if (2 0) is a solution of linear equation 2x-5y=k then find the value of k​

Step-by-step explanation:

To solve the equation 4e^(2x) - 1 = 13 for x, we will follow these steps:

Step 1: Add 1 to both sides of the equation to isolate the term with the exponential:

4e^(2x) = 14

Step 2: Divide both sides of the equation by 4 to isolate the exponential term:

e^(2x) = 14/4

Simplifying the right side:

e^(2x) = 7/2

Step 3: Take the natural logarithm (ln) of both sides of the equation to eliminate the exponential:

ln(e^(2x)) = ln(7/2)

By the properties of logarithms, the ln and e^(2x) cancel each other out:

2x = ln(7/2)

Step 4: Divide both sides of the equation by 2 to solve for x:

x = (1/2) ln(7/2)

Thus, the solution to the equation 4e^(2x) - 1 = 13 is x = (1/2) ln(7/2).

The value of dx / dt is -3 when x= -4.

Given that, `x y

= -8` and `dy / dt

= -6`To find: `dx / dt` when `x

= -4

`The product rule states that for two functions u(x) and v(x), the derivative of the product of these two functions is given as `(uv)' = u'v + uv'`If we differentiate the given equation `x y = -8` with respect to time, we get:

`d(x y) / dt

= d(-8) / dt``d(x y) / dt

= 0` (derivative of a constant is zero)`

x (dy / dt) + y (dx / dt)

= 0`

Given, `x = -4` and `y = -8 / x

`Therefore, `y = 2`Substituting the values in the above equation, we get:`-4 (dx / dt) + 2 (-6) = 0`Simplifying, we get:`-4 (dx / dt) = 12`Dividing by -4 on both sides, we get:`dx / dt = -3`Therefore, `dx / dt` is `-3` when `x = -4`.

To know more about value visit:-

https://brainly.com/question/30145972

#SPJ11

The curl of vector field F is -i - j and the divergence of the vector field F is 0.

We have to follow the below steps:

Step 1: Find curl.

Step 2: Find divergence.

Curl :

Curl is the vector operator that defines the cross product of two vectors. It is applied to a vector field to generate another vector field.

Step 1:

Find the curl of vector field

Curl of the vector field F is defined as ,F = (P, Q, R)

Curl F = (Ry - Qz)i + (Pz - Rx)j + (Qx - Py)k

Where, Ry denotes the partial derivative of R with respect to y. Qz denotes the partial derivative of Q with respect to z, Pz denotes the partial derivative of P with respect to z .Rx denotes the partial derivative of R with respect to x, Qx denotes the partial derivative of Q with respect to x. Py denotes the partial derivative of P with respect to y.

Here,P = -x, Q = y, and R = -2.Then

Ry = 0, Qz = 0, Pz = 0, Rx = 0, Qx = 0, and Py = 0

Therefore, the curl of F = (Ry - Qz)i + (Pz - Rx)j + (Qx - Py)k= -i - j + 0k= -i - j

Then, the curl of vector field F is -i - j.

Divergence:

Divergence is a vector operator that operates on a vector field to produce a scalar value. It measures the magnitude of the outward flux of the vector field from an infinitesimal region around a particular point. Divergence is given by,

Divergence of the vector field F = (P, Q, R)div F = ∂P/∂x + ∂Q/∂y + ∂R/∂z

Here, P = -x, Q = y, and R = -2.Then, ∂P/∂x = -1, ∂Q/∂y = 1, and ∂R/∂z = 0

Therefore, the divergence of the vector field F is

div F = ∂P/∂x + ∂Q/∂y + ∂R/∂z= -1 + 1 + 0= 0

Learn more about divergence from:

https://brainly.com/question/32770589

#SPJ11

The solution of the system of equations is [tex]\[x=3,\text{ }y=0,\text{ and }z=2\][/tex].

As per data the system of linear equations,

[tex]\[ \left\{\begin{array}{c} x+2 y+3 z=9 \\ 2 x-y+z=8 \\ 3 x-z=3 \end{array}\right. \] 1)[/tex]

Write the system in the matrix form [tex]\( A . X=B \)[/tex]

We know that the matrix form of the system of linear equations is as follows.

[tex]\[A. X = B\][/tex]

Where

[tex]\[A=\begin{pmatrix} 1 & 2 & 3 \\ 2 & -1 & 1 \\ 3 & 0 & -1 \end{pmatrix}\[X=\begin{pmatrix} x \\ y \\ z \end{pmatrix}\][/tex]

and

[tex]\[B=\begin{pmatrix} 9 \\ 8 \\ 3 \end{pmatrix}\]2)[/tex]

To solve the system, we can use row reduction method.

[tex]\[\begin{pmatrix} 1 & 2 & 3 & 9 \\ 2 & -1 & 1 & 8 \\ 3 & 0 & -1 & 3 \end{pmatrix}\][/tex]

Applying the elementary row operations

[tex]\[R_{2}\to R_{2}-2R_{1}\][/tex]

and

[tex]\[R_{3}\to R_{3}-3R_{1}\][/tex]

we get,

[tex]\[\begin{pmatrix} 1 & 2 & 3 & 9 \\ 0 & -5 & -5 & -10 \\ 0 & -6 & -10 & -24 \end{pmatrix}\][/tex]

Now applying the elementary row operations

[tex]\[R_{3}\to R_{3}-(6/5)R_{2}\][/tex]

we get,

[tex]\[\begin{pmatrix} 1 & 2 & 3 & 9 \\ 0 & -5 & -5 & -10 \\ 0 & 0 & -1 & -2 \end{pmatrix}\][/tex]

Now, we need to apply back substitution method. Using the third row, we can get the value of z as z = 2.

