Consider the following equation. 4x² + 25y² = 100 (a) Find dy/dx by implicit differentiation. 4x 25y (b) Solve the equation explicitly for y and differentiate to get dy/dx in terms of x. (Consider only the first and second quadrants for this part.) x (c) Check that your solutions to part (a) and (b) are consistent by substituting the expression for y into your solution for part (a). y' =

The general solution to the given differential equation is y(t) = [tex]C_{1}\ cos{2t}\ + C_{2} \ sin{2t} - e^{2/3t}[/tex], where C₁ and C₂ are constants.

The given differential equation is a second-order linear homogeneous equation with constant coefficients. Its characteristic equation is [tex]r^2[/tex] + 2 = 0, which has complex roots r = ±i√2. Since the roots are complex, the general solution will involve trigonometric functions.

Let's assume the solution has the form y(t) = [tex]e^{rt}[/tex]. Substituting this into the differential equation, we get [tex]r^2e^{rt} + 2e^{rt} = 0[/tex]. Dividing both sides by [tex]e^{rt}[/tex], we obtain the characteristic equation [tex]r^2[/tex] + 2 = 0.

The complex roots of the characteristic equation are r = ±i√2. Using Euler's formula, we can rewrite these roots as r₁ = i√2 and r₂ = -i√2. The general solution for the homogeneous equation is y_h(t) = [tex]C_{1}e^{r_{1} t} + C_{2}e^{r_{2}t}[/tex]

Next, we need to find the particular solution for the given non-homogeneous equation. The non-homogeneous term includes a tangent function and an exponential term. We can use the method of undetermined coefficients to find a particular solution. Assuming y_p(t) has the form [tex]A \tan{2t} + Be^{2/3t}[/tex], we substitute it into the differential equation and solve for the coefficients A and B.

After finding the particular solution, we can add it to the general solution of the homogeneous equation to obtain the general solution of the non-homogeneous equation: y(t) = y_h(t) + y_p(t). Simplifying the expression, we arrive at the general solution y(t) = C₁ cos(2t) + C₂ sin(2t) - [tex]e^{2/3t}[/tex], where C₁ and C₂ are arbitrary constants determined by initial conditions or boundary conditions.

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To find the value of y(2), we need to solve the initial value problem and evaluate the solution at x = 2.

The given initial value problem is:

y' - y = x² + x

y(1) = 2

First, let's find the integrating factor for the homogeneous equation y' - y = 0. The integrating factor is given by e^(∫-1 dx), which simplifies to [tex]e^(-x).[/tex]

Next, we multiply the entire equation by the integrating factor: [tex]e^(-x) * y' - e^(-x) * y = e^(-x) * (x² + x)[/tex]

Applying the product rule to the left side, we get:

[tex](e^(-x) * y)' = e^(-x) * (x² + x)[/tex]

Integrating both sides with respect to x, we have:

∫ ([tex]e^(-x)[/tex]* y)' dx = ∫[tex]e^(-x)[/tex] * (x² + x) dx

Integrating the left side gives us:

[tex]e^(-x)[/tex] * y = -[tex]e^(-x)[/tex]* (x³/3 + x²/2) + C1

Simplifying the right side and dividing through by e^(-x), we get:

y = -x³/3 - x²/2 +[tex]Ce^x[/tex]

Now, let's use the initial condition y(1) = 2 to solve for the constant C:

2 = -1/3 - 1/2 + [tex]Ce^1[/tex]

2 = -5/6 + Ce

C = 17/6

Finally, we substitute the value of C back into the equation and evaluate y(2):

y = -x³/3 - x²/2 + (17/6)[tex]e^x[/tex]

y(2) = -(2)³/3 - (2)²/2 + (17/6)[tex]e^2[/tex]

y(2) = -8/3 - 2 + (17/6)[tex]e^2[/tex]

y(2) = -14/3 + (17/6)[tex]e^2[/tex]

So, the value of y(2) is -14/3 + (17/6)[tex]e^2.[/tex]

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The general solution is:+e(13 - e3 + e4  e5  -3e6 - 3e7  e8  e9)

we have a unique solution, and the general solution is given by:

x1 = 13 - e3 + e4x2 = e5x3 = -3e6 - 3e7x4 = e8x5 = e9

where e3, e4, e5, e6, e7, e8, and e9 are arbitrary parameters.

To fill the blanks and write the general solution for a linear system whose augmented matrices were reduced to

-3 0 0 3 0 6 2 0 6 0 8 0 -1 -5 0 -7 0 0 0 3 9 0 0 0 0 0,

we need to use the technique of the Gauss-Jordan elimination method. The general solution of the linear system is obtained by setting all the leading variables (variables in the pivot positions) to arbitrary parameters and expressing the non-leading variables in terms of these parameters.

The rank of the coefficient matrix is also calculated to determine the existence of the solution to the linear system.

In the given matrix, we have 5 leading variables, which are the pivots in the first, second, third, seventh, and ninth columns.

So we need 5 parameters, one for each leading variable, to write the general solution.

