Consider the function z = f(x,y) = 6x^2 - 2x + 2y^2-3y+7xy+3

a. Compute the partial derivatives
dz/dx:
dz/dy:

b. Find the critical point of f
(x,y) =

c. starting at (x,y) = (3,4); a small increase in y with x held constant will lead to a ___ increase in z than a small increase in x with y held constant

d. suppose a and b are fixed positive numbers and that a is greater than b. Which of the following is larger?

the slope of f(a,y) at y =3
the slope of f(b,y) at y = 3

The length of X=10 in a right angle triangle .

what is right angle triangle ?

A right-angled triangle, also known as a right triangle, is a triangle in which one of the angles is a right angle, which is exactly 90 degrees. The side opposite to the right angle is called the hypotenuse, while the other two sides are called legs or catheti. The Pythagorean Theorem is a fundamental theorem that applies to right triangles and states that the square of the length of the hypotenuse is equal to the sum of the squares of the lengths of the legs. Right triangles have many applications in mathematics and other fields, such as engineering and physics.

According to the question:
Since the line drawn from B is perpendicular to AC, we have two similar triangles: BAE and BCE.

Using the Pythagorean Theorem on triangle BAE, we get:

AB² + AE² = BE²

AB² + 12² = BE²

AB² = BE² - 12²

Using the Pythagorean Theorem on triangle BCE, we get:

BC² + CE² = BE²

10² + CE² = BE²

CE² = BE² - 10²

Since AB = AC - BC and AB² = AC² - 2AC * BC + BC², we can substitute and simplify:

(AC - 10)² + 12² = BE²

AC² - 20AC + 100 + 144 = BE²

AC² - 20AC + 244 = BE²

Substituting the previous expressions for AB² and CE² into the equation CE² = BE² - 10², we get:

AC² - 20AC + 244 = AB² - 100

AC² - AB² - 20AC + 344 = 0

(AC - AB - 4)(AC + AB - 16) = 0

Since AC > AB, we have AC + AB - 16 = 0, which implies AC = 16 - AB. Substituting into the equation 10² + CE² = BE², we get:

100 + CE² = AB² - 144

CE² = AB² - 244

CE² = (16 - AC)² - 244

CE² = (16 - (16 - AB))² - 244

CE² = AB² - 244

Substituting the expression for AB² from above, we get:

CE² = BE² - 12² - 244

CE² = 10² + CE² - 12² - 244

CE² - CE² = -100

CE = sqrt(100) = 10

Therefore, EC = 10. so the length of X=10


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Answer: E

Step-by-step explanation:

To cancel out the Y you need to have both multiplied by each other

so the first equation get multiplied by 7 and the second one by 4

28y + -28y = 0 they cancel out

Answer:

Circumference= 44 in

Area= 154 in²

Step-by-step explanation:

\(\boxed{circumference \: of \: circle = 2\pi r}\)

Radius

= diameter ÷2

= 14 ÷2

= 7 in

Circumference of circle

\( = 2( \frac{22}{7} )(7)\)

= 44 in

\(\boxed{area \: of \: circle = \pi {r}^{2} }\)

Area of circle

\( = \frac{22}{7} ( {7}^{2} )\)

= 154 in²

Answer:

Step-by-step explanation:

a)

First I will calculate the equation of the line that goes trough the points

C (0, 0) and (-4, 3).

y= mx + b

y = [(3-0)/ (-4 -0) ] x + 0

y = (-3/4)x  →the slope of tis line is (-3/4)

The slope of line l is the negative reciprocal of (-3/4) the slop of line that goes trough points C (0, 0) and (-4, 3) because the lines are ⊥.  

