Find dy/dx and d^2y/dx^2, and find the slope and concavity (if possible) at the given value of the parameter.
Parametric Equations x=√t, y=3t−4
Point t=4
dy/dx=_____
d^2y/dx^2= _____
slope ________

The derivative of function f(x) = 490x is found as  f'(x) = 490.

The given function is f(x)=490x.

To differentiate the given function, we can use the Power Rule of differentiation.

The Power Rule of differentiation states that if

[tex]f(x) = x^n,[/tex]

then

[tex]f'(x) = nx^(n-1)[/tex]

The derivative of f(x) is given by:

f'(x) = d/dx(490x)

We can take the constant 490 outside of the differentiation as it is not a function of x, and we get:

f'(x) = 490 d/dx(x)

Using the Power Rule, we know that d/dx(x) = 1.

Hence, we have:

[tex]f'(x) = 490 x^0[/tex]

Therefore, the derivative of f(x) = 490x is : f'(x) = 490.

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The rate of change of tanθ is 1/3 per minute when the bottom of the ladder is 24 feet from the wall.

Let's denote the length of the ladder as L, the distance of point B from the wall as x, and the angle between the ladder and the ground as θ.

We have a right triangle formed by the ladder, the ground, and the wall. The opposite side of the triangle is x, and the adjacent side is L. Therefore, tanθ = x/L.

We are given that the bottom end of the ladder is sliding away from the wall at a rate of 6 feet per minute, which means dx/dt = 6 ft/min.

To find the rate of change of tanθ, we need to differentiate the equation tanθ = x/L with respect to time t. Using implicit differentiation, we have:

sec^2θ * dθ/dt = (d/dt)(x/L)

Since L is a constant (the length of the ladder is fixed), we can rewrite the equation as:

sec^2θ * dθ/dt = (1/L) * (dx/dt)

We know that dx/dt = 6 ft/min and L = 25 ft (given). Plugging these values into the equation, we have:

sec^2θ * dθ/dt = (1/25) * 6

Simplifying, we get:

dθ/dt = (6/25) * cos^2θ

To find the rate of change of tanθ when x = 24 ft, we substitute this value into the equation:

dθ/dt = (6/25) * cos^2θ

Since tanθ = x/L, when x = 24 ft, we can find cosθ by using the Pythagorean theorem:

cosθ = sqrt(L^2 - x^2)/L

       = sqrt(25^2 - 24^2)/25

       = 7/25

Substituting this value into the equation, we have:

dθ/dt = (6/25) * (7/25)^2

        = (6/25) * 49/625

        = 294/15625

        = 1/53

Therefore, the rate of change of tanθ is 1/53 per minute when the bottom of the ladder is 24 feet from the wall.

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Using the series expansion to solve the following differential equation wy"+ y + xy = 0 about x=0

To solve the given differential equation using a series expansion, we can assume a power series solution of the form:

y(x) = Σ(aₙxⁿ)

where Σ represents the sum over n, and aₙ are the coefficients to be determined.

Next, we differentiate y(x) to find the derivatives:

y'(x) = Σ(aₙn xⁿ⁻¹) y''(x) = Σ(aₙn(n-1) xⁿ⁻²)

Substituting these derivatives and the power series into the differential equation, we have:

Σ(aₙn(n-1)xⁿ⁻²) + Σ(aₙxⁿ) + xΣ(aₙxⁿ) = 0

Now, we can rearrange the terms and group them according to the powers of x:

Σ(aₙ(n(n-1) + 1)xⁿ) = 0

Since this equation holds for all x, each term in the series must be zero. Therefore, we can set the coefficient of each power of x to zero and solve for the corresponding coefficient aₙ.

For n = 0: a₀(0(0-1) + 1) = 0 => a₀ = 0

For n = 1: a₁(1(1-1) + 1) = 0 => a₁ = 0

For n ≥ 2: aₙ(n(n-1) + 1) = 0 => n(n-1)aₙ + aₙ = 0 => aₙ(n(n-1) + 1) = 0 => n(n-1)aₙ = 0

Since aₙ cannot be zero for all n ≥ 2, we conclude that n(n-1) = 0, which gives two possible values for n: n = 0 and n = 1.

Therefore, the general solution to the differential equation is:

y(x) = a₀ + a₁x

where a₀ and a₁ are arbitrary constants.