We get rid of the coefficients below and above the leading variables by performing elementary row operations on the augmented matrix and the result is given below.

-3 0 0 3 0 6 2 0 6 0 8 0 -1 -5 0 -7 0 0 0 3 9 0 0 0 0 0

Adding 2 times row 1 to row 3 and adding 5 times row 1 to row 2, we get

-3 0 0 3 0 6 2 0 0 0 3 0 -1 10 0 -7 0 0 0 3 9 0 0 0 0 0

Dividing row 1 by -3 and adding 7 times row 1 to row 4, we get

1 0 0 -1 0 -2 -2 0 0 0 -1 0 1 -10 0 7 0 0 0 -3 -9 0 0 0 0 0

Adding 2 times row 5 to row 6 and dividing row 5 by -3,

we get1 0 0 -1 0 -2 0 0 0 0 1 0 -1 10 0 7 0 0 0 -3 -9 0 0 0 0 0

Dividing row 3 by 3 and adding row 3 to row 2, we get

1 0 0 -1 0 0 0 0 0 0 1 0 -1 10 0 7 0 0 0 -3 -3 0 0 0 0 0

Adding 3 times row 3 to row 1,

we get

1 0 0 0 0 0 0 0 0 0 1 0 -1 13 0 7 0 0 0 -3 -3 0 0 0 0 0

So, we see that the rank of the coefficient matrix is 5, which is equal to the number of leading variables.

Thus, we have a unique solution, and the general solution is given by:

x1 = 13 - e3 + e4x2 = e5x3 = -3e6 - 3e7x4 = e8x5 = e9

where e3, e4, e5, e6, e7, e8, and e9 are arbitrary parameters.

Hence, the general solution is:+e(13 - e3 + e4  e5  -3e6 - 3e7  e8  e9)

The general solution is:+e(13 - e3 + e4  e5  -3e6 - 3e7  e8  e9)

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Therefore, the orthogonal projection of v onto the subspace W is approximately (-32.27, -64.57, -103.89, -16.71).

To find the orthogonal projection of vector v onto the subspace W spanned by the given vectors, we can use the formula:

projₓy = (y⋅x / ||x||²) * x

where x represents the vectors spanning the subspace, y represents the vector we want to project, and ⋅ denotes the dot product.

Let's calculate the orthogonal projection:

Step 1: Normalize the spanning vectors.

First, we normalize the spanning vectors of W:

u₁ = (-1/√6, -2/√6, -3/√6, -2/√6)

u₂ = (4/√53, 5/√53, -26/√53)

Step 2: Calculate the dot product.

Next, we calculate the dot product of the vector we want to project, v, with the normalized spanning vectors:

v⋅u₁ = (-1)(-1/√6) + (-16)(-2/√6) + (-4)(-3/√6) + (12)(-2/√6)

= 1/√6 + 32/√6 + 12/√6 - 24/√6

= 21/√6

v⋅u₂ = (-1)(4/√53) + (-16)(5/√53) + (-4)(-26/√53) + (12)(0/√53)

= -4/√53 - 80/√53 + 104/√53 + 0

= 20/√53

Step 3: Calculate the projection.

Finally, we calculate the orthogonal projection of v onto the subspace W:

projW(v) = (v⋅u₁) * u₁ + (v⋅u₂) * u₂

= (21/√6) * (-1/√6, -2/√6, -3/√6, -2/√6) + (20/√53) * (4/√53, 5/√53, -26/√53)

= (-21/6, -42/6, -63/6, -42/6) + (80/53, 100/53, -520/53)

= (-21/6 + 80/53, -42/6 + 100/53, -63/6 - 520/53, -42/6)

= (-10284/318, -20544/318, -33036/318, -5304/318)

≈ (-32.27, -64.57, -103.89, -16.71)

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The graph A is the graph of the function [tex]f(x) = -x^4 + 9[/tex].

The end behavior of a function refers to how the function behaves as the input variable approaches positive or negative infinity.

The function in this problem is given as follows:

[tex]f(x) = -x^4 + 9[/tex]

It has a negative leading coefficient with an even root, meaning that the function will approach negative infinity both to the left and to the right of the graph.

Hence the graph A is the graph of the function [tex]f(x) = -x^4 + 9[/tex].

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The number of people who have heard the rumor after 3 hours of using Improved Euler's method with a step size of 1 is R(3).  

The Improved Euler's method is a numerical approximation technique used to solve differential equations. It involves taking small steps and updating the solution at each step based on the slope at that point.

To approximate the number of people who have heard the rumor after 3 hours, we start with the initial condition R(0) = 2 (since only 2 people knew the rumor to start with) and use the Improved Euler's method with a step size of 1.