Now, using the second row,

[tex]\[-5y - 5z = -10\]\\\-5y - 5(2) = -10\][/tex]

Solving this equation, we get y = 0.

Finally, using the first row, we can get the value of x as

[tex]\[x + 2y + 3z = 9\]\\x = 3\][/tex]

Hence, the solution of the system of equations is [tex]\[x=3,\text{ }y=0,\text{ and }z=2\][/tex].

To learn more about linear equations from the given link.

https://brainly.com/question/2030026

#SPJ11

The cubic model is y = -5x³ - 9x² - 3x - 4.

To find a cubic and quartic model for the given set of values, we can use polynomial regression. Polynomial regression is a type of regression analysis where the relationship between the independent variable (x) and the dependent variable (y) is modeled as an nth degree polynomial.

Let's start by finding the cubic model. The general equation for a cubic model is

y = ax³ + bx² + cx + d,

where a, b, c, and d are the coefficients we need to find.

To find the coefficients, we need to solve a system of equations using the given data points:

When x = -2, y = -65:

-65 = a(-2)³ + b(-2)² + c(-2) + d

When x = -1, y = -14:

-14 = a(-1)³ + b(-1)² + c(-1) + d

When x = 0, y = -4:

-4 = a(0)³ + b(0)² + c(0) + d

When x = 1, y = 2:

2 = a(1)³ + b(1)² + c(1) + d

When x = 2, y = 90:

90 = a(2)³ + b(2)² + c(2) + d

Simplifying these equations, we get:
-8a + 4b - 2c + d = -65
-a + b - c + d = -14
d = -4
a + b + c + d = 2
8a + 4b + 2c + d = 90

Solving this system of equations, we find:
a = -5
b = -9
c = -3
d = -4
Therefore, the cubic model is y = -5x³ - 9x² - 3x - 4.

Learn more about cubic model visit:

brainly.com/question/29133798

#SPJ11

The given integral is:

S™ ⁹x: 9x sin²(x) dx

To use the Midpoint Rule with the given value of n to approximate the integral, we need to perform the following steps:

First, we have to find the width of each subinterval, which is given by:

Δx = (b - a)/n

where 'b' is the upper limit of integration and 'a' is the lower limit of integration.

Substituting the given values, we get:

Δx = (π/2 - 0)/4

= π/8

Next, we have to find the midpoint of each subinterval, which is given by:

xᵢ = a + (i - 1/2)Δx

where 'i' is the number of the subinterval.

Substituting the given values, we get the following midpoints:

x₁ = 0 + (1 - 1/2)

Δx = π/16

x₂ = 0 + (2 - 1/2)

Δx = 3π/16

x₃ = 0 + (3 - 1/2)

Δx = 5π/16

x₄ = 0 + (4 - 1/2)

Δx = 7π/16

Next, we have to evaluate the function at each midpoint, which is given by:

f(xᵢ) = 9xᵢ sin²(xᵢ)

Substituting the midpoints, we get the following values:

f(π/16) ≈ 0.1468f(3π/16)

≈ 1.0595f(5π/16)

≈ 2.9305f(7π/16)

≈ 3.6867

Finally, we can approximate the integral using the Midpoint Rule, which is given by:

∫[a, b] f(x) dx ≈ Δx [f(x₁) + f(x₂) + ... + f(xₙ)]

Substituting the values, we get:

∫[0, π/2] 9x sin²(x) dx ≈ Δx [f(π/16) + f(3π/16) + f(5π/16) + f(7π/16)]

≈ (π/8) [0.1468 + 1.0595 + 2.9305 + 3.6867]

≈ 1.9123

Rounded to four decimal places, the approximation is 1.9123.

Therefore, the correct answer is 1.9123.

To know more about approximate visit:

https://brainly.com/question/31695967

#SPJ11

When a young patient's forearm and elbow are immobilized by a cast for several weeks, it would result in a reduction in bone density in the bones of the upper limb. Bones have the ability to remodel themselves in response to the stress they receive.

Bone remodeling refers to the process by which old bone tissue is replaced by new bone tissue. This process helps to maintain bone strength and shape. However, when a limb is immobilized, the bones in that area experience reduced stress. As a result, bone density decreases and the bones become weaker. This phenomenon is known as disuse atrophy.

Disuse atrophy is more common in the elderly, but it can also occur in young patients who have been immobilized for an extended period of time. To prevent disuse atrophy, patients are often encouraged to perform range-of-motion exercises while the limb is immobilized.

This helps to maintain bone density and prevent muscle atrophy.

#SPJ11

Learn more about upper limb and bones https://brainly.com/question/27961339

all critical points of the function f(x,y)=8xy−2x^4−2y^4 are (0, 0), (1, 1), (-1, -1).

To find the critical points of the function \( f(x, y) = 8xy - 2x^4 - 2y^4 \), we need to calculate its partial derivatives with respect to \( x \) and \( y \) and solve the resulting system of equations.