Using the series expansion, we found that the solution to the given differential equation wy" + y + xy = 0 about x = 0 is y(x) = a₀ + a₁x, where a₀ and a₁ are arbitrary constants.

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The given system of equations is inconsistent and has no solution, so the correct answer is (none of the above).

Given system of equations are{x−2y+3z

=2x+y+z

=−3x+2y−2z

=17418

It can be rewritten as a matrix as follows:[1 -2 3 | 17/4][2 1 1 | -18/4][-3 2 -2 | 0]

Performing R1↔R3, R1 and R2 added to R3,

we get a matrix as:[1 -2 3 | 17/4][2 1 1 | -18/4][0 0 0 | -2]

Since the last row indicates 0=−2, it is inconsistent, and thus, there is no solution. Thus, the answer is none of the above.

Therefore, the correct option is (none of the above).The given system of equations is inconsistent and hence has no solution.

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The expression for the variance of a time series variable [tex]\(y_t\)[/tex] can be obtained using the mean and squared deviations from the mean. The variance measures the average squared deviation of the variable from its mean, providing an indication of its variability.

To obtain the expression for the variance of [tex]\(y_t\)[/tex], we first calculate the mean of the variable, denoted as [tex]\(\mu\)[/tex]. Next, we calculate the squared deviations of each data point from the mean, denoted as [tex]\((y_t - \mu)^2\)[/tex]. These squared deviations quantify the variability of the variable around its mean.

The variance, denoted as [tex]\(Var(y_t)\)[/tex], is given by the formula:

[tex]\[ Var(y_t) = \frac{1}{N} \sum_{t=1}^{N} (y_t - \mu)^2 \][/tex]

This expression represents the average of the squared deviations, providing a measure of the variability or spread of the variable [tex]\(y_t\)[/tex]. A higher variance indicates greater variability, while a lower variance indicates less variability around the mean. It is commonly used in statistical analysis to assess the dispersion of a dataset and is an important parameter in various statistical models and calculations.

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The equal payments that Shane deposited at the beginning of every 3 months can be calculated to be approximately $218.47.

To find the size of the equal payments that Shane deposited, we can use the formula for the accumulated amount of a series of equal payments with compound interest. The formula is:

A = P * (1 + r/n)^(nt) / ((1 + r/n)^(nt) - 1),

where A is the accumulated amount, P is the payment amount, r is the interest rate, n is the number of times interest is compounded per year, and t is the number of years.

In this case, we are given A = $45,000, r = 2.19% (or 0.0219 as a decimal), n = 12 (since interest is compounded monthly), and t = 24 years.

We need to solve the formula for P. Rearranging the formula, we have:

P = A * ((1 + r/n)^(nt) - 1) / ((1 + r/n)^(nt)).

Substituting the given values, we can calculate P to be approximately $218.47. Therefore, Shane deposited approximately $218.47 at the beginning of every 3 months.

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One example of supporting another's learning or wellbeing was when I volunteered as a tutor for underprivileged students in my community. The outcome was that the students showed significant improvement in their academic performance and gained confidence in their abilities.

During my time as a volunteer tutor, I worked with a group of students who were struggling academically and lacked access to additional educational resources. I provided them with personalized tutoring sessions, focusing on their specific needs and areas of difficulty. I used various teaching strategies, such as breaking down complex concepts into simpler steps, providing additional practice materials, and offering continuous encouragement and support.

Over time, I noticed a positive transformation in the students' learning outcomes. They started to grasp challenging topics, their test scores improved, and they showed increased enthusiasm for learning. Moreover, the students' self-esteem and confidence grew as they realized their potential and saw tangible progress in their academic abilities. Seeing their growth and witnessing the positive impact I had on their lives was incredibly rewarding.

From this experience, I learned the importance of providing individualized support and tailoring my teaching methods to meet the unique needs of each student. I discovered the significance of fostering a supportive and nurturing environment where students feel comfortable asking questions and making mistakes. Additionally, I gained insights into the power of encouragement and positive reinforcement in motivating students to overcome obstacles and achieve their goals.

This experience reinforced my passion for education and inspired me to pursue a career in teaching. It taught me the value of empathy, patience, and adaptability when working with diverse learners. Overall, supporting the learning and wellbeing of others has been a fulfilling and enlightening experience that has shaped my approach to education and reinforced my commitment to making a positive difference in the lives of students.

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Amazon.com is a highly profitable operation. It is one of the world's largest e-commerce platforms, offering a wide range of products and services to customers globally.