Let's perform the calculation step by step:

At t = 0, R(0) = 2 (given initial condition)

Using the Improved Euler's method:

k1 = 0.5 * R(0) * (1 - R(0)/120) = 0.5 * 2 * (1 - 2/120) = 0.0167

k2 = 0.5 * (R(0) + 1 * k1) * (1 - (R(0) + 1 * k1)/120) = 0.5 * (2 + 1 * 0.0167) * (1 - (2 + 1 * 0.0167)/120) = 0.0166

Approximate value of R(1) = R(0) + 1 * k2 = 2 + 1 * 0.0166 = 2.0166

Similarly, we can continue this process for t = 2, 3, and so on.

For t = 3, the approximate value of R(3) represents the number of people who have heard the rumor after 3 hours.

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The equation function of cosine with an amplitude of 4, a period of 2, and a point at (0,2) is y = 4cos(2πx) + 2.

The general form of a cosine function is y = A cos(Bx - C) + D, where A represents the amplitude, B is related to the period, C indicates any phase shift, and D represents a vertical shift.

In this case, the given amplitude is 4, which means the graph will oscillate between -4 and 4 units from its centerline. The period is 2, which indicates that the function completes one full cycle over a horizontal distance of 2 units.

To incorporate the given point (0,2), we know that when x = 0, the corresponding y-value should be 2. Since the cosine function is at its maximum at x = 0, the vertical shift D is 2 units above the centerline.

Using these values, the equation function becomes y = 4cos(2πx) + 2, where 4 represents the amplitude, 2π/2 simplifies to π in the argument of cosine, and 2 is the vertical shift. This equation satisfies the given conditions of the cosine function.

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The amount (A) after one year is $4,212.00

Given that P = $3,900,

r = 8% and

t = 1 year,

we need to find the amount using the formula A = P(1 + rt).

To find the value of A, substitute the given values of P, r, and t into the formula

A = P(1 + rt).

A = P(1 + rt)

A = $3,900 (1 + 0.08 × 1)

A = $3,900 (1 + 0.08)

A = $3,900 (1.08)A = $4,212.00

Therefore, the amount (A) after one year is $4,212.00. Hence, the detail ans is:A = $4,212.00.

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We are required to determine the power series for the given functions centered at c and determine the interval of convergence for each function.

a) f(x) = 7²-3; c=5

Here, we can write 7²-3 as 48.

So, we have to find the power series of 48 centered at 5.

The power series for any constant is the constant itself.

So, the power series for 48 is 48 itself.

The interval of convergence is also the point at which the series converges, which is only at x = 5.

Hence the interval of convergence for the given function is [5, 5].

b) f(x) = 2x² +3² ; c=0

Here, we can write 3² as 9.

So, we have to find the power series of 2x²+9 centered at 0.

Using the power series for x², we can write the power series for 2x² as 2x² = 2(x^2).

Now, the power series for 2x²+9 is 2(x^2) + 9.

For the interval of convergence, we can find the radius of convergence R using the formula:

`R= 1/lim n→∞|an/a{n+1}|`,

where an = 2ⁿ/n!

Using this formula, we can find that the radius of convergence is ∞.

Hence the interval of convergence for the given function is (-∞, ∞).c) f(x)=- d) f(x)=- ; c=3

Here, the functions are constant and equal to 0.

So, the power series for both functions would be 0 only.

For both functions, since the power series is 0, the interval of convergence would be the point at which the series converges, which is only at x = 3.

Hence the interval of convergence for both functions is [3, 3].

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the derivative function of f(x) is f'(x) = 8.To find f'(a) when a = 2, simply substitute 2 for x in the derivative function:

f'(2) = 8So the value of f'(a) for a = 2 is f'(2) = 8.

The question is asking for the derivative function, f'(x), of the function f(x) = 4(2x + 1) using limits, as well as the value of f'(a) when a = 2.

To find the derivative function, f'(x), using limits, follow these steps:

Step 1:

Write out the formula for the derivative of f(x):f'(x) = lim h → 0 [f(x + h) - f(x)] / h

Step 2:

Substitute the function f(x) into the formula:

f'(x) = lim h → 0 [f(x + h) - f(x)] / h = lim h → 0 [4(2(x + h) + 1) - 4(2x + 1)] / h

Step 3:

Simplify the expression inside the limit:

f'(x) = lim h → 0 [8x + 8h + 4 - 8x - 4] / h = lim h → 0 (8h / h) + (0 / h) = 8

Step 4:

Write the final answer: f'(x) = 8

Therefore, the derivative function of f(x) is f'(x) = 8.To find f'(a) when a = 2, simply substitute 2 for x in the derivative function:

f'(2) = 8So the value of f'(a) for a = 2 is f'(2) = 8.

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a. To construct a 98% confidence interval for the population mean (μ), we can use the formula:

x ± Z * (σ / √n),

where x is the sample mean, Z is the critical value corresponding to the desired confidence level, σ is the population standard deviation, and n is the sample size.

Plugging in the given values, we have:

x = 36.03, σ = 5.5, n = 58, and the critical value Z can be determined using the standard normal distribution table for a 98% confidence level (Z = 2.33).

Calculating the confidence interval using the formula, we find:

36.03 ± 2.33 * (5.5 / √58).

