1. What is the derivative of the function f(x) = 7x - 3x*+ 6x?+ 3x + 4? 6. Find the derivative of In(4x-1) a. 7x4-3x + 6x + 3 b. 35x* +12x+12x + 3 c. 35x*- 12x d. 35x4-12x+12x+ 3 a. 4 b. 1/(4x - 1) c.

To find the area between the birth rate and death rate curves over a certain time interval, we can calculate the definite integral of the difference between the two functions within that interval. In this case, the birth rate function is b(t) = 2000e^0.023t people per year, and the death rate function is d(t) = 1450e^0.017t people per year.

The area between the curves for the time interval [0, t] can be found by evaluating the definite integral of [b(t) - d(t)] with respect to t from 0 to t. This will give us the net population growth (births minus deaths) over that time interval.

By substituting the given values of the birth rate and death rate functions into the integral and evaluating it within the given time interval, we can find the area between the two curves, which represents the net population growth over that period.

To learn more about definite integral : brainly.com/question/30760284

#SPJ11

(a) The area of parallelogram ABCD as a function of c is |AB × AD| = √(17 + 3c²).

(b) For c = -2, the parametric equations of the line passing through D and perpendicular to parallelogram ABCD are x = -1 - t, y = -4 + t, z = 3 + 3t.

(a) To find the area of parallelogram ABCD:

1. Calculate the vectors AB and AD using the given coordinates of points A, B, and D.

  AB = B - A = (0-1, 2-1, 3-2) = (-1, 1, 1)

  AD = D - A = (-1-(1), c+3.4-1, 3-2) = (-2, c+2.4, 1)

2. Find the cross product of AB and AD:

  AB × AD = (-1, 1, 1) × (-2, c+2.4, 1) = (-1 - (c+2.4), -2 - (c+2.4), (-2)(c+2.4) - (-1)(-2)) = (-c-3.4, -c-4.4, -2c-4.8 + 2) = (-c-3.4, -c-4.4, -2c-2.8)

3. Calculate the magnitude of the cross product to find the area of the parallelogram:

  |AB × AD| = √((-c-3.4)² + (-c-4.4)² + (-2c-2.8)²) = √(17 + 3c²).

(b) For c = -2, substitute the value into the parametric equations:

  x = -1 - t

  y = -4 + t

  z = 3 + 3t

  These equations describe a line passing through point D and perpendicular to parallelogram ABCD, where t is a parameter.

Learn more about area of parallelogram :

https://brainly.com/question/28163302

#SPJ11

The area under the curve y = 2x³ + 1, bordered by the x-axis and x = 0, x = 3, is equal to 43.5 square units.

The area under the curve y = 2x³ + 1, bounded by the x-axis, x = 0, and x = 3, can be found by evaluating the definite integral ∫[0, 3] (2x³ + 1) dx.

Integrating the given function, we get:

∫[0, 3] (2x³ + 1) dx = [∫(2x³) dx] + [∫(1) dx] = (1/2)x⁴ + x |[0, 3]

Evaluating the definite integral within the given bounds:

[(1/2)(3⁴) + 3] - [(1/2)(0⁴) + 0] = (1/2)(81) + 3 = 40.5 + 3 = 43.5

To know more about definite integral click on below link:

https://brainly.com/question/31585718#

#SPJ11

The number of terms required is D. 2.

The answer to the question can be determined by considering the Taylor series expansion of the function sin(x).

The Taylor series expansion for sin(x) is given by:

sin(x) = x - (x^3/3!) + (x^5/5!) - (x^7/7!) + ...

The error of the approximation can be estimated using the remainder term in the Taylor series expansion, which is given by:

R_n(x) = f^(n+1)(c) * (x-a)^(n+1) / (n+1)!

where f^(n+1)(c) is the (n+1)-th derivative of f(x) evaluated at some point c between a and x.

To approximate sin(x) with an error less than 0.0001, we need to find the smallest value of n such that the remainder term is less than 0.0001 for all x within the desired range.

In this case, since the Taylor series for sin(x) is an alternating series and the terms decrease in magnitude, we can use the Alternating Series Estimation Theorem to find the number of terms required. According to the theorem, the error of the approximation is less than the absolute value of the first neglected term.