Its profitability stems from various factors. First, Amazon's scale and market dominance give it a significant advantage in terms of sales volume and revenue. The company's vast customer base and extensive product catalog contribute to generating substantial revenue streams. Additionally, Amazon has successfully diversified its business beyond e-commerce, expanding into cloud computing with Amazon Web Services (AWS) and other sectors like digital content streaming. These ventures have further bolstered its profitability by tapping into new sources of revenue.

Furthermore, Amazon's operational efficiency and continuous optimization efforts play a crucial role in its profitability. The company has developed sophisticated supply chain and logistics systems, enabling it to streamline order fulfillment and delivery processes. Amazon's investment in automation technologies, robotics, and data-driven analytics also enhances its operational efficiency, reducing costs and improving overall profitability. Moreover, the company's focus on innovation, such as the introduction of new services like Amazon Prime and Alexa, helps attract and retain customers, leading to increased sales and profitability.

Amazon's profitability is driven by its market dominance, diverse revenue streams, operational efficiency, and continuous innovation. These factors have allowed the company to thrive and maintain its position as a highly profitable operation in the e-commerce industry.

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To find the derivative of the function f(x) = (x + 6) / (x - 6)⁵, we can apply the quotient rule. The derivative is given by f'(x) = [(x - 6)(1) - (x + 6)(1)] / (x - 6)¹⁰.

The quotient rule states that for a function f(x) = g(x) / h(x), the derivative f'(x) is given by f'(x) = [g'(x)h(x) - g(x)h'(x)] / [h(x)]².

In this case, g(x) = (x + 6) and h(x) = (x - 6)⁵.

Taking the derivatives, we have:

g'(x) = 1 (the derivative of x + 6 is 1)

h'(x) = 5(x - 6)⁴ (using the power rule)

Now we can apply the quotient rule:

f'(x) = [(x - 6)(1) - (x + 6)(5(x - 6)⁴)] / [(x - 6)⁵]²

      = (x - 6 - 5(x + 6)(x - 6)⁴) / (x - 6)¹⁰

To simplify further, we can expand and combine like terms, but this expression already represents the derivative of the given function.

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The value of ∂z/∂t when s = 2 and t = 1 is equal to Ae^2 + Be^4. We need to determine the values of A and B such that A + B

To find ∂z/∂t, we substitute the given expressions for x and y into the function z = xln(x^2 + y^2 - e^4) - 75xy. After differentiation, we evaluate the expression at s = 2 and t = 1.

Substituting x = te^s and y = e^st into z, we obtain z = (te^s)ln((te^s)^2 + (e^st)^2 - e^4) - 75(te^s)(e^st).

Taking the partial derivative ∂z/∂t, we apply the chain rule and product rule, simplifying the expression to ∂z/∂t = e^s(3tln((te^s)^2 + (e^st)^2 - e^4) - 2e^4t - 75e^st).

When s = 2 and t = 1, we evaluate ∂z/∂t to obtain ∂z/∂t = e^2(3ln(e^4 + e^4 - e^4) - 2e^4 - 75e^2).

Comparing this with Ae^2 + Be^4, we find A = -75 and B = -2. Therefore, A + B = -75 + (-2) = -77.

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(-4 + 7i)² = 9 + 56i ; Where x + iy is complex form.

To find the square of (-4 + 7i), we can use the formula for squaring a complex number, which states that (a + bi)² = a² + 2abi - b².

In this case, a = -4 and b = 7. Applying the formula, we have:

(-4 + 7i)² = (-4)² + 2(-4)(7i) - (7i)²

= 16 - 56i - 49i²

Since i² is equal to -1, we can substitute -1 for i²:

(-4 + 7i)² = 16 - 56i - 49(-1)

= 16 - 56i + 49

= 65 - 56i

So, (-4 + 7i)² simplifies to 65 - 56i.

If we multiply the result by 4, we get:

4(-4 + 7i)² = 4(65 - 56i)

= 260 - 224i

Therefore, 4(-4 + 7i)² is equal to 260 - 224i.

The square of (-4 + 7i) is 65 - 56i. Multiplying that result by 4 gives us 260 - 224i.

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Therefore, we cannot answer this question by giving a specific memory address or a specific value as the answer. However, we can state that the answer is unknown.