Let's begin by finding the partial derivative of \( f \) with respect to \( x \):

\( \frac{\partial f}{\partial x} = 8y - 8x^3 \)

Next, we find the partial derivative of \( f \) with respect to \( y \):

\( \frac{\partial f}{\partial y} = 8x - 8y^3 \)

To find the critical points, we need to solve the system of equations:

\[

\begin{align*}

\frac{\partial f}{\partial x} &= 0 \\

\frac{\partial f}{\partial y} &= 0

\end{align*}

\]

Setting \( \frac{\partial f}{\partial x} = 0 \), we have:

\( 8y - 8x^3 = 0 \)

Simplifying, we get:

\( y = x^3 \)

Now, substituting this into \( \frac{\partial f}{\partial y} = 0 \), we have:

\( 8x - 8(x^3)^3 = 0 \)

Simplifying further:

\( 8x - 8x^9 = 0 \)

Factoring out \( 8x \), we get:

\( 8x(1 - x^8) = 0 \)

This equation has two solutions:

\( x = 0 \) or \( x^8 = 1 \)

If \( x = 0 \), then \( y = 0 \) since \( y = x^3 \).

If \( x^8 = 1 \), then \( x = 1 \) or \( x = -1 \), and in both cases, \( y = x^3 \), so \( y = 1 \) or \( y = -1 \).

Therefore, the critical points are:

\( (0, 0) \), \( (1, 1) \), and \( (-1, -1) \).

So, the correct answer is:

A. The critical point(s) is/are (0, 0), (1, 1), (-1, -1).

To know more about Critical Points, visit

https://brainly.com/question/30459381

#SPJ11

The polynomial 8r^2 - 8r^5 is classified as a quintic binomial. It is a binomial because it has two terms and quintic because the highest power of the variable is 5.

Degree refers to the highest power of the variable in a polynomial, while the number of terms indicates the count of distinct terms in the expression.

In this case, the polynomial has two distinct terms: 8r^2 and -8r^5. Thus, we can classify it as a binomial since it consists of two terms.

Now, let's determine the degree of the polynomial. The highest power of the variable r in the polynomial is 5 (found in the term -8r^5). Hence, we classify the polynomial as a quintic binomial.

To break it down further, the term "quintic" denotes the degree of 5, while "binomial" refers to the presence of two terms.

In summary, the polynomial 8r^2 - 8r^5 is classified as a quintic binomial. It is a binomial because it has two terms and quintic because the highest power of the variable is 5.

For more such questions on quintic trinomial.

https://brainly.com/question/3145552

#SPJ11

The magnitude of the electric field at a radial distance r from the central axis of 6.1 cm for points far from the ends of the cylinders can be calculated as follows:

Formula: `E = kλ / r` Where;`k = 9 x 109 Nm2 / C2` is Coulomb's constant.`λ` is the linear charge density in C/m.`r` is the radial distance from the central axis in m.

The electric field at the center of the cylinder can be written as follows;`E1 = kλ / R1`

Here, the electric field `E1` is given by `E1 = 625 V/m`

We need to find the electric field at a radial distance of `r = 6.1 cm = 0.061 m` from the central axis, where `r` is much less than the length of the cylinder.

So, the electric field `E2` can be calculated as follows;`E2 = kλ / r = E1 * R1 / r``E2 = 625 x 0.021 / 0.061 = 216.39 V/m.

`Therefore, the magnitude of the electric field at a radial distance of `r = 6.1 cm` from the central axis of the cylinder is `216.39 V/m`.

#SPJ11

Learn more about electric field and magnitude https://brainly.com/question/24256733

The equality sin(x + 21) = sin(x) is confirmed.

We want to demonstrate that sin(x + 21) = sin(x) which is to say that two angles, x and x + 21, have the same sine.

The formula for the sine of the sum of two angles states that:

sin(a + b) = sin(a)cos(b) + cos(a)sin(b)

Therefore, sin(x + 21) = sin(x) cos(21) + cos(x) sin(21)

To demonstrate that sin(x + 21) = sin(x), we must demonstrate that

sin(x) cos(21) + cos(x) sin(21) = sin(x) sin(1) cos(21) + cos(x) sin(1) sin(21), because the right-hand side is equivalent to sin(x) sin(21 + x).

Recall that sin(21) = sin(180° - 159°) = sin(159°), thus:

sin(x + 21) = sin(x) cos(21) + cos(x) sin(21)

= sin(x) sin(1) cos(21) + cos(x) sin(1) sin(21)

= sin(x) sin(21 + x).

Therefore, the equality sin(x + 21) = sin(x) is confirmed.

Learn more about equality from:

https://brainly.com/question/30339264

#SPJ11

The given expression, sin(pi/12)cos(7pi/12) - cos(pi/12)sin(7pi/12), simplifies to -sin(2pi/3).

To simplify the given expression, we can use the trigonometric identity for the sine of a difference of angles. The identity states that sin(A - B) = sin(A)cos(B) - cos(A)sin(B). In this case, we have A = pi/12 and B = 7pi/12.

Using the identity, we can rewrite the given expression as sin(pi/12)cos(7pi/12) - cos(pi/12)sin(7pi/12) = sin(pi/12 - 7pi/12).

Simplifying further, we have sin(pi/12 - 7pi/12) = sin(-6pi/12) = sin(-pi/2) = -sin(pi/2).

Since sine of pi/2 is equal to 1, we have -sin(pi/2) = -1.