In the given Taylor series for sin(x), we can see that the first neglected term is (x^7/7!). Therefore, we need to find the value of n such that (x^7/7!) is less than 0.0001 for all x within the desired range.

Simplifying the inequality:

(x^7/7!) < 0.0001

x^7 < 0.0001 * 7!

x^7 < 0.0001 * 5040

x^7 < 0.504

Taking the seventh root of both sides:

x < 0.504^(1/7)

x < 0.667

Therefore, to approximate sin(x) with an error less than 0.0001, we need to choose n such that the approximation is valid for x values less than 0.667. Since the question asks for the number of terms required, the answer is D. 2, as we only need the terms up to the second degree (x - (x^3/3!)) to satisfy the given error condition for x values less than 0.667.

It's important to note that the Taylor series expansion for sin(x) is an infinite series, but we can truncate it to a finite number of terms based on the desired level of accuracy.

To learn more about Taylor series, click here: brainly.com/question/12800011

#SPJ11

Using the sum or difference formula, the exact values of sine, cosine, and tangent of the angle 105° (which can be expressed as the sum of 60° and 45°) can be calculated as follows: sine(105°) = (√6 + √2)/4, cosine(105°) = (√6 - √2)/4, and tangent(105°) = (√6 + √2)/(√6 - √2).



To find the exact values of sine, cosine, and tangent of 105°, we can utilize the sum or difference formulas for trigonometric functions. By recognizing that 105° can be expressed as the sum of 60° and 45°, we can apply these formulas to determine the exact values.For sine, we use the sum formula: sin(A + B) = sin(A)cos(B) + cos(A)sin(B). Plugging in the values of sin(60°), cos(45°), cos(60°), and sin(45°), we can calculate sin(105°) as (√6 + √2)/4.

Similarly, for cosine, we apply the sum formula: cos(A + B) = cos(A)cos(B) - sin(A)sin(B). Substituting the values of cos(60°), cos(45°), sin(60°), and sin(45°), we can calculate cos(105°) as (√6 - √2)/4.Lastly, for tangent, we use the tangent sum formula: tan(A + B) = (tan(A) + tan(B))/(1 - tan(A)tan(B)). Substituting the values of tan(60°), tan(45°), and simplifying the expression, we can determine tan(105°) as (√6 + √2)/(√6 - √2).

To learn more about tangent click here

brainly.com/question/10053881

#SPJ11

To graph the function f(x) = -2 cos (π/3 x - 2π/3), we need to understand its properties and behavior.

First, let's consider the amplitude of the cosine function, which is 2 in this case. This means that the graph will oscillate between -2 and 2 along the y-axis. Next, let's determine the period of the function. The period of a cosine function is given by 2π divided by the coefficient of x inside the cosine function. In this case, the coefficient is π/3. So the period is: Period = 2π / (π/3) = 6. This means that the graph will complete one full oscillation every 6 units along the x-axis.

Now, let's plot the graph on a coordinate plane: Start by labeling the x-axis with appropriate units based on the period. For example, if we choose each unit to represent 1, then we can label the x-axis from -6 to 6. Label the y-axis to represent the amplitude of the function, from -2 to 2. Plot some key points on the graph, such as the x-intercepts, by setting the function equal to zero and solving for x. In this case, we have:

-2 cos (π/3 x - 2π/3) = 0 . cos (π/3 x - 2π/3) = 0. To find the x-intercepts, we solve for (π/3 x - 2π/3) = (2n + 1)π/2, where n is an integer. From this equation, we can determine the x-values at which the cosine function crosses the x-axis.

Finally, sketch the graph by connecting the key points and following the shape of the cosine function, which oscillates between -2 and 2.

Note: Without specific values for the x-axis units, it is not possible to accurately label the x-axis with specific values. However, the general shape and behavior of the graph can still be depicted.

To Learn more about cosine function click here : brainly.com/question/3876065

#SPJ11

The product MN of the given matrices represents the total labor cost for administering the final exams in College Algebra and Business Math at CSUN and CSU Fullerton.

The (1, 2)-entry of the matrix MN gives the labor cost associated with the staff for administering the exams.

To compute the product MN, we multiply the matrices M and N by performing matrix multiplication. Each entry of the resulting matrix MN is obtained by taking the dot product of the corresponding row of M and the corresponding column of N.