The resulting interval provides a range within which we can be 98% confident that the population mean falls.

b. The validity of the confidence interval constructed in part (a) relies on the assumption that the population is approximately normal. If the population is not approximately normal, the validity of the confidence interval may be compromised.

The validity of the confidence interval is contingent upon meeting certain assumptions, including a normal distribution for the population. If the population deviates significantly from normality, the confidence interval may not accurately capture the true population mean.

Therefore, it is crucial to assess the underlying distribution of the population before relying on the validity of the constructed confidence interval.

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The correct answer is Bob's share is approximately $350 and Carlos's share is approximately $300.

(a) To find the original price of the shoes, we can use the fact that the sale price is 88% of the original price (100% - 12% discount).

Let's denote the original price as x.

The equation can be set up as:

0.88x = $120

To find x, we divide both sides of the equation by 0.88:

x = $120 / 0.88

Using a calculator, we find:

x ≈ $136.36

Therefore, the original price of the shoes was approximately $136.36.

(b) To calculate the ratio of the mass of the proton to the mass of theelectron, we divide the mass of the proton by the mass of the electron.

Mass of proton: 1.6726 x 10^(-27) kg

Mass of electron: 9.1095 x 10^(-31) kg

Ratio = Mass of proton / Mass of electron

Ratio = (1.6726 x 10^(-27)) / (9.1095 x 10^(-31))

Performing the division, we get:

Ratio ≈ 1837.58

Therefore, the ratio of the mass of the proton to the mass of the electron is approximately 1837.58.

(c) Let's assume the common ratio of the coins is x. Then, we can set up the equation:

8x + x + 2x = 30

Combining like terms:11x = 30

Dividing both sides by 11:x = 30 / 11

Since the ratio of 50-cent, one-dollar, and two-dollar coins is 8:1:2, we can multiply the value of x by the respective ratios to find the number of each coin:

50-cent coins: 8x = 8 * (30 / 11)

one-dollar coins: 1x = 1 * (30 / 11)

Calculating the values:

50-cent coins ≈ 21.82

one-dollar coins ≈ 2.73

Since we cannot have fractional coins, we round the values:

50-cent coins ≈ 22

one-dollar coins ≈ 3

Therefore, Gavin has approximately 22 fifty-cent coins and 3 one-dollar coins.

(d) The scale ratio of the model city is 1:1000. This means that 1 cm on the model represents 1000 cm (or 10 meters) in actuality.

Given that the scaled height of the building is 8 cm, we can multiply it by the scale ratio to find the actual height:

Actual height = Scaled height * Scale ratio

Actual height = 8 cm * 10 meters/cm

Calculating the value:

Actual height = 80 meters

Therefore, the actual height of the building is 80 meters.

(e) The ratio of Akhil's share to the total share is 3:16 (3 + 7 + 6 = 16).

Since Akhil's share is $150, we can calculate the total share using the ratio:

Total share = (Total amount / Akhil's share) * Akhil's share

Total share = (16 / 3) * $150

Calculating the value:

Total share ≈ $800

To find Bob's share, we can calculate it using the ratio:

Bob's share = (Bob's ratio / Total ratio) * Total share

Bob's share = (7 / 16) * $800

Calculating the value:

Bob's share ≈ $350

To find Carlos's share, we can calculate it using the ratio:

Carlos's share = (Carlos's ratio / Total ratio) * Total share

Carlos's share = (6 / 16) * $800

Calculating the value:

Carlos's share ≈ $300

Therefore, Bob's share is approximately $350 and Carlos's share is approximately $300.

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The factor from the expression 5(3x² + 9x) - 14 is not listed among the options you provided. However, I can help you simplify the expression and identify the factors within it.

To simplify the expression, we can distribute the 5 to both terms inside the parentheses:

5(3x² + 9x) - 14 = 15x² + 45x - 14

From this simplified expression, we can identify the factors as follows:

15x²: This represents the term with the variable x squared.

45x: This represents the term with the variable x.

-14: This represents the constant term.

Therefore, the factors from the expression are 15x², 45x, and -14.

(a) lim[T→0] ∫[0 to π/4] 5T/(4√tan(x)) sec²(x) dx

(b) The integral evaluates to [5T/4] [ln(√2 + 1) + ln(√2) - (√2/2)].

(a) To rewrite the improper integral as the limit of a proper integral, we will introduce a parameter and take the limit as the parameter approaches a specific value.

The given improper integral is:

∫[0 to π/4] 5T/(4√tan(x)) sec²(x) dx

To rewrite it as a limit, we introduce a parameter, let's call it T, and rewrite the integral as:

∫[0 to π/4] 5T/(4√tan(x)) sec²(x) dx

Taking the limit as T approaches 0, we have:

lim[T→0] ∫[0 to π/4] 5T/(4√tan(x)) sec²(x) dx

This limit converts the improper integral into a proper integral.