Therefore, the expression sin(pi/12)cos(7pi/12) - cos(pi/12)sin(7pi/12) is equivalent to -sin(2pi/3), or simply -1.

Learn more about trigonometric identity here:

https://brainly.com/question/12537661

#SPJ11

The marginal average cost function is C'(x) = -189/x²

The given cost function is C(x)= 189 +8.4x and the revenue function is R(x) = 8x -0.04x².

The profit function P(x) is given by;

P(x) = R(x) - C(x)P(x) = (8x - 0.04x²) - (189 + 8.4x)

P(x) = -0.04x² - 0.4x - 189

The marginal profit function P'(x) is given by;

P'(x) = -0.08x - 0.4

Let the average cost be given by

C(x)/xC(x)/x

= (189 + 8.4x)/x

= 189/x + 8.4

Let's differentiate the above equation with respect to x to get the marginal average cost function.

C'(x) = d/dx [189/x + 8.4]C'(x)

= -189/x²

The marginal average cost function is C'(x) = -189/x²

To know more about profit, visit:

https://brainly.com/question/23262297

#SPJ11

Therefore the marginal average cost function is given by C'(x) = 8.4

The given functions are C(x) = 189 + 8.4x and R(x) = 8x - 0.04x², and we are required to find the marginal average cost function, which is C'(x).

We know that the marginal cost is the derivative of the cost function and that the average cost is given by the cost divided by the number of units produced.

Marginal cost function The marginal cost is given by the derivative of the cost function C(x), therefore we find the derivative of C(x) using the power rule of derivatives.

C(x) = 189 + 8.4x Taking the derivative of both sides of the above equation with respect to x, we get;

C'(x) = 0 + 8.4

Differentiating 189 with respect to x gives zero since 189 is a constant.

The marginal cost function C'(x) is therefore given by:

C'(x) = 8.4

Revenue function The revenue function R(x) is given as R(x) = 8x - 0.04x²

We know that the marginal revenue is given by the derivative of the revenue function, which is the first derivative of R(x).

R(x) = 8x - 0.04x²

Differentiating the revenue function R(x) with respect to x gives;

R'(x) = 8 - 2(0.04)x

Simplifying the above expression by multiplying by 2, we get;

R'(x) = 8 - 0.08x

The marginal revenue function R'(x) is therefore given by:

R'(x) = 8 - 0.08x

Marginal average cost function The marginal average cost is the ratio of the marginal cost to the number of units produced, thus;

C'(x)/x = (8.4/x) And C'(x) = (8.4/x) * x

C'(x) = 8.4.

To know more about power rule , visit:

https://brainly.in/question/50326454

#SPJ11

Arithmetic SequenceAn arithmetic sequence is an ordered list of numbers where the difference between consecutive numbers remains constant. For example, the following numbers are an arithmetic sequence: 2, 4, 6, 8, 10, 12.

The difference between consecutive terms is 2; this is the common difference. We can determine the nth term of an arithmetic sequence using the following formula:an = a1 + (n – 1)dwhere an is the nth term, a1 is the first term, n is the term number, and d is the common difference. Formula for the general term (the nth term) of the arithmetic sequence:

We are given

a₁ = 4 and an = an-1 + 5For the 1st term (n = 1)

:a₁ = 4and for the 2nd term (n = 2):a₂ = a₁ + 5 = 4 + 5 = 9For the

3rd term (n = 3):a₃ = a₂ + 5 = 9 + 5 = 14

For the 4th term (n = 4):a₄ = a₃ + 5 = 14 + 5 = 19

Thus, we can see that the common difference is 5. Hence, the nth term of the arithmetic sequence is given by:

an = a₁ + (n – 1)d = 4 + (n – 1)5 = 5n – 1Therefore, the formula for the general term (the nth term) of the arithmetic sequence is: an = 5n – 1.

To know more about Arithmetic Sequence visit:-

https://brainly.com/question/28999301

#SPJ11

Answer:

12545454/100000000

Step-by-step explanation:

To express 0.12545454 in the form of p/q, where p and q are integers, we can follow these steps:

1. Let x = 0.12545454.

2. Multiply x by a power of 10 to remove the repeating decimal part. Since there are two digits repeating, we will multiply by 100 to get rid of the repetition.

  x * 100 = 12.545454.

3. Subtract x from the product obtained in step 2 to eliminate the repeating part.

  100x - x = 12.545454 - 0.12545454 = 12.42.

4. Now, we have the equation 99x = 12.42. Divide both sides by 99 to solve for x.

  x = 12.42 / 99 = 0.125454545...

5. Notice that the decimal part in step 4 is the same as the original decimal part (0.12545454). Therefore, we can express 0.12545454 as a fraction p/q.

  p/q = 0.12545454/1 = 12545454/100000000.

So, 0.12545454 can be expressed in the form of p/q as 12545454/100000000.

We find that the probability that a winner receives a puppy, a kitten, or a unicorn is 41/80.

To find the probability that a winner receives a puppy, a kitten, or a unicorn, we need to determine the total number of possible outcomes and the number of favorable outcomes.

The total number of possible outcomes is the sum of the number of puppies, kittens, frogs, snakes, and unicorns, which is

15 + 16 + 14 + 25 + 10 = 80.

The number of favorable outcomes is the sum of the number of puppies, kittens, and unicorns, which is

15 + 16 + 10 = 41.