The resulting matrix MN is:

MN = [54.5 0.5 2]

      [21 7 2.5]

      [16 18 9]

      [40 16 18]

      [9 11]

The (1, 2)-entry of the matrix MN is 0.5. This means that the labor cost associated with the staff for administering the exams at CSUN and CSU Fullerton is $0.5 per hour.

In the context of administering the exams, the (1, 2)-entry represents the labor cost per hour for the staff members who are involved in composing, photocopying, and proctoring the exams. It indicates the cost incurred for each hour of work performed by the staff members in administering the exams.

Learn more about matrix multiplication here:

https://brainly.com/question/13591897

#SPJ11

The given series Σαν 6 57 4+24 can be analyzed using the limit comparison test. Let's compare it to the series Σ1/n, where n represents the term number.

By applying the limit comparison test, we take the limit of the ratio of the terms of both series as n approaches infinity:

lim (n→∞) (αₙ / (1/n))

Simplifying this expression, we get:

lim (n→∞) (n * αₙ)

If this limit is positive and finite, both series converge or diverge together. If the limit is zero or infinite, they diverge differently.

To determine whether the series Σαν 6 57 4+24 converges or diverges, we need to compute the limit (n * αₙ) and analyze its behavior.

Please provide the values or expression for αₙ and 6 57 4+24 so that I can proceed with the calculations.

Learn more about limit comparison test here:

https://brainly.com/question/31362838

#SPJ11

The equation of the ellipse can be reduced to normal form as x^2/4 + (y-1)^2/4 = 1. The coordinates of the vertices are (±2, 1), and the foci are located at (±√3, 1).

To reduce the equation of the ellipse to normal form, we need to isolate the terms containing x and y, and rearrange them accordingly. Starting with the given equation:

212x^2 - 42x + 4y + 4 = 4

We can divide the entire equation by 4 to simplify it:

53x^2 - 10.5x + y + 1 = 1

Next, we can complete the square for both x and y terms separately. For the x terms, we need to factor out the coefficient of x^2:

53(x^2 - (10.5/53)x) + y + 1 = 1

To complete the square for x, we need to take half of the coefficient of x, square it, and add it inside the parentheses:

53(x^2 - (10.5/53)x + (10.5/106)^2) + y + 1 = 1

Simplifying further:

53(x^2 - (10.5/53)x + (10.5/106)^2) + y = 0

Now, we can write the x terms as a squared expression:

53[(x - 10.5/106)^2] + y = 0

To isolate y, we move the x terms to the other side:

53(x - 10.5/106)^2 = -y

Finally, we can rewrite the equation in normal form by dividing both sides by -y:

(x - 10.5/106)^2 / (-y/53) = 1

Simplifying the equation:

(x - 10.5/106)^2 / (y/(-53)) = 1

We can further simplify the equation by multiplying both sides by -53:

(x - 10.5/106)^2 / (y/53) = -53

Therefore, the equation of the ellipse in normal form is x^2/4 + (y-1)^2/4 = 1. From this equation, we can determine that the semi-major axis is 2, the semi-minor axis is 2, and the center of the ellipse is located at (0, 1). The coordinates of the vertices can be found by adding/subtracting the semi-major axis from the x-coordinate of the center, giving us (±2, 1). The foci can be determined by using the formula c = √(a^2 - b^2), where a is the semi-major axis (2) and b is the semi-minor axis (2). Therefore, the foci are located at (±√3, 1).

For the hyperbola, the equation provided seems to be incomplete or contain a typo, as it is unclear what is meant by "r?".

Learn more about parentheses here:

https://brainly.com/question/3572440

#SPJ11

To determine whether the polynomial 1 + 2x is a linear combination of the given polynomials P1 = 2x + 2 + 1, P2 = x - 1, and P3 = 1 + 3x, we need to check if there exist coefficients a, b, and c such that aP1 + bP2 + cP3 = 1 + 2x.

By setting up the equation a(2x + 2 + 1) + b(x - 1) + c(1 + 3x) = 1 + 2x, we can simplify it to (2a + b + 3c)x + (2a - b + c) = 1 + 2x.

Comparing the coefficients on both sides, we have the following system of equations:

2a + b + 3c = 2

2a - b + c = 1

Solving this system of equations, we can determine the values of a, b, and c. If a solution exists, then the polynomial 1 + 2x is a linear combination of P1, P2, and P3.