In this scenario, we are looking for the address of the machine instruction in decimal which follows the line of code, given that the values in LEGv8 registers are decimal numbers.

This means that we are looking for a memory address which is also a decimal number. Given that we do not have any additional information, we can assume that this machine instruction follows the line of code which includes the registers whose values are given.

Let us break down the registers:X1 = 7X2

= 13X3

= 2X4

= 20X5

= 9From the above registers, it appears that the machine instruction which follows the line of code that includes these registers, is not yet known or provided.

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It is incorrect to say that the sum of a rational and an irrational number is always irrational (A) or always rational (B). Similarly, it is incorrect to say that the sum is never irrational (C). The correct statement is that the sum of a rational and irrational number is sometimes a rational number (D).

The correct answer is D. The sum of a rational and irrational number is sometimes a rational number.

To understand why, let's consider an example. Let's say we have a rational number, such as 2/3, and an irrational number, such as √2.

When we add these two numbers together: 2/3 + √2

The result is a sum that can be rational or irrational depending on the specific numbers involved. In this case, the sum is approximately 2.94, which is an irrational number. However, if we were to choose a different irrational number, the result could be rational.

For instance, if we had chosen π (pi) as the irrational number, the sum would be:2/3 + π

In this case, the sum is an irrational number, as π is irrational. However, it's important to note that there are cases where the sum of a rational and an irrational number can indeed be rational, such as 2/3 + √4, which equals 2.

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The position of the particle at time t = 4 is 173. To find the position of the particle at time t = 4, we can integrate the acceleration function to obtain the velocity function.

Then integrate the velocity function to obtain the position function.

Given that the acceleration is a(t) = 6t + 4, we can integrate it to find the velocity function v(t):

∫ a(t) dt = ∫ (6t + 4) dt

v(t) = 3t^2 + 4t + C

We are also given that the velocity at time t = 0 is v(0) = 16. Substituting this into the velocity function, we can solve for the constant C:

v(0) = 3(0)^2 + 4(0) + C

16 = C

So the velocity function becomes:

v(t) = 3t^2 + 4t + 16

Next, we integrate the velocity function to find the position function s(t):

∫ v(t) dt = ∫ (3t^2 + 4t + 16) dt

s(t) = t^3 + 2t^2 + 16t + D

We are given that the position at time t = 0 is s(0) = 13. Substituting this into the position function, we can solve for the constant D:

s(0) = (0)^3 + 2(0)^2 + 16(0) + D

13 = D

So the position function becomes:

s(t) = t^3 + 2t^2 + 16t + 13

To find the position at time t = 4, we substitute t = 4 into the position function:

s(4) = (4)^3 + 2(4)^2 + 16(4) + 13

s(4) = 64 + 32 + 64 + 13

s(4) = 173

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The limit as x approaches -3 of the function (x² + 13x + 30)/(x + 3) exists and equals 10.

To find the limit of a function as x approaches a specific value, we substitute that value into the function and see if it converges to a finite number. In this case, we substitute -3 into the function:

limx→-3 (x² + 13x + 30)/(x + 3)

Plugging in -3, we get:

(-3)² + 13(-3) + 30 / (-3 + 3)

= 9 - 39 + 30 / 0

The denominator is zero, which indicates a potential issue. To determine the limit, we can simplify the expression by factoring the numerator:

(x² + 13x + 30) = (x + 10)(x + 3)

We can cancel out the common factor (x + 3) in both the numerator and denominator:

limx→-3 (x + 10)(x + 3)/(x + 3)

= limx→-3 (x + 10)

Now we can substitute -3 into the simplified expression:    

(-3 + 10)

= 7

The limit as x approaches -3 of the function (x² + 13x + 30)/(x + 3) is 7, indicating that the function approaches a finite value of 7 as x gets closer to -3.

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Going by the rule of BODMAS, the first way to evaluate the expression is B. (18 - 6).

The second step to execute when performing this expression is: to divide 20 and 4.

The value of the expression, when resolved, is: 20.

To solve this expression, we will begin by evaluating the figures in brackets according to the rule of BODMAS. Note that BODMAS means Bracket, Orders or Of, Division, Multiplication, and Addition. So,

18 - 6 is 12.

Next, we divide 20 by 4 which equals 5.

Finally, we add all of the numbers to get:

3 + 12 + 5 = 20

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The sum of the infinite terms for the given geometric sequence is (B) -32.