Therefore, the probability of receiving a puppy, a kitten, or a unicorn is given by the ratio of the number of favorable outcomes to the total number of possible outcomes, which is 41/80.

To simplify the fraction, we can divide both the numerator and denominator by their greatest common divisor, which is 1 in this case.

Thus, the probability that a winner receives a puppy, a kitten, or a unicorn is 41/80.

In summary, the probability of receiving a puppy, a kitten, or a unicorn in the ring toss game at the carnival is 41/80.

Learn more about the probability from the given link-

https://brainly.com/question/13604758

#SPJ11

To make the expression x^2 + 20x + n a perfect square trinomial, we need to add and subtract the square of half the coefficient of the x term. The coefficient of the x term is 20, so half of it is 10. The square of 10 is 100.

By adding and subtracting 100, we can rewrite the expression as:

[tex]x^2 + 20x + n = x^2 + 2(10)x + 10^2 + n - 100 = (x + 10)^2 + (n - 100)[/tex]

To make it a perfect square trinomial, we need the last term, (n - 100), to be zero.

Therefore, n = 100.

When n = 100, the trinomial becomes [tex](x + 10)^2[/tex], which is a perfect square trinomial.

Hence, the correct answer is option (a) n = 100.

To know more about trinomial visit:

https://brainly.com/question/11379135

#SPJ11

The aspects of the function f(x, y) = x² + y⁴ + 2xy are: local maximum value(s) at (0, 1), local minimum value(s) DNE, saddle point(s) DNE.

To analyze the aspects of the given function, we need to find its critical points and classify them based on their behavior.

To find the critical points, we differentiate f(x, y) with respect to x and y, and set both partial derivatives equal to zero:

∂f/∂x = 2x + 2y = 0

∂f/∂y = 4y³ + 2x = 0

Solving these equations simultaneously, we find that the only critical point is (0, 1).

To determine the nature of this critical point, we examine the second partial derivatives. The second partial derivatives of f(x, y) are:

∂²f/∂x² = 2

∂²f/∂y² = 12y²

∂²f/∂x∂y = 2

Evaluating these second partial derivatives at the critical point (0, 1), we find that ∂²f/∂x² = 2, ∂²f/∂y² = 12, and ∂²f/∂x∂y = 2.

Since the second partial derivative test is inconclusive at this point, we cannot definitively classify it as a local maximum, local minimum, or saddle point. Therefore, local minimum value(s) and saddle point(s) DNE.

However, the point (0, 1) is a critical point, so we can confirm it as a local maximum value.

Learn more about Critical points

brainly.com/question/32810485

#SPJ11

The coterminal angle with 415° between 0° and 360° is -305°.

To find the coterminal angle with 415° between 0° and 360°, we can subtract or add multiples of 360° until we get an angle within that range.

To find the coterminal angle greater than 415°, we can subtract multiples of 360°. Let's start by subtracting 360°:
415° - 360° = 55°

Since 55° is still greater than 360°, we can subtract another 360°:
55° - 360° = -305°

Now we have a negative angle, -305°, which is within the range of 0° and 360°. Therefore, the coterminal angle with 415° between 0° and 360° is -305°.

Learn more about the coterminal angle from the given link;

https://brainly.com/question/23093580

#SPJ11

The answer is given as follows: `(2 + √−36) + (8 + √−16) = 10 + 10i` and `(3 − 2i) − (2 + 9i) = 1 − 11i`. Hence addation and substraction is simplified.

In order to solve the given addition expression `(2 + √−36) + (8 + √−16)`, we will use the following rule of addition of complex numbers. `(a + bi) + (c + di) = (a + c) + (b + d)i`So, `2 + √−36 = 2 + 6i` (i.e. √−36 = 6i)

Similarly, `8 + √−16 = 8 + 4i` (i.e. √−16 = 4i)

Therefore, `(2 + √−36) + (8 + √−16) = (2 + 6i) + (8 + 4i)`= `2 + 8 + 6i + 4i`=`10 + 10i`

Hence, `(2 + √−36) + (8 + √−16) = 10 + 10i`.  

In order to solve the given subtraction expression `(3 − 2i) − (2 + 9i)`, we will use the following rule of subtraction of complex numbers. `(a + bi) − (c + di) = (a − c) + (b − d)i`So, `(3 − 2i) − (2 + 9i) = 3 − 2i − 2 − 9i`=`1 − 11i`

Hence, `(3 − 2i) − (2 + 9i) = 1 − 11i`.

Therefore, the answer is given as follows: `(2 + √−36) + (8 + √−16) = 10 + 10i` and `(3 − 2i) − (2 + 9i) = 1 − 11i`.

To know more about addation and substraction visit:
brainly.com/question/16614146

#SPJ11

The equation of the vertical line passing through (-1, 2) is x = -1. Therefore, the correct option is d) x = -1.

A vertical line is a line that is perpendicular to the x-axis, and its slope is undefined because the denominator of the slope formula (Δx) is zero. So, the equation of a vertical line always has the form of x = a where a is the x-coordinate of any point that lies on that line.

We're given that a point (-1, 2) lies on the vertical line that we need to find its equation, thus the equation is x = -1.