Slope of line l is 4/3

b)

The equation of l is

y= (4/3)x + b

for point (x= -4, y = 3) that belongs to the line l, we have

3 = (4/3) (-4) + b

3 = (-16/3) + b

3 + (16/3) = b

(9+16)/3 = b

25/3 = b

The equation of the line l in slope intercept form is :

y = (4/3)x + (25/3)

c)

The radius of circle C is the distance between C (0, 0) and (-4, 3).

d² = (x2-x1)² +(y2-y2)²

d = √(-4-0)² +(3-0)²

d = √16+ 9

d = 5 → the radius of the circle C is 5

d)

Line l intersects the y-axis at (25/3) as we previously found out.

The radius is 5.

...therefore the distance between the line l's y-intercept and the circumference of the circle is

(25/3 ) - 5  = (25 - 5·3)/3 = 10/3

The yy-component of vector a a is -y-direction, which is equal to 0 m/s2.the product of the second term of the vector with unit vector j^j^, which is in the direction of the y-axis.

Given a vector, a⃗ a→ = (5.0 m/s2, −y−direction)Find the yy -component of the given vector a⃗ a→.Solution: The yy-component of the given vector a⃗ a→ = -y-directionTo find the yy-component of the given vector a⃗ a→, we have to take the product of the second term of the vector with unit vector j^j^, which is in the direction of the y-axis.Therefore, the yy-component of vector a⃗ a→ is :a_yy = -y-direction = (-1)(0) = 0 m/s²  Answer: 0 m/s².

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The yy-component of vector a⃗ a→ is -y-direction.

The yy-component of a vector represents the magnitude of the vector in the y-direction.

In this case, the vector a⃗ a→ is given as (5.0 m/s2, −y-direction).

Since the yy-component is the magnitude in the y-direction, the yy-component of the vector a⃗ a→ is -y-direction.

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Answer:

add all of the vegetables she bought

Step-by-step explanation:

Answer:

1.Pick the point x= 0, y=3 as your starting point.

2. write the slope as a fraction, in this case 2/1. Move up as many times the numerator says (2), and forward as many times the denominator says (1). If either is negative, you move the opposite direction.

3. You got a second point. Put a ruler between them - or any straight edge - and draw the line.

Answer:

16.384x10^6

Step-by-step explanation:

Using the formula of area of a circle, about 226.08in² has been eaten

The diameter of the pizza is given as 24 inches. To calculate the area of the entire pizza, we need to use the formula for the area of a circle:

Area = π * r²

where π is approximately 3.14 and r is the radius of the circle.

Given that the diameter is 24 inches, the radius (r) would be half of the diameter, which is 12 inches.

Let's calculate the area of the entire pizza first:

Area = 3.14 * 12²

Area = 3.14 * 144

Area ≈ 452.16 square inches

Now, if half of the pizza is consumed, we need to calculate the area of half of the pizza. To do that, we divide the area of the entire pizza by 2:

Area of half of the pizza = 452.16 / 2

Area of half of the pizza ≈ 226.08 square inches

Therefore, if half of the pizza is consumed, approximately 226.08 square inches of pizza would be eaten.

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The points on the curve where the tangent line has a slope of -1 are (2/√3, -(2/√3)) and (-2/√3, 2/√3). None of the given answer choices matches this solution, so the correct option is (E) None of the above.

For the points on the curve where the tangent line has a slope equal to -1, we need to find the points where the derivative of the curve with respect to x is equal to -1. Let's find the derivative:

Differentiating both sides of the equation x^2 - xy + y^2 = 4 with respect to x:

2x - y - x(dy/dx) + 2y(dy/dx) = 0

Rearranging and factoring out dy/dx:

(2y - x)dy/dx = y - 2x

Now we can solve for dy/dx:

dy/dx = (y - 2x) / (2y - x)

We want to find the points where dy/dx = -1, so we set the equation equal to -1 and solve for the values of x and y:

(y - 2x) / (2y - x) = -1

Cross-multiplying and rearranging:

y - 2x = -2y + x

3x + 3y = 0

x + y = 0

y = -x

Substituting y = -x back into the original equation:

x^2 - x(-x) + (-x)^2 = 4

x^2 + x^2 + x^2 = 4

3x^2 = 4

x^2 = 4/3

x = ±sqrt(4/3)

x = ±(2/√3)

When we substitute these x-values back into y = -x, we get the corresponding y-values:

For x = 2/√3, y = -(2/√3)

For x = -2/√3, y = 2/√3

Therefore, the points on the curve where the tangent line has a slope of -1 are (2/√3, -(2/√3)) and (-2/√3, 2/√3).