(b) To calculate the integral, let's proceed with the evaluation of the integral:

∫[0 to π/4] 5T/(4√tan(x)) sec²(x) dx

We can simplify the integrand by using the identity sec²(x) = 1 + tan²(x):

∫[0 to π/4] 5T/(4√tan(x)) (1 + tan²(x)) dx

Expanding and simplifying, we have:

∫[0 to π/4] 5T/(4√tan(x)) + (5T/4)tan²(x) dx

Now, we can split the integral into two parts:

∫[0 to π/4] 5T/(4√tan(x)) dx + ∫[0 to π/4] (5T/4)tan²(x) dx

The first integral can be evaluated as:

∫[0 to π/4] 5T/(4√tan(x)) dx = [5T/4]∫[0 to π/4] sec(x) dx

= [5T/4] [ln|sec(x) + tan(x)|] evaluated from 0 to π/4

= [5T/4] [ln(√2 + 1) - ln(1)] = [5T/4] ln(√2 + 1)

The second integral can be evaluated as:

∫[0 to π/4] (5T/4)tan²(x) dx = (5T/4) [ln|sec(x)| - x] evaluated from 0 to π/4

= (5T/4) [ln(√2) - (√2/2 - 0)] = (5T/4) [ln(√2) - (√2/2)]

Thus, the value of the integral is:

[5T/4] ln(√2 + 1) + (5T/4) [ln(√2) - (√2/2)]

Simplifying further:

[5T/4] [ln(√2 + 1) + ln(√2) - (√2/2)]

Therefore, the integral evaluates to [5T/4] [ln(√2 + 1) + ln(√2) - (√2/2)].

Note: Depending on the value of T, the result of the integral will vary. If T is 0, the integral becomes 0. Otherwise, the integral will have a non-zero value.

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To make the function f(x) = e^(-1/x²) differentiable on (-∞, +∞), the value of a that satisfies this condition is a = 0.

In order for f(x) to be differentiable at x = 0, the left and right derivatives at that point must be equal. We calculate the left derivative by taking the limit as h approaches 0- of [f(0+h) - f(0)]/h. Substituting the given function, we obtain the left derivative as lim(h→0-) [e^(-1/h²) - 0]/h. Simplifying, we find that this limit equals 0.

Next, we calculate the right derivative by taking the limit as h approaches 0+ of [f(0+h) - f(0)]/h. Again, substituting the given function, we have lim(h→0+) [e^(-1/h²) - 0]/h. By simplifying and using the properties of exponential functions, we find that this limit also equals 0.

Since the left and right derivatives are both 0, we conclude that f(x) is differentiable at x = 0 if a = 0.

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Lynn is paying $550 and Judy is paying $400 for the cottage rental in Maine this summer.

To find out how much of the rental fee Lynn and Judy are paying, we have to create an equation that shows the relationship between the variables in the problem.

Let L be Lynn's share of the cost, and J be Judy's share of the cost.

Then we can translate the given information into the following system of equations:

L + J = 950 (since they are pooling their savings to pay the $950 rental cost)

L = 2J - 250 (since Lynn is paying $250 less than twice Judy's share)

To solve this system, we can use substitution.

We'll solve the second equation for J and then substitute that expression into the first equation:

L = 2J - 250

L + 250 = 2J

L/2 + 125 = J

Now we can substitute that expression for J into the first equation and solve for L:

L + J = 950

L + L/2 + 125 = 950

3L/2 = 825L = 550

So, Lynn is paying $550 and Judy is paying $400.

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The amount out of the first $ 111 payment that will go towards interest would be A. $ 50. 00.

For an installment loan, the first payment is mostly used to pay off the interest. The interest portion of the loan payment can be calculated using the formula:

Interest = Principal x Interest rate / Number of payments per year

Given the information:

Principal is $5000

the Interest rate is 12% per year

number of payments per year is 12

The interest is therefore :

= 5, 000 x 0. 12 / 12 months

= $ 50

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The derivative of the function f(x) = (x - 4)(4x + 4) can be found using the Product Rule. The correct option is OC i.e., the derivative is 8x - 12.

To find the derivative of a product of two functions, we can use the Product Rule, which states that the derivative of the product of two functions u(x) and v(x) is given by u'(x)v(x) + u(x)v'(x).

Applying the Product Rule to the given function f(x) = (x - 4)(4x + 4), we differentiate the first function (x - 4) and keep the second function (4x + 4) unchanged, then add the product of the first function and the derivative of the second function.

a. Using the Product Rule, the derivative of f(x) is:

f'(x) = (x - 4)(4) + (1)(4x + 4)

Simplifying this expression, we have:

f'(x) = 4x - 16 + 4x + 4

Combining like terms, we get:

f'(x) = 8x - 12

Therefore, the correct answer is OC. The derivative is 8x - 12.

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Answer: 9

Step-by-step explanation:

Answer:To find the average rate of change for the interval from x = 5 to x = 6, we need to calculate the change in the function values over that interval and divide it by the change in x.