Learn more about vertical line visit:

brainly.com/question/29349507

#SPJ11

The Pythagorean identity and the double and half angle identities indicates that we get;

cos(2·a) = 288/337

cos((1/2)·a) = √((676 - 25·√(674))/1348)

The Pythagorean identity for tangent indicates that we get; tan²a + 1 = sec²a

Therefore; sec²a = (7/25)² + 1 = 674/625

Which indicates that we get; cos²(a) = 625/674

The double angle identity for cosines, indicates;

cos(2·a) = 2·cos²(a) - 1 = 2 × (625/674) - 1 = 288/337

The half angle identity indicates that we get; cos(a/2) = ±√((1 + cos(a))/2)

sec²a = 625/674

cos²(a) = 625/674

cos(a) = √(625/674) = -25/√(674)

Therefore; cos(a/2) = ±√((1 + (-25/√(674)))/2) = √((674 - 25·√(674))/1348)

Learn more on the half angle identity here: https://brainly.com/question/14308845

#SPJ1

The integral can be written as:∫sec x tan x(sec x + tan x) dx= ∫t dt= (1/2)t² + C= (1/2)(sec x + tan x)² + C

The integrals can be evaluated as follows:

1) ∫xe−xdxThe integral can be evaluated using the integration by parts.∫udv = uv − ∫vduHere,u = xdv = e^(-x)dxdu = dxv = -e^(-x)The integral can be written as:∫xe−xdx = -xe^(-x) + ∫e^(-x)dx= -xe^(-x) - e^(-x) + Cwhere C is the constant of integration

2) ∫ln(2x+1)dxThe integral can be evaluated by using the substitution method. Let u = 2x + 1. Then, du/dx = 2dx and dx = (1/2)du.∫ln(2x+1)dx=∫lnudu/2= (u ln u - u)/2= [(2x+1)ln(2x+1)-(2x+1)]/23) ∫xsec²(2x)dxThe integral can be evaluated using the integration by substitution. Let u = 2x, then du/dx = 2 and dx = du/2.∫xsec²(2x)dx= ∫ (1/2) u sec² u du= (1/2) tan u + C= (1/2)tan(2x) + Cwhere C is the constant of integration4) ∫(1/2)x²lnxdxThe integral can be evaluated using integration by parts.∫udv = uv - ∫vduHere,u = ln(x)dv = (1/2)x²dxdx = du/vThe integral can be written as:∫(1/2)x²lnxdx= (1/2)x² ln(x) - ∫x (1/2)dx= (1/2)x² ln(x) - (1/4)x² + Cwhere C is the constant of integration1) ∫sin³x cos²x dxWe can convert the integral to product form as follows:∫sin³x cos²x dx= ∫(sin²x) sin x(cos²x) dx= ∫(1-cos²x) sin x(cos²x) dxLet t = cos x, then dt/dx = -sin x dx and dx = -dt/sin x.Then, the integral can be written as:∫(1-cos²x) sin x(cos²x) dx= -∫(1-t²) t² dt= -∫(t² - t^4) dt= -[(1/3)t³ - (1/5)t⁵] + C= -(1/3)cos³x + (1/5)cos⁵x + C2) ∫cos⁵x dxWe can convert the integral to product form as follows:∫cos⁵x dx= ∫cos⁴x cos x dx= ∫(1-sin²x)² cos x dxLet t = sin x, then dt/dx = cos x dx and dx = dt/cos x.Then, the integral can be written as:∫(1-sin²x)² cos x dx= ∫(1-t²)² dt= ∫(1 - 2t² + t⁴) dt= t - (2/3)t³ + (1/5)t⁵ + C= sin x - (2/3)sin³x + (1/5)sin⁵x + C3) ∫sin⁴(3x) dxWe can convert the integral to product form as follows:∫sin⁴(3x) dx= ∫(1-cos²(3x))² dxLet t = cos(3x), then dt/dx = -3sin(3x) dx and dx = -dt/(3sin(3x)).Then, the integral can be written as:∫(1-cos²(3x))² dx= ∫(1-t²)² dt/(3sin(3x))= (-1/3) ∫(t² - 2t² + t⁴) dt/(sin(3x))= (-1/3) ∫(1 - 2t² + t⁴) dt/(sin(3x))= (-1/3) [t - (2/3)t³ + (1/5)t⁵] /(sin(3x)) + C= (-1/3) [cos(3x) - (2/3)cos³(3x) + (1/5)cos⁵(3x)] /(sin(3x)) + C4) ∫sec²x tanx dxWe can convert the integral to product form as follows:∫sec²x tanx dx= ∫sec x tan x(sec x + tan x) dxLet t = sec x + tan x, then dt/dx = sec x (sec x + tan x) and dx = dt/sec x (sec x + tan x).Then, the integral can be written as:∫sec x tan x(sec x + tan x) dx= ∫t dt= (1/2)t² + C= (1/2)(sec x + tan x)² + C

Learn more about integral :

https://brainly.com/question/18125359

#SPJ11

Given: f(x) = 3/x, a = 4 We have to find the Taylor polynomials of orders 0, 1, 2, and 3 generated by f at x = a.

