[/tex].

The characteristic equation is:[tex]r^2 + 2 = 0[/tex]

Which has the roots:[tex]r = ±√(-2)[/tex]

Since the roots are complex, we can write them as:[tex]r1 = i√2[/tex]

and [tex]r2 = -i\sqrt{2}[/tex]

Thus, the complementary function is: y_c(x) = [tex]c1cos(\sqrt{2x} )[/tex] + [tex]c2sin(\sqrt{2}x )[/tex]

To find the particular solution, we assume a solution of the form:[tex]y_p(x) = Ax^2e^2x[/tex]

Taking the first and second derivatives of y_p(x), we get:

[tex]y'_p(x) = 2Axe^2x + 2Ax^2e^2x[/tex]

[tex]y''_p(x) = 4Axe^2x + 4Ax^2e^2x + 4Ae^2x[/tex]

Substituting y_p(x), y'_p(x), and y''_p(x) back into the original differential equation, we get:

[tex](4Axe^2x + 4Ax^2e^2x + 4Ae^2x) + 2(Ax^2e^2x) = -18x^2e^2x[/tex]

Simplifying and collecting like terms, we get:[tex](6A + 4Ax)xe^2x + (4A + 2A)x^2e^2x = -18x^2e^2x[/tex]

Equating coefficients of like terms, we get:[tex]6A + 4Ax = 0, 4A + 2A = -18[/tex]

Solving for A, we get:

A =[tex]\frac{-3}{2}[/tex]

Therefore, the particular solution is:[tex]y_p(x) = -3/2*x^2e^2x[/tex]

The general solution is the sum of the complementary function and the particular solution:

[tex]y(x) = y_c(x) + y_p(x)[/tex]

[tex]y(x) = c1cos(√2x) + c2sin(√2x) - 3/2*x^2e^2x[/tex]

Where c1 and c2 are constants determined by initial conditions.

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