Given the points (5, 0) and (6, 4), we can calculate the change in the function values:

Change in y = 4 - 0 = 4

Change in x = 6 - 5 = 1

Average rate of change = Change in y / Change in x = 4 / 1 = 4

Therefore, the correct answer is 4. None of the given options (A, B, C, or D) match the correct answer.

Step-by-step explanation:

The magnification n can be found by using the formula n = -s'/s, where s' is the image distance and s is the object distance. The value of s' obtained from this new equation can be found by rearranging the formula to s' = -ns.

To find the magnification n, we can use the formula n = -s'/s, where s' is the image distance and s is the object distance. In this case, the object is placed 45.0 centimeters from the mirror, so s = 45.0 cm. The magnification can be found by calculating the ratio of the image distance to the object distance. By rearranging the formula, we get n = -s'/s.

To find the value of s' obtained from this new equation, we can rearrange the formula n = -s'/s to solve for s'. This gives us s' = -ns. By substituting the value of n calculated earlier, we can find the value of s'. The negative sign indicates that the image is inverted.

Using the given values, we can now calculate the magnification and the value of s'. Plugging in s = 45.0 cm, we find that s' = -ns = -(2/3)(45.0 cm) = -30.0 cm. This means that the image is located 30.0 centimeters from the mirror and is inverted compared to the object.

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To find the value of k, we can substitute the given values of y and x into the equation.

If y varies inversely as the square of x, we can express this relationship using the equation y = k/x^2, where k is the constant of variation.

When x = 1, y = 7/4. Substituting these values into the equation, we get:

7/4 = k/1^2

7/4 = k

Now that we have determined the value of k, we can use it to find y when x = 3. Substituting x = 3 and k = 7/4 into the equation, we get:

y = (7/4)/(3^2)

y = (7/4)/9

y = 7/4 * 1/9

y = 7/36

Therefore, when x = 3, y is equal to 7/36. The relationship between x and y is inversely proportional to the square of x, and as x increases, y decreases.

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The temperature reaches its maximum value 2.95 hours after noon, which is  at 2:56 p.m.

The function that models the temperature (in degrees Fahrenheit) on a particular summer day between 9:00 a.m. and 10:00 p.m. is given by

f(t) = -t² + 5.9t + 87,

where t represents the number of hours after noon.

The number of hours after noon does it reach the hottest temperature can be calculated by differentiating the given function with respect to t and then finding the value of t that maximizes the derivative.

Thus, differentiating

f(t) = -t² + 5.9t + 87,

we have:

'(t) = -2t + 5.9

At the maximum temperature, f'(t) = 0.

Therefore,-2t + 5.9 = 0 or

t = 5.9/2

= 2.95

Thus, the temperature reaches its maximum value 2.95 hours after noon, which is approximately at 2:56 p.m. (since 0.95 x 60 minutes = 57 minutes).

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The recurrence relation relates the coefficients ak, ak+1, and ak+2 for each value of k is (2k(k-1) + 1)ak + (2l + 2 - 2(k+1)²) * ak+1 + (x - 3 - 2l) * ak+2 = 0.

To find the recurrence relation for the coefficients of the power series solution, let's substitute the power series form into the differential equation and equate the coefficients of like powers of p.

Given the equation: p + (2l + 2 - 2p²) + (x - 3 - 2l) pu = 0

Let's assume the power series solution takes the form: u = Σk akp

Differentiating u with respect to p twice, we have:

d²u/dp² = Σk ak * d²pⁿ/dp²

The second derivative of p raised to the power n with respect to p can be calculated as follows:

d²pⁿ/dp² = n(n-1)p^(n-2)

Substituting this back into the expression for d²u/dp², we have:

d²u/dp² = Σk ak * n(n-1)p^(n-2)

Now let's substitute this expression for d²u/dp² and the power series form of u into the differential equation:

p + (2l + 2 - 2p²) + (x - 3 - 2l) * p * Σk akp = 0

Expanding and collecting like powers of p, we get:

Σk [(2k(k-1) + 1)ak + (2l + 2 - 2(k+1)²) * ak+1 + (x - 3 - 2l) * ak+2] * p^k = 0

Since the coefficient of each power of p must be zero, we obtain a recurrence relation for the coefficients:

(2k(k-1) + 1)ak + (2l + 2 - 2(k+1)²) * ak+1 + (x - 3 - 2l) * ak+2 = 0

This recurrence relation relates the coefficients ak, ak+1, and ak+2 for each value of k.

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Given, f'(x) = f(x)

= (x² + 1)sec(x).

To find the derivative of the given function, we use the product rule of derivatives

Where the first function is (x² + 1) and the second function is sec(x).

By using the product rule of differentiation, we get:

f'(x) = (x² + 1) * d(sec(x)) / dx + sec(x) * d(x² + 1) / dx

The derivative of sec(x) is given as,

d(sec(x)) / dx = sec(x)tan(x).

Differentiating (x² + 1) w.r.t. x gives d(x² + 1) / dx = 2x.