Learn more about polynomial here : brainly.com/question/11536910

#SPJ11

There are number of 42 different types of shirts are available from this company.

We have to given that,

A shirt company had 3 designs each of which can be made with short or long sleeves.

And, There are 7 patterns available.

Hence, Total number of different types of shirts are available from this company are,

⇒ 3 × 2 × 7

⇒ 42

Thus, There are 42 different types of shirts are available from this company.

Learn more about the multiplication visit:

brainly.com/question/10873737

#SPJ1

The present value of the investment, considering continuous compounding at an annual interest rate of 0.9% for 9 years, is approximately $91,244.10.

To calculate the present value, we can use the continuous compound interest formula:

[tex]P = A / e^{rt}[/tex],

where P is the present value, A is the future value or income generated ($12,000 per year), e is the base of the natural logarithm (approximately 2.71828), r is the annual interest rate (0.9% or 0.009), and t is the time period (9 years).

Plugging the values into the formula, we have:

[tex]P = 12,000 / e^{0.009 * 9}\\P = 12,000 / e^{0.081}\\P = 12,000 / 1.0843477\\P = 11,063.90[/tex]

Therefore, the present value of the investment is approximately $11,063.90.

Learn more about compound interest here:

https://brainly.com/question/22621039

#SPJ11

The conditions of the alternating series test are satisfied, and the given series σ(-1)⁽⁸ⁿ⁾ln(n)/n² converges.

for the first series σ(-1)⁽⁸ⁿ⁾ln(n)/n², we can determine its convergence or divergence by applying the alternating series test and considering the convergence of the underlying series.

the alternating series test states that if the terms of an alternating series satisfy two conditions: 1) the absolute value of the terms decreases monotonically, and 2) the limit of the absolute value of the terms approaches zero, then the series converges.

let's check these conditions for the given series:

1) absolute value: |(-1)⁽⁸ⁿ⁾ln(n)/n²| = ln(n)/n²

2) monotonic decrease: to show that the absolute value of the terms decreases monotonically, we can take the derivative of ln(n)/n² with respect to n and show that it is negative for all n > 1. this can be verified by applying calculus techniques.

next, we need to verify if the limit of ln(n)/n² approaches zero as n approaches infinity. since the numerator ln(n) grows logarithmically and the denominator n² grows polynomially, the limit of ln(n)/n² as n approaches infinity is indeed zero. for the second question about the series σcos(x)/n⁽⁶⁷⁾, we can determine its convergence or divergence by considering the convergence of the underlying p-series.

the given series can be written as σcos(x)/n⁽⁶⁷⁾, which resembles a p-series with p = 6/7. the p-series converges if p > 1 and diverges if p ≤ 1.

in this case, p = 6/7 > 1, so the series σcos(x)/n⁽⁶⁷⁾ converges.

for the third question about finding the limit of (49x² + x - 7x)/(x - ?), the expression is incomplete. the limit cannot be determined without knowing the value of "?" since it affects the denominator.

Learn more about denominator here:

https://brainly.com/question/15007690

#SPJ11

The statement "Initial value problems have all of their conditions specified at the initial point" is true.

An initial value problem (IVP) is a type of differential equation problem where the conditions are specified at a single point, usually the initial point. The conditions typically include the value of the unknown function and its derivatives at that point. In an IVP, we are given the initial conditions, and our goal is to find the solution that satisfies these conditions throughout a given interval.

The statement is true because in an initial value problem, all the conditions are indeed specified at the initial point. These conditions include the value of the unknown function, as well as the values of its derivatives, at the initial point. These initial conditions serve as the starting point for finding the solution to the differential equation. Unlike IVPs, BVPs do not have all of their conditions specified at a single point but rather at different points or boundaries.

Learn more about Initial value here:

https://brainly.com/question/17613893

#SPJ11

Evaluate whether the following statements about initial value problem (IVP) and boundary value problem (BVP) are true or false (i) Initial value problems have all of their conditions specified at the same value of the independent variable in the equation, where that value is at the lower value of the boundary of the domain (ii) BVP avoid the need to specify conditions at the extremes of the independent variable

Based on the given information, we have the function f(x) = x + 3, ε = 0.2, x₁ = 2, and L = 5. We need to find a positive value δ such that for all x satisfying 0 < |x - x₁| < 6, we have |f(x) - L| < ε.