None of the given answer choices matches this solution, so the correct option is:

E) None of the above.

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The probability of getting stuck in traffic on any given day when leaving the city is 70%. When considering two separate days, we can use the multiplication rule of probability to find the probability of getting stuck in traffic on both days.

The multiplication rule of probability states that the probability of two independent events occurring together is the product of their individual probabilities. In this case, the events of getting stuck in traffic on two separate days are independent, meaning that the occurrence of one does not affect the probability of the other.

To find the probability of getting stuck in traffic on both days, we can multiply the probability of getting stuck on the first day (0.7) by the probability of getting stuck on the second day (also 0.7):

P(getting stuck on both days) = P(getting stuck on day 1) x P(getting stuck on day 2)
P(getting stuck on both days) = 0.7 x 0.7
P(getting stuck on both days) = 0.49 or 49%

Therefore, the probability of getting stuck in traffic on both days is 49%. This means that there is a less than 50% chance of getting stuck in traffic on both days, despite the 70% chance of getting stuck on each individual day.

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Answer:B

Step-by-step explanation:

This is because -3 is going to multiple numbers, which cannot happen in a function.

The optimal values of these parameters are:

a. β₁ = 0

b. β₂ = 0

The linear regression y = β1 + β2x + e, assuming that the sum of squared errors (SSE) takes the following form:

SSE = 382 + 681 + 382 + 18β1β2

Derive the partial derivatives of SSE with respect to β1 and β2 and solve the optimal values of these parameters.

Given that SSE = 382 + 681 + 382 + 18β1β2 ∂SSE/∂β1 = 0 ∂SSE/∂β2 = 0

Now, we need to find the partial derivative of SSE with respect to β1.

∂SSE/∂β1 = 0 + 0 + 0 + 18β2 ⇒ 18β2 = 0 ⇒ β2 = 0

Therefore, we obtain the optimal value of β2 as 0.

Now, we need to find the partial derivative of SSE with respect to β2. ∂SSE/∂β2 = 0 + 0 + 0 + 18β1 ⇒ 18β1 = 0 ⇒ β1 = 0

Therefore, we obtain the optimal value of β1 as 0. Hence, the partial derivative of SSE with respect to β1 is 18β2 and the partial derivative of SSE with respect to β2 is 18β1.

Thus, the optimal values of β1 and β2 are 0 and 0, respectively.

Therefore, the answers are: a. β₁ = 0 b. β₂ = 0

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According to the question, the points are \((-3,-10)\) and \((-1,-7)\) for the function \(f(x).\)

The calculated slope for these values are as follows:

\(m = \frac{(y(2)-y(1))}{(x(2)-x(1))} = \frac{(-7-(-10))}{(-1-(-3))} = \frac{3}{2} = 1.5\)

The slope for the function \(f(x)\) is: \(m=1.5\)

Now, the function \(g(x)=\frac{1}{2} x+3\)

Using the standard formula that is \(y = mx+c\) to calculate the slope for the function \(g(x).\)

Slope(m) for \(g(x) = \frac{1}{2} = 0.5\)

As per the question, lance gains \(1\) level each time he plays the video game.

Therefore, \(slope = \frac{1}{1} =1\)

Hence, the slope for the function \(h(x)\) is \(1\).

Therefore, the function \(f(x)\) has the largest slope and it is \(1.5.\)

What is the equation of the line?

To represent the set of points in an algebraic form is known as the equation of line. The set of points represent lines in the coordinate system. The parameters can be calculated and those parameters are slope and y-intercept.