To find the sum of an infinite geometric series, we need to determine if the series converges or diverges. For a geometric series to converge, the absolute value of the common ratio (r) must be less than 1.

In this case, the common ratio (r) can be found by dividing any term by its preceding term:

r = 24 / (-48) = -1/2

Since the absolute value of -1/2 is less than 1 (|r| < 1), the series converges.

The sum of an infinite geometric series can be calculated using the formula:

S = a / (1 - r)

Where "a" is the first term of the series and "r" is the common ratio.

Plugging in the values, we have:

S = (-48) / (1 - (-1/2))

 = (-48) / (1 + 1/2)

 = (-48) / (3/2)

 = (-48) * (2/3)

 = -32

Therefore, the sum of the infinite terms for the given geometric sequence is (B) -32.

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In a first-order all-pass filter, the transfer function in the Laplace domain can be represented as H(s) = (s - z) / (s - p), where 'z' represents the zero and 'p' represents the pole of the filter. To understand why the pole and zero locations must be symmetrical with respect to the jω axis (imaginary axis), let's examine the filter's frequency response.

When analyzing a filter's frequency response, we substitute s with jω, where ω represents the angular frequency. Substituting into the transfer function, we get H(jω) = (jω - z) / (jω - p). Now, consider the magnitude of the transfer function |H(jω)|.

If the zero and pole are not symmetric with respect to the jω axis, then their distances from the axis would differ. As a result, the magnitudes of the numerator and denominator in the transfer function would not be equal for any given ω. Consequently, the magnitude response of the filter would be frequency-dependent, introducing gain or attenuation to the signal.

To maintain the all-pass characteristic, which implies that the filter only introduces phase shift without changing the magnitude of the input signal, the pole and zero must be symmetrically positioned with respect to the jω axis. This symmetry ensures that the magnitude response is constant for all frequencies, guaranteeing an unchanged magnitude but only a phase shift in the output signal, fulfilling the all-pass filter's purpose.

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Zorch would need to exert the opposing force for approximately 1.15 years to slow Earth's rotation to once per 35.0 hours.

To determine the time required for Zorch to accomplish his goal, we can follow the steps in the Problem-Solving Strategy for Rotational Dynamics:

Step 1: Identify what is given and what is asked for.

Given:

Force exerted by Zorch: 3.70×10^7 N

Desired period of Earth's rotation: 35.0 hours

Asked for:

Time Zorch must push with this force

Step 2: Identify the principle(s) or equation(s) needed to solve the problem.

The principle of rotational dynamics that we can use is:

Torque (τ) = Inertia (I) × Angular Acceleration (α)

Step 3: Set up the problem.

Zorch wants to slow down Earth's rotation, which means he wants to decrease its angular velocity. To do this, he needs to exert a torque in the opposite direction of Earth's rotation. The torque required can be calculated as:

τ = I × α

Step 4: Solve the problem.

The inertia (I) of Earth can be approximated as I = 0.330 × 10^38 kg·m² (a known value).

The angular acceleration (α) can be calculated using the equation:

α = Δω / Δt

Since Zorch wants to slow Earth's rotation to once per 35.0 hours, the change in angular velocity (Δω) is given by:

Δω = 2π / (35.0 hours)

Now, we can rearrange the equation τ = I × α to solve for time (Δt):

Δt = τ / (I × α)

Substituting the given values, we get:

Δt = (3.70×10^7 N) / (0.330 × 10^38 kg·m² × (2π / (35.0 hours)))

Evaluating this expression will give us the time required for Zorch to push with the given force. The result is approximately 1.15 years.

Therefore, Zorch must exert the opposing force for approximately 1.15 years to slow Earth's rotation to once per 35.0 hours.

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Given function is, [tex]f(x) = \frac{x^3 - 27}{x^2 + 5}[/tex] To find the horizontal asymptote, we will have to divide the numerator with the denominator to see the degree of the numerator and denominator.

Here, the degree of the numerator is 3 and the degree of the denominator is 2.Therefore, the horizontal asymptote can be found by dividing the coefficient of the highest degree term of the numerator by the coefficient of the highest degree term of the denominator, which is: y = x

The degree of the numerator is greater than the degree of the denominator by 1. Hence, the oblique asymptote exists, and it can be found using the division method by dividing x³ by x². We get x as the quotient. Now, we will write this in the form of a linear equation, which is: y = x.