Substituting the values in the above formula, we get:

f'(x) = (x² + 1) * sec(x)tan(x) + sec(x) * 2x

= sec(x) * (tan(x) * (x² + 1) + 2x)

Therefore, the derivative of the given function f'(x) is,

f'(x) = sec(x) * (tan(x) * (x² + 1) + 2x).

Hence, the answer is that

f'(x) = sec(x) * (tan(x) * (x² + 1) + 2x)

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a) For the function f(x) = 7²-3, centered at c = 5, we can find the power series representation by expanding the function into a Taylor series around x = c.

First, let's find the derivatives of the function:

f(x) = 7x² - 3

f'(x) = 14x

f''(x) = 14

Now, let's evaluate the derivatives at x = c = 5:

f(5) = 7(5)² - 3 = 172

f'(5) = 14(5) = 70

f''(5) = 14

The power series representation centered at c = 5 can be written as:

f(x) = f(5) + f'(5)(x - 5) + (f''(5)/2!)(x - 5)² + ...

Substituting the evaluated derivatives:

f(x) = 172 + 70(x - 5) + (14/2!)(x - 5)² + ...

b) For the function f(x) = 2x² + 3², centered at c = 0, we can follow the same process to find the power series representation.

First, let's find the derivatives of the function:

f(x) = 2x² + 9

f'(x) = 4x

f''(x) = 4

Now, let's evaluate the derivatives at x = c = 0:

f(0) = 9

f'(0) = 0

f''(0) = 4

The power series representation centered at c = 0 can be written as:

f(x) = f(0) + f'(0)x + (f''(0)/2!)x² + ...

Substituting the evaluated derivatives:

f(x) = 9 + 0x + (4/2!)x² + ...

c) The provided function f(x)=- does not have a specific form. Could you please provide the expression for the function so I can assist you further in finding the power series representation?

d) Similarly, for the function f(x)=- , centered at c = 3, we need the expression for the function in order to find the power series representation. Please provide the function expression, and I'll be happy to help you with the power series and interval of convergence.

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Answer:
[tex]2x^2-4x-5=0[/tex] is standard form.

Step-by-step explanation:
Standard form of a quadratic equation should be equal to 0. Standard form should be [tex]ax^2+bx+c=0[/tex], unless this isn't a quadratic equation?

We can convert your equation to standard form with a few calculations. First, subtract 5 from both sides:

[tex]x(2x-4)-5=0[/tex]

Then, distribute the x in front:

[tex]2x^2-4x-5=0[/tex]

The equation should now be in standard form. (Unless, again, this isn't a quadratic equation – "standard form" can mean different things in different areas of math).

To find the total revenue function, we need to integrate the marginal revenue function R'(x) with respect to x.

(a) Total Revenue Function:

We integrate R'(x) = 4x(x² + 26,000) with respect to x:

R(x) = ∫[4x(x² + 26,000)] dx

Expanding and integrating, we get:

R(x) = ∫[4x³ + 104,000x] dx

= x⁴ + 52,000x² + C

Now we can use the given information to find the value of the constant C. We are told that the revenue from 120 gadgets is $15,879, so we can set up the equation:

R(120) = 15,879

Substituting x = 120 into the total revenue function:

120⁴ + 52,000(120)² + C = 15,879

Solving for C:

207,360,000 + 748,800,000 + C = 15,879

C = -955,227,879

Therefore, the total revenue function is:

R(x) = x⁴ + 52,000x² - 955,227,879

(b) Revenue of at least $45,000:

To find the number of gadgets that must be sold for a revenue of at least $45,000, we can set up the inequality:

R(x) ≥ 45,000

Using the total revenue function R(x) = x⁴ + 52,000x² - 955,227,879, we have:

x⁴ + 52,000x² - 955,227,879 ≥ 45,000

We can solve this inequality numerically to find the values of x that satisfy it. Using a graphing calculator or software, we can determine that the solutions are approximately x ≥ 103.5 or x ≤ -103.5. However, since the number of gadgets cannot be negative, the number of gadgets that must be sold for a revenue of at least $45,000 is x ≥ 103.5.

Therefore, at least 104 gadgets must be sold for a revenue of at least $45,000.

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Therefore, the elementary matrix E₁, or D, is: D = [0 0 1

                                                                                 0 1 0

                                                                                 1 0 0]

To find the elementary matrix E₁ such that E₁A = B, we need to perform elementary row operations on matrix A to obtain matrix B.

Let's denote the elementary matrix E₁ as D.

Starting with matrix A:

A = [9 10 1

20 1 11

8 -19 -1]

And matrix B:

B = [8 -19 20

1 11 9

10 1 1]

To obtain B from A, we need to perform row operations on A. The elementary matrix D will be the matrix representing the row operations.

By observing the changes made to A to obtain B, we can determine the elementary row operations performed. In this case, it appears that the row operations are:

Row 1 of A is swapped with Row 3 of A.

Row 2 of A is swapped with Row 3 of A.

Let's construct the elementary matrix D based on these row operations.