[tex]Taylor PolynomialThe Taylor polynomial of order n generated by a function f at x = a is given by P (x) = f(a) + f'(a)(x - a) + f''(a)(x - a)²/2! + ... + fⁿ(a)(x - a)ⁿ/n![/tex]

[tex]Taylor PolynomialThe Taylor polynomial of order n generated by a function f at x = a is given by P (x) = f(a) + f'(a)(x - a) + f''(a)(x - a)²/2! + ... + fⁿ(a)(x - a)ⁿ/n![/tex]

[tex]Where fⁿ(a) denotes the nth derivative of f evaluated at a.[/tex]

[tex]We know that, f(x) = 3/xDifferentiating both sides of f(x) with respect to x, we getf'(x) = -3/x²[/tex]

[tex]Again differentiating both sides of f'(x) with respect to x, we getf''(x) = 6/x³[/tex]

[tex]Again differentiating both sides of f''(x) with respect to x, we getf'''(x) = -18/x⁴[/tex]

[tex]Thus,f(4) = 3/4f'(4) = -3/4²f''(4) = 6/4³f'''(4) = -18/4⁴[/tex]

[tex]The Taylor polynomial of order 0 is given byP₀(x) = f(a) = f(4) = 3/4[/tex]

[tex]The Taylor polynomial of order 1 is given byP₁(x) = f(a) + f'(a)(x - a) = f(4) + f'(4)(x - 4) = 3/4 - 3/4²(x - 4)[/tex]

[tex]The Taylor polynomial of order 2 is given byP₂(x) = f(a) + f'(a)(x - a) + f''(a)(x - a)²/2! = f(4) + f'(4)(x - 4) + f''(4)(x - 4)²/2! = 3/4 - 3/4²(x - 4) + 6/4³(x - 4)²/2![/tex]

[tex]The Taylor polynomial of order 3 is given byP₃(x) = f(a) + f'(a)(x - a) + f''(a)(x - a)²/2! + f'''(a)(x - a)³/3! = f(4) + f'(4)(x - 4) + f''(4)(x - 4)²/2! + f'''(4)(x - 4)³/3! = 3/4 - 3/4²(x - 4) + 6/4³(x - 4)²/2! - 18/4⁴(x - 4)³/3![/tex]

[tex]Hence, the required Taylor polynomials are as follows.P₀(x) = 3/4P₁(x) = 3/4 - 3/4²(x - 4)P₂(x) = 3/4 - 3/4²(x - 4) + 6/4³(x - 4)²/2!P₃(x) = 3/4 - 3/4²(x - 4) + 6/4³(x - 4)²/2! - 18/4⁴(x - 4)³/3![/tex]

To know more about the word polynomials visits :

https://brainly.com/question/25566088

#SPJ11

Answer:

Step-by-step explanation:

y=- (1/3)cos 3θ

Period:  2π/b= 2π/3 since b=3 here.

Range: The range of a cosine function is always [-1, 1].

Amplitude:  amplitude of a cosine function = the absolute value of the coefficient in front of the cosine function. In this case, the coefficient is -(1/3), so the amplitude is |-(1/3)| = 1/3.

We have been given the following function:[tex]`f(x) = log(x)`[/tex] and a value of x: `x = 1/4`We need to evaluate `f(x)` at the given value of `x`.

For that we will replace the value of `x` in `f(x)` and use a calculator to calculate the result.[tex]`f(x) = log(1/4) = -1.386`[/tex]Rounding to three decimal places, we get `-1.386` as the final result.

We are given the following expression:[tex]`4 log(x) + 3 log(y) - 2 log(z)`[/tex]We need to condense the expression to the logarithm of a single quantity. To do that, we use the following properties of logarithms:

1. `log a + log b = log(ab)`2. `log a - log b = log(a/b)`We can use property 1 to condense the first two terms of the expression as follows[tex]:`4 log(x) + 3 log(y) = log(x^4) + log(y^3) = log(x^4y^3)`[/tex]Now, we have:`log(x^4y^3) -

2 log(z)`Using property 2, we can write the above expression as:

[tex]`log(x^4y^3/z^2)`[/tex]Hence, we have condensed the expression to the logarithm of a single quantity:[tex]`log(x^4y^3/z^2)`[/tex]

To know more about evaluate visit:

https://brainly.com/question/33104289

#SPJ11

The most accurate statement about the p-value for this test is: 0.05 < p-value < 0.10.

In a hypothesis test, the null hypothesis can be rejected at the .10 level but cannot be rejected at the .05 level. This means that the p-value for the test falls between these two levels.

A p-value is a measure of the strength of evidence against the null hypothesis. It represents the probability of obtaining a test statistic as extreme as, or more extreme than, the observed value, assuming that the null hypothesis is true.

In this case, the fact that the null hypothesis can be rejected at the .10 level means that the p-value is less than .10. However since the null hypothesis cannot be rejected at the .05 level, it implies that the p-value is greater than .05.

This indicates that the evidence against the null hypothesis is not strong enough to reject it at the .05 level, but it is strong enough to reject it at the .10 level. The p-value falls between these two values, providing a range of uncertainty in the results of the hypothesis test.

In summary, the p-value in this scenario is greater than .05 but less than .10, indicating moderate evidence against the null hypothesis.

Learn more about  p-value

https://brainly.com/question/33325466

#SPJ11

In a hypothesis test, the null hypothesis can be rejected at the 0.10 level but cannot be rejected at the 0.05 level. This means that the p-value obtained in the test falls between these two levels.

The p-value is a measure of the strength of evidence against the null hypothesis. It represents the probability of observing the data or more extreme data, assuming that the null hypothesis is true.

When the p-value is less than the chosen significance level (alpha), typically 0.05, it indicates that the data provides strong evidence against the null hypothesis. In this case, we reject the null hypothesis in favor of the alternative hypothesis.