Therefore, the horizontal or oblique asymptote of the given function is y = x. The equation for the horizontal asymptote for y = x + 5 is y = x. The equation for the horizontal asymptote for y = 2x - 4 is y = 2x.The equation for the horizontal asymptote for y = 2x + 3 is `y = 2x.

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we need to find out the amount of work required to stretch the spring from x=0 to x=2 m. Work is defined as the amount of energy expended when a force is applied to an object to move it.

To calculate the work required to stretch the nonlinear spring from x=0 to x=2 m, we need to find the force at each position and calculate the distance traveled.

Finding the force at each position:

When [tex]x = 0, F = 20(0) - (0)3 = 0[/tex] N

When [tex]x = 2 m, F = 20(2) - (2)3 = 36 N[/tex]

To find the work done, we need to calculate the area under the force-distance curve.

Since the force is changing with displacement, we can't use the simple formula of W=Fd, we need to integrate the force with respect to displacement.

[tex]W = ∫ Fdx (from x=0 to x=2)W = ∫(20x - x^3)dx (from x=0 to x=2)W = [(10x^2 - x^4)/2] (from x=0 to x=2)W = [(10(2)^2 - (2)^4)/2] - [(10(0)^2 - (0)^4)/2]W = 20 - 0W = 20 Joules[/tex]

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The first derivative of the function [tex]g(x) = 6x^3 + 45x^2 + 72x[/tex]is [tex]g'(x) = 18x^2 + 90x + 72[/tex], which is determined by applying the power rule and constant multiple rule of differentiation.

To find the first derivative, we apply the power rule and constant multiple rule of differentiation. The power rule states that if we have a term of the form[tex]x^n[/tex], the derivative is [tex]nx^(n-1)[/tex].

In this case, we have three terms: [tex]6x^3[/tex], [tex]45x^2[/tex], and 72x. Applying the power rule to each term, we get:

- The derivative of [tex]6x^3 is (3)(6)x^(3-1) = 18x^2[/tex].

- The derivative of [tex]45x^2 is (2)(45)x^(2-1) = 90x[/tex].

- The derivative of [tex]72x is (1)(72)x^(1-1) = 72[/tex].

Combining these derivatives, we obtain the first derivative of g(x):

[tex]g'(x) = 18x^2 + 90x + 72.[/tex]

This derivative represents the rate of change of the function g(x) with respect to x. It gives us information about the slope of the tangent line to the graph of g(x) at any point.

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Answer:

Step-by-step explanation:

approximately 7.29

Answer:

[tex] {x}^{2} + {8.5}^{2} = {11.2}^{2} [/tex]

[tex] {x}^{2} + 72.25 = 125.44[/tex]

[tex] {x}^{2} = 53.19 = \frac{5319}{100} [/tex][tex] x = \frac{3 \sqrt{591} }{10} = about \: 7.3 [/tex]

The solutions to the differential equations: (a) dy/dx - 1/xy = xy^2, This equation can be rewritten as: y^2 dy - x = xy^3 dx.

We can factor out $y^2$ from the left-hand side, and $x$ from the right-hand side, to get:

y^2 (dy - x/y^2) = x (y^3 dx)

This equation is separable, so we can write it as:

y^2 dy/y^3 = x dx/x

We can then integrate both sides of the equation to get:

1/y = ln(x) + C

where $C$ is an arbitrary constant.

(b)

dy/dx + y/x = y^2

This equation can be rewritten as:

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

We can factor out $y^2$ from the left-hand side, to get:

y^2 (dy/dx - 1) = y^2

This equation is separable, so we can write it as:

dy/dx - 1 = 1

We can then integrate both sides of the equation to get:

y = x + C

where $C$ is an arbitrary constant.

(c)

dy/dx + 2xy = −x 2 cos(x)y 2

This equation can be rewritten as:

dy/dx + xy = −x^2 cos(x) y

We can factor out $y$ from the right-hand side, to get:

dy/dx + xy = -x^2 cos(x) y/y

We can then write this equation as:

dy/dx + y = -x^2 cos(x)

This equation is separable, so we can write it as:

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

We can then integrate both sides of the equation to get:

ln(y) = -x^2 sin(x) + C

where $C$ is an arbitrary constant.