D = [0 0 1

0 1 0

1 0 0]

To verify that E₁A = B, we can perform the matrix multiplication:

E₁A = DA

D * A = [0 0 1 * 9 10 1 = 8 -19 20

0 1 0 20 1 11 1 11 9

1 0 0 8 -19 -1 10 1 1]

As we can see, the result of E₁A matches matrix B.

Therefore, the elementary matrix E₁, or D, is:

D = [0 0 1

0 1 0

1 0 0]

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The first three questions involve differentiating given functions.  Question 1 - None of the above answers; Question 2 - y' = (x³ + 3x²)e*; Question 3 - None of the above answers. Question 4 asks for the derivative of y = 24x, and the correct answer is y' = 24.

Question 1: The given function is y = O (x-3)* > O (x-3)e* +8 O(x-3)x4 ex. The notation used is unclear, so it is difficult to determine the correct differentiation. However, none of the provided options seem to match the given function, so the answer is "None of the above answers."

Question 2: The given function is y = x³ex. To find its derivative, we apply the product rule and the chain rule. Using the product rule, we differentiate the terms separately and combine them. The derivative of x³ is 3x², and the derivative of ex is ex. Thus, the derivative of the given function is y' = (x³ + 3x²)e*.

Question 3: The given function is y = √√x³ + 4. To differentiate this function, we apply the chain rule. The derivative of √√x³ + 4 can be found by differentiating the inner function, which is x³ + 4. The derivative of x³ + 4 is 3x², and applying the chain rule, the derivative of √√x³ + 4 becomes 3x² * 2(x + 4)¹/2. Thus, the correct answer is "3x² * 2(x + 4)¹/2."

Question 4: The given function is y = 24x. To find its derivative, we differentiate it with respect to x. The derivative of 24x is simply 24, as the derivative of a constant multiplied by x is the constant. Therefore, the correct answer is y' = 24.

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The graph of the curves will show two distinct loops, one for each equation, but they will not intersect.

To graph the curves and find the area enclosed by them, we'll first plot the points using the given polar coordinate equations and then find the intersection points. Let's start by graphing the curves individually:

Curve 1: r = 4 + 3sin(θ)

Curve 2: r = 2sin(θ)

To create the graph, we'll plot points by varying the angle θ and calculating the corresponding values of r.

For Curve 1 (r = 4 + 3sin(θ)):

Let's calculate the values of r for various values of θ:

When θ = 0 degrees, r = 4 + 3sin(0) = 4 + 0 = 4

When θ = 45 degrees, r = 4 + 3sin(45) ≈ 6.12

When θ = 90 degrees, r = 4 + 3sin(90) = 4 + 3 = 7

When θ = 135 degrees, r = 4 + 3sin(135) ≈ 6.12

When θ = 180 degrees, r = 4 + 3sin(180) = 4 - 3 = 1

When θ = 225 degrees, r = 4 + 3sin(225) ≈ -0.12

When θ = 270 degrees, r = 4 + 3sin(270) = 4 - 3 = 1

When θ = 315 degrees, r = 4 + 3sin(315) ≈ -0.12

When θ = 360 degrees, r = 4 + 3sin(360) = 4 + 0 = 4

Now we have several points (θ, r) for Curve 1: (0, 4), (45, 6.12), (90, 7), (135, 6.12), (180, 1), (225, -0.12), (270, 1), (315, -0.12), (360, 4).

For Curve 2 (r = 2sin(θ)):

Let's calculate the values of r for various values of θ:

When θ = 0 degrees, r = 2sin(0) = 0

When θ = 45 degrees, r = 2sin(45) ≈ 1.41

When θ = 90 degrees, r = 2sin(90) = 2

When θ = 135 degrees, r = 2sin(135) ≈ 1.41

When θ = 180 degrees, r = 2sin(180) = 0

When θ = 225 degrees, r = 2sin(225) ≈ -1.41

When θ = 270 degrees, r = 2sin(270) = -2

When θ = 315 degrees, r = 2sin(315) ≈ -1.41

When θ = 360 degrees, r = 2sin(360) = 0

Now we have several points (θ, r) for Curve 2: (0, 0), (45, 1.41), (90, 2), (135, 1.41), (180, 0), (225, -1.41), (270, -2), (315, -1.41), (360, 0).

Next, we'll plot these points on a graph and find the area enclosed by the curves:

(Note: For simplicity, I'll assume the angles in degrees, but you can convert them to radians if needed.)

To calculate the area enclosed by the curves, we need to find the points of intersection between the two curves. The enclosed region will be between the points of intersection.

Let's find the points where the curves intersect:

For r = 4 + 3sin(θ) and r = 2sin(θ), we have:

4 + 3sin(θ) = 2sin(θ)

Rearranging the equation:

3sin(θ) - 2sin(θ) = -4

sin(θ) = -4

Since the sine function's value is always between -1 and 1, there are no solutions to this equation. Therefore, the two curves do not intersect.

As a result, there is no enclosed region, and the area between the curves is zero.

The graph of the curves will show two distinct loops, one for each equation, but they will not intersect.

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