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Answer:

no

Step-by-step explanation:

Answer: A) -2m^3-m^2+4m+2

Answer:

2 inches would be left

Step-by-step explanation:

The limit is 0.

We will first understand what limit is and ten calculate it :

A limit is a boundary beyond which something may not extend. Limit has many meanings such as "the point at which something stops", or "a boundary". It can also mean "to restrict" or "restrict".

We can use the word "limit" to describe what we don't want, but if it is to get used with time-sensitive items we have to make sure it's accurate.

To evaluate the limit, we can use L'Hopital's rule since we have an indeterminate form of "infinity/infinity". Differentiating the numerator and denominator with respect to x, we get:

lim x→[infinity] 7 - x / 7(2ex) = lim x→[infinity] (-1) / (14ex)

Since the denominator goes to infinity as x goes to infinity, and the numerator remains constant, we can see that the limit is equal to zero:

lim x→[infinity] 7 - x / 7(2ex) = lim x→[infinity] (-1) / (14ex) = 0

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Answer:

944 cm²

Step-by-step explanation:

get the area of each side: 12*10+10*16+12*16= 472

2 sets of each side: 944

Answer:

3 candies for each friend

Step-by-step explanation:

(7+8)÷4=c

15 pieces of candy

4 people

Answer:

Step-by-step explanation:

the number is 1300 if we have 40% from it will give us 520

The monthly payment for the 20-year mortgage on the farming operation is approximately $5,118.42. and The new monthly payment would be approximately $5,901.25, but refinancing would result in higher costs overall

To calculate the monthly payment for a 20-year mortgage on a farming operation in Gates County, NC, we can use the formula for calculating the monthly mortgage payment:

\(M = P * (r * (1 + r)^n) / ((1 + r)^n - 1)\)

Where:

M is the monthly payment

P is the principal amount (loan amount)

r is the monthly interest rate

n is the total number of monthly payments

Given:

Loan amount (P) = $900,000 - 10% down payment = $810,000

Interest rate (r) = 5% per annum = 5% / 12 months = 0.4167% per month

Total number of monthly payments (n) = 20 years * 12 months = 240 months

Substituting these values into the formula:

\(M = $810,000 * (0.004167 * (1 + 0.004167)^240) / ((1 + 0.004167)^240 - 1)\)

Calculating the monthly payment:

M ≈ $5,118.42

Therefore, the monthly payment for the 20-year mortgage on the farming operation is approximately $5,118.42.

Now, let's consider the situation where the interest rates drop to 4% after five years. We need to calculate the new monthly payment for the 15-year mortgage.

Loan amount (P) for the new mortgage = $810,000 remaining balance after five years

Interest rate (r) = 4% per annum = 4% / 12 months = 0.3333% per month

Total number of monthly payments (n) = 15 years * 12 months = 180 months

To determine if refinancing is beneficial, we need to compare the savings in monthly payments with the cost of refinancing.

To calculate the new payment, we use the same formula:

\(M = P * (r * (1 + r)^n) / ((1 + r)^n - 1)\)

\(M = $810,000 * (0.003333 * (1 + 0.003333)^180) / ((1 + 0.003333)^180 - 1)\)

Calculating the new monthly payment:

M ≈ $5,901.25

To determine if refinancing is beneficial, we compare the savings in monthly payments:

Monthly savings = $5,118.42 - $5,901.25 ≈ -$782.83

However, there is also a cost associated with refinancing, which is a $2,000 fee plus 1 point ($810,000 * 1% = $8,100) financed by the bank.

Total refinancing cost = $2,000 + $8,100 = $10,100

Considering the negative monthly savings and the cost of refinancing, it is not advisable to refinance in this scenario.

The new monthly payment would be approximately $5,901.25, but refinancing would result in higher costs overall. Therefore, it is not recommended to refinance in this case.

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The car will cost $ 800 after a depreciation time of approximately 6 years.