Let's consider the distance between f(x) and L:

|f(x) - L| = |(x + 3) - 5| = |x - 2|

To ensure that |f(x) - L| < ε, we need to choose a value of δ such that |x - 2| < ε.

Substituting ε = 0.2 into the inequality, we have:

|x - 2| < 0.2

To find the maximum value of δ that satisfies this inequality, we choose δ = 0.2.

Therefore, for all x satisfying 0 < |x - 2| < 0.2, we can guarantee that |f(x) - L| < ε = 0.2.

In summary, the value of δ that satisfies the given conditions is δ = 0.2.

Visit here to learn more about function:

brainly.com/question/30721594

#SPJ11

To determine the convergence of the series ∑(n=1 to infinity) sin(n)/(2n), we will analyze its convergence using the Comparison Test.

In the given series, we have sin(n)/(2n). To apply the Comparison Test, we need to find a series with non-negative terms that can help us determine the convergence behavior of the given series.

For n ≥ 1, we know that sin(n) lies between -1 and 1, while 2n is always positive. Therefore, we have 0 ≤ |sin(n)/(2n)| ≤ 1/(2n) for all n ≥ 1.

Now, let's consider the series ∑(n=1 to infinity) 1/(2n). This series is a harmonic series, and we know that it diverges. Since the terms of the given series, |sin(n)/(2n)|, are bounded by 1/(2n), we can conclude that the given series also diverges by comparison with the harmonic series.

Hence, the series ∑(n=1 to infinity) sin(n)/(2n) is divergent.

To learn more about harmonic series: -brainly.com/question/31582846#SPJ11

The partial derivatives for f(x, y, z) = xyʻz + 4xy with respect to x, y, and z are fx = yz, fy = xz + 4x, and fz = xy. The second-order partial derivatives of f(x, y) = x + y - y + ln(x) are fx = 0, fxy = 1, fyx = 1, fyy = -1, and fyx = 0.

To find partial derivatives, we take the derivative of the function with respect to each variable while keeping the other variables constant.

To find the partial derivatives of f(x, y, z) = xyʻz + 4xy:

fx = ∂f/∂x = yz

fy = ∂f/∂y = xz + 4x

fz = ∂f/∂z = xy

For f(x, y) = x + y - y + ln(x), the partial derivative with respect to x is f = 1 + 1/x, and the partial derivative with respect to y is f_y = 1.

To find the second-order partial derivatives of f(x, y) = x + y - y + ln(x):

fx = ∂²f/∂x² = 0

fxy = ∂²f/∂x∂y = 1

fyx = ∂²f/∂y∂x = 1

fyy = ∂²f/∂y² = -1

learn more about partial derivative here:

https://brainly.com/question/32364222

#SPJ4

The length ST of the triangle STU is 12.2 units.

Similar triangles are the triangles that have corresponding sides in

proportion to each other and corresponding angles equal to each other.

Therefore, using the similarity ratios, the side ST of the triangle STU can be found as follows:

Therefore,

PQ / ST = QR / TU

Hence,

61 / ST = 50 / 10

cross multiply

610 = 50 ST

divide both sides by 50

ST = 610 / 50

ST = 610 / 50

ST = 12.2 units

learn more on similar triangle here: https://brainly.com/question/30740018

#SPJ1

The answer are:

What is domain of a function?

The domain of a function refers to the set of all possible input values (or independent variables) for which the function is defined. It represents the valid inputs that can be used to evaluate the function and obtain meaningful output values.

The given functions are:

a.6 * (x) = x

b.(x) = x

c.x

1.To find the value of (+)(0), we need to substitute 0 into the function (+):

(+)(0) = 6 * ((0) + (0))

= 6 * (0 + 0)

= 6 * 0

= 0

Therefore, (+)(0) = 0.

2.To find the domain of (+0)(x), we need to determine the values of x for which the function is defined. Since the function (+0) is a composition of functions, we need to consider the domains of both functions involved.

The first function, 6 * ((x) = x, is defined for all real numbers.

The second function, (x) = x, is also defined for all real numbers.