However, if the p-value is greater than the significance level, such as 0.10, it means that the data does not provide enough evidence to reject the null hypothesis. We fail to reject the null hypothesis and do not have sufficient evidence to support the alternative hypothesis.

Therefore, for this given hypothesis test, since the null hypothesis can be rejected at the 0.10 level but not at the 0.05 level, the most accurate statement about the p-value is:

0.05 < p-value < 0.10

This means that the p-value falls between 0.05 and 0.10, indicating moderate evidence against the null hypothesis, but not strong enough to reject it at the 0.05 level.

Learn more about null hypothesis

https://brainly.com/question/30821298

#SPJ11

The accompanying game is a 2x2 matrix in which two firms, Firm 1 and Firm 2, have to make independent decisions on whether to charge a high or low price.

The accompanying game's payoff matrix is as follows:

Firm One/Firm Two

|High Price| Low Price

|High Price| (10,10)| (-50,50)|Low Price| (50,-50)| (0,0)

|The strategy Firm 1 should choose is a high price.

When Firm 2 charges a high price, choosing a high price would give Firm 1 a payoff of 10. If Firm 2 chooses to charge a low price, selecting a high price would result in a payoff of 50 for Firm 1.

Regardless of whether Firm 2 chooses to charge a high or low price, selecting a high price is the optimal decision. Similarly, Firm 2 should select a high price.

When Firm 1 charges a high price, choosing a high price would give Firm 2 a payoff of 10. If Firm 1 chooses to charge a low price, selecting a high price would result in a payoff of 50 for Firm 2.

Regardless of whether Firm 1 chooses to charge a high or low price, choosing a high price is the optimal decision.

To learn more about accompanying game from the given link.

https://brainly.com/question/29979503

#SPJ11

Both the firms that is firm 1 and firm 2 have to chose high prices .

The accompanying game is a 2x2 matrix in which two firms, Firm 1 and Firm 2, have to make independent decisions on whether to charge a high or low price.

The accompanying game's payoff matrix is as follows:

Firm One/Firm Two

|High Price| Low Price

|High Price| (10,10)| (-50,50)|Low Price| (50,-50)| (0,0)

|The strategy Firm 1 should choose is a high price.

When Firm 2 charges a high price, choosing a high price would give Firm 1 a payoff of 10. If Firm 2 chooses to charge a low price, selecting a high price would result in a payoff of 50 for Firm 1.

Regardless of whether Firm 2 chooses to charge a high or low price, selecting a high price is the optimal decision. Similarly, Firm 2 should select a high price.

When Firm 1 charges a high price, choosing a high price would give Firm 2 a payoff of 10. If Firm 1 chooses to charge a low price, selecting a high price would result in a payoff of 50 for Firm 2.

Regardless of whether Firm 1 chooses to charge a high or low price, choosing a high price is the optimal decision.

Know more about accompanying game,

brainly.com/question/29979503

#SPJ4

The total derivative of Z with respect to t is given by

[tex]Z' = (-2e^{-2t})(t + 3/(3e^{-2t} + 3t^2 - t)).[/tex]

We have,

To find the total derivative of Z with respect to t, we can apply the chain rule and the product rule.

Let's calculate it step by step:

Given:

z = tx + ln(3x + y)

[tex]x = e^{-2t}y = 3t^2 - t[/tex]

We'll start by finding the derivative of z with respect to x, and then multiply it by the derivative of x with respect to t:

dz/dx = t + 1/(3x + y) * (3)

= t + 3/(3x + y)

Next, we'll find the derivative of x with respect to t:

[tex]dx/dt = d/dt(e^{-2t})\\= -2e^{-2t}[/tex]

Now, applying the chain rule, we can find the derivative of z with respect to t:

dz/dt = dz/dx * dx/dt

[tex]= (t + 3/(3x + y)) * (-2e^{-2t})[/tex]

Substituting the expressions for x and y, we have:

[tex]dz/dt = (t + 3/(3e^{-2t} + 3t^2 - t)) * (-2e^{-2t})[/tex]

Simplifying further, we can rewrite it as the total derivative of Z with respect to t:

[tex]Z' = (-2e^{-2t})(t + 3/(3e^{-2t} + 3t^2 - t))[/tex]

Thus,

The total derivative of Z with respect to t is given by

[tex]Z' = (-2e^{-2t})(t + 3/(3e^{-2t} + 3t^2 - t)).[/tex]

Learn more about derivatives here:

https://brainly.com/question/29020856

#SPJ4

To determine the number of solutions for each system, we can analyze the equations provided. In the first equation, x+y+z = 7, we have three variables (x, y, and z) and one equation. This means that we have an infinite number of solutions, as we have more variables than equations.

In the second equation, y+z = ?, we have two variables (y and z) and one equation. Since we don't have enough information to solve for both variables, we cannot determine the number of solutions. The given equation is incomplete.

Similarly, in the third equation, y+z = ?, we also have two variables (y and z) and one equation. Again, since we don't have enough information to solve for both variables, we cannot determine the number of solutions. The given equation is also incomplete.

Therefore, for the given system, the number of solutions is infinite for the first equation, and the number of solutions cannot be determined for the second and third equations as they are incomplete.

To know more about number of solutions visit:

https://brainly.com/question/30072305

#SPJ11