Herein we are informed about the case of a car bought in 2011 at a cost of $ 36,000 and that depreciates linearly every year. Then, the depreciation function is described below:

c(t) = c' + m · t

Where:

If we know that c(0) = 36,000, c(4) = 12,000 and c(t) = 800, then the depreciation rate is:

m = (12,000 - 36,000) / (4 - 0)

m = - 24,000 / 4

m = - 6,000

800 = 36,000 - 6,000 · t

6,000 · t = 35,200

t = 35,200 / 6,000

t = 5.867

The expected depreciation time is approximately 6 years.

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Answer:

0.50146267 i think thats it hun

There is exactly one more real solution or there is exactly one more complex solution

Given,

One solution to a quadratic equation is x = -5/3

We have to find the correct statements from the given ones;

A polynomial of degree two is a quadratic equation.

This implies that a polynomial has two solutions.

We already have an answer to the query, which is a true root.

Then, the additional response that we lack can appear in the form of two responses.

It may be a real root or a complicated root, depending on the situation.

The following is the response to this query:

There is exactly one more real solution or there is exactly one more complex solution

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The answer is:
   " m∠BAC = 45° ;  m∠ABC, = 64° ; and:  m∠ACB = 70° ."
__________

Step-by-step explanation:

To find the measure of ∠BCA {"m∠BCA"} :

<BCA is supplementary to " 110 " ; that is, both, or all, of all the measurements that form a "line or line segment";  or any portions thereof,

that can be written in:  "slope-intercept format"

  that is, on Cartesian plane:  " y = mx + b " ;

  in which:  "y" is isolated on the "left-hand side" of the equation; as a single, isolated variable;

  and in which:  "m" is the "slope" (if applicable); and the coefficient to the "x"  value'

  and in which "b" is the "y-intercept"; or more precisely"
    the value of the "y-coordinate"—in: "(x, y)" format —at which:
    " x = 0" ; that is, in the format:  " (0, b)."
_________________
So, to find
m∠BCA :  Subtract our given value, "110" ; from "180" ; as follows:
_________________

  " (180 − 110) = 70" ;
__________________
Now:   Note a triangle, by definition, has 3 (three) sides and 3 (three) angles—and all three (3) angles of a triangle add up to 180°.

So:  We just solved for  m∠ACB ; which is:  70° .

Now shall solve for the value of "x" ; & then find the value of all 3 (three) sides of this triangle

Since all angles within a triangle add up to 180: Let's add them up:
  " 70  +  3x + 20 + 3x = 180 " ;

 Combine the "like terms" on the "left-hand side" of the equation:
  " +3x + 3x = 6x " ;  "+ 70 + 20 = 90 ";
And rewrite the equation:
  "  6x + 90 = 180 " ;

Now, subtract "90" from Each Side of the equation:
  "  6x + 90 − 90 = 180 − 90 " ;
to get:   " 6x = 90 " ;

Now, divide Each side of the equation by "6" ;

  to isolate "x" on one side of the equation ;
    and to solve for "x" :
   " 6x / 6 = 90 / 6 ;

to get:  " x = 15 " ;
________________
Now, we are asked to find:
1)  " m∠BAC = 3x " ;  Plug in "15" for "x" ; and solve:
                     "3x = 3(15) = 45 " ;
2)  " m∠ABC = "(3x + 20)" ; Plug in "15" for "x"; and solve:
                     "3(15) + 20 = 45 + 20 = 65 " ;
3)  " m∠ACB  = 70 " ; (as per our calculated value);

To check our work—do these 3 (three) numbers add up to 180?

   "45 + 65 + 70 =? 180 ? " ;

   " 110 + 70 = " 180 "? Yes!
__________
So: The answer is as follows:
   " m∠BAC = 45° ;  m∠ABC, = 64° ; and:  m∠ACB = 70° ."
__________
Hope this is helpful to you!  Best wishes!