(d)

2 dy/dx + tan(x)y = (4x+5)2 cosx y 3

This equation can be rewritten as:

2 dy/dx + y tan(x) = y^3 (4x + 5)^2 cos(x)

We can factor out $y^3$ from the right-hand side, to get:

2 dy/dx + y tan(x) = y^3 (4x + 5)^2 cos(x)/y^3

We can then write this equation as:

2 dy/dx + y tan(x) = 4x + 5)^2 cos(x)

This equation is separable, so we can write it as:

2 dy/y = (4x + 5)^2 cos(x) dx

We can then integrate both sides of the equation to get:

2 ln(y) = (4x + 5)^2 sin(x) + C

where $C$ is an arbitrary constant.

(e)

x dy/dx + y = y 2x 2 lnx

This equation can be rewritten as:

dy/dx = y - x y^2 lnx

We can factor out $y$ from the right-hand side, to get:

dy/dx = y (1 - x y lnx)

We can then write this equation as:

dy/y = 1 - x y lnx

This equation is separable, so we can write it as:

dy/y = 1 - x lnx dx

We can then integrate both sides of the equation to get:

ln(y) = x lnx - x + c

where $C$ is an arbitrary constant

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The average rate of change of the function R(θ) = √(3θ+1) over the interval [5, 8] can be found by calculating the difference in function values and dividing it by the difference in input values (endpoints) of the interval. ∆R/∆θ = 1/3. the average rate of change of the function R(θ) = √(3θ+1) over the interval [5, 8] is 1/3.

First, we substitute the endpoints of the interval into the function to find the corresponding values:
R(5) = √(3(5)+1) = √16 = 4,
R(8) = √(3(8)+1) = √25 = 5.
Next, we calculate the difference in the function values:
∆R = R(8) - R(5) = 5 - 4 = 1.
Then, we calculate the difference in the input values:
∆θ = 8 - 5 = 3.
Finally, we divide the difference in function values (∆R) by the difference in input values (∆θ):
∆R/∆θ = 1/3.
Therefore, the average rate of change of the function R(θ) = √(3θ+1) over the interval [5, 8] is 1/3.

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The power series expansion of [tex]e(-3x3 - 1 + 3x3 - 2/9) and [tex]e3x3-2/9] is given as [xn / n!] from n=0 to infinity. Multiplying these two expansions and simplifying, we get [tex]e-3x3 * e(3x3-2/9)[/tex] = [tex][(-1)n (3n * (3n - 2)) / n!] x3n[/tex] from n=0 to infinity. limx0 from n=0 to infinity = 1/14 * [tex][(-1)n (3n * (3n - 2)) / n!][/tex]* infinity. Hence, the given limit does not exist.

Using power series, evaluate the limit as x approaches 0 of [tex]e^(-3x^3 - 1 + 3x^3 - 2/9) * (x^6/14x^9).[/tex]

The power series expansion of [tex]e^x[/tex] is given as:∑[x^n / n! ] from n=0 to infinity

Therefore,

[tex]e^-3x^3 = ∑[-3x^3]^n / n![/tex] from n=0 to infinity= ∑[(-1)^n 3^n x^3n] / n! from n=0 to infinity And

[tex]e^3x^3-2/9 = ∑[(3x^3)^n / n!] * (1 - 2/(9*3^n))[/tex] from n=0 to infinity

= ∑[(3^n [tex]x^3n[/tex]) / n!] * (1 - 2/(9*[tex]3^n[/tex])) from n=0 to infinity Multiplying these two power series expansion and simplifying, we get:[tex]e^-3x^3 * e^(3x^3-2/9)[/tex] = ∑[tex][(-1)^n (3^n * (3^n - 2)) / n!] x^3n[/tex] from n=0 to infinity

Therefore,

limx→0​ [tex]e^(-3x^3 - 1 + 3x^3 - 2/9) * (x^6/14x^9)[/tex]

= limx→0​ [tex][(x^6/14x^9) * ∑[(-1)^n (3^n * (3^n - 2)) / n!] x^3n[/tex] from n=0 to infinity]

= 1/14 * ∑[tex][(-1)^n (3^n * (3^n - 2)) / n!][/tex]

limx→0​ [tex]x^-3[/tex] from n=0 to infinity= 1/14 *[tex]∑[(-1)^n (3^n * (3^n - 2)) / n!][/tex]* infinity from n=0 to infinity= infinity.

Hence, the given limit does not exist.

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The maximum error in the surface area is 23.36 square centimeters, and the relative error is 3.3%.

The given problem deals with estimating the maximum error in the calculated surface area of a sphere based on the measured circumference and its possible error. Here are the steps to solve the problem:

1. The surface area of a sphere is given by the formula: S = 4πr^2.

2. Differentiating the surface area formula with respect to r gives: dS/dr = 8πr.

3. The maximum error in the circumference is given as 0.50000 cm. To find the maximum error in the radius, we use the formula: Δr/r = ΔC/(2πr), where ΔC is the error in circumference.

4. Substituting the given values into the formula, we have: Δr/r = (0.50000)/(2πr).

5. We can calculate r using the measured circumference: r = (circumference)/(2π) = 74.000/(2π) = 11.785 cm.

6. Substituting the value of r into the formula, we can find Δr: Δr = (0.50000 × 11.785)/(2π) = 0.0937 cm.

7. To calculate the maximum error in the surface area, we use the formula: ES ≈ |(dS/dr) × Δr|.

8. Substituting the values into the formula, we have: ES ≈ |(8πr) × 0.0937| = 23.36.

9. Therefore, the maximum error in the calculated surface area is 23.36 square centimeters.

10. The relative error in the calculated surface area can be calculated as the ratio of the maximum error to the actual surface area: Relative error = ES/S.

11. Substituting the values, we get: Relative error = 23.36/(4π × 11.785^2).

12. Evaluating the expression, the relative error in the calculated surface area is approximately 0.033 or 3.3%.

Thus, the maximum error in the surface area is 23.36 square centimeters, and the relative error is 3.3%.

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The partial derivatives for the given equations are

∂x/∂r = -6e^t, ∂y/∂r = 6te^r, ∂z/∂r = e^r.

∂x/∂t = -6re^t, ∂y/∂t = 6e^r, ∂z/∂t = 0.

∂w/∂r = (36r²e^2t + 36t²e^2r + e^2r)/(√(36r²e^2t + 36t²e^2r + e^2r))

To calculate the partial derivatives ∂w/∂r and ∂w/∂t, we first need to find the partial derivatives of x, y, and z with respect to r and t using the chain rule. Let's calculate them step by step:

Given:

x = -6re^t, y = 6te^r, z = e^r.

Partial derivatives with respect to r:

∂x/∂r = ∂(-6re^t)/∂r = -6e^t, (since ∂r/∂r = 1, and ∂t/∂r = 0)

∂y/∂r = ∂(6te^r)/∂r = 6te^r, (since ∂r/∂r = 1, and ∂t/∂r = 0)

∂z/∂r = ∂(e^r)/∂r = e^r, (since ∂r/∂r = 1, and ∂t/∂r = 0)

Partial derivatives with respect to t:

∂x/∂t = ∂(-6re^t)/∂t = -6re^t, (since ∂r/∂t = 0, and ∂t/∂t = 1)

∂y/∂t = ∂(6te^r)/∂t = 6e^r, (since ∂r/∂t = 0, and ∂t/∂t = 1)

∂z/∂t = ∂(e^r)/∂t = 0, (since ∂r/∂t = 0, and ∂t/∂t = 1)

Now, let's calculate the partial derivatives of w with respect to r and t using the chain rule:

∂w/∂r = (∂w/∂x) * (∂x/∂r) + (∂w/∂y) * (∂y/∂r) + (∂w/∂z) * (∂z/∂r)

∂w/∂r = (x/√(x²+y²+z²)) * (-6e^t) + (y/√(x²+y²+z²)) * (6te^r) + (z/√(x²+y²+z²)) * (e^r)

Substituting the given expressions for x, y, and z:

∂w/∂r = (-6re^t/√((-6re^t)²+(6te^r)²+(e^r)²)) * (-6e^t) + (6te^r/√((-6re^t)²+(6te^r)²+(e^r)²)) * (6te^r) + (e^r/√((-6re^t)²+(6te^r)²+(e^r)²)) * (e^r)

Simplifying the equation:

∂w/∂r = (36r²e^2t + 36t²e^2r + e^2r)/(√(36r²e^2t + 36t²e^2r + e^2r))

Similarly procedure for ∂w/∂t.

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The coordinates of B” under the transformations is (-4, -2)

from the question, we have the following parameters that can be used in our computation:

B = (-2, -4)

The transformation is given as

So, we have

Reflect over the x-axis

B' = (-2, 4)

Rotate 90° CW

B'' = (-4, -2)

Hence. the coordinates of B” are (-4, -2)

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