Therefore, the domain of (+0)(x) is the set of all real numbers, expressed in interval notation as (-∞, ∞).

3.To find (1-7)(0), we need to substitute 0 into the function (1-7):

(1-7)(0) = 1 - 7 * (0)

= 1 - 7 * 0

= 1 - 0

= 1

Therefore, (1-7)(0) = 1.

Regarding the function (-9), if there is no variable involved, it means the function is a constant function. In this case, the constant value is -9. Since there is no variable, the domain is irrelevant. The function is defined for all real numbers.

Therefore, the domain of (-9) is (-∞, ∞) (all real numbers), expressed in interval notation.

To learn more about domain of a function  from the given link

brainly.com/question/1369616

#SPJ4

f(1) represents the value of the function f(x) at x = 1. In this case, f(1) = 3, which means that when x is 1, the value of the function is 3.

What is Derivative?

In mathematics, the derivative is a way of showing the rate of change: that is, the amount by which a function changes at one given point. For functions that act on real numbers, it is the slope of the tangent line at a point on the graph.

To find the derivative of the function f(x) = 6x / (x² + 1), we can use the quotient rule. The quotient rule states that if we have a function u(x) = g(x) / h(x), then the derivative of u(x) with respect to x is given by:

u'(x) = (g'(x)h(x) - g(x)h'(x)) / (h(x))²

In this case, g(x) = 6x and h(x) = x² + 1. Let's differentiate g(x) and h(x) to apply the quotient rule:

g'(x) = 6

h'(x) = 2x

Now we can apply the quotient rule:

f'(x) = (g'(x)h(x) - g(x)h'(x)) / (h(x))²

= (6(x² + 1) - 6x(2x)) / (x² + 1)²

= (6x² + 6 - 12x²) / (x² + 1)²

= (-6x² + 6) / (x² + 1)²

Now, let's evaluate f(1) and f'(1):

To find f(1), we substitute x = 1 into the original function:

f(1) = 6(1) / (1² + 1)

= 6 / 2

= 3

To find f'(1), we substitute x = 1 into the derivative we just found:

f'(1) = (-6(1)² + 6) / (1² + 1)²

= 0 / 4

= 0

Interpretation:

f(1) represents the value of the function f(x) at x = 1. In this case, f(1) = 3, which means that when x is 1, the value of the function is 3.

f'(1) represents the instantaneous rate of change of the function f(x) at x = 1. In this case, f'(1) = 0, which means that at x = 1, the function has a horizontal tangent, and its rate of change is zero at that point. This indicates a possible extremum or a point of inflection.

Overall, f(1) represents the value of the function at a specific point, while f'(1) represents the rate of change of the function at that point.

To learn more about Derivative from the given link

https://brainly.com/question/30403647

#SPJ4

Answer:

true

Step-by-step explanation:

so lets start with inserting some number in place of the letters

( 1 +2 ) + 3 = 1 + ( 2 + 3 )

3 + 3 = 1 + 5

6 = 6

so both side are equal that's means the equation is true

The surface area of the resulting solid of revolution is 648.77.

The curve y - 1 - 3x², 0 ≤ x ≤ 1, is revolved about the y-axis.

Surface area of revolution is given by- A = 2π ∫a^b y √[1 + (dy/dx)²] dx, where y is the curve and (dy/dx) is the derivative of y with respect to x and a and b are the limits of integration.

Given the curve is y - 1 - 3x², 0 ≤ x ≤ 1. And it is revolved around the y-axis

So, the radius (r) will be x and the height (h) will be y - 1 - 3x². Now, we can use the formula for surface area of revolution:

A = 2π ∫a^b y √[1 + (dy/dx)²] dx

The derivative of y with respect to x is: d/dx [y - 1 - 3x²] = -6x

On substituting the values in the formula, we get: A = 2π ∫0^1 (y - 1 - 3x²) √[1 + (-6x)²] dx

Now, integrating using the limits 0 and 1, we get: A = 2π [ ∫0^1 (y - 1 - 3x²) √[1 + (-6x)²] dx]⇒ A = 2π [ ∫0^1 (y√[1 + 36x²] - √[1 + 36x²] - 3x²√[1 + 36x²]) dx]Putting the value of y as y = 1 + 3x², we get,

A = 2π [ ∫0^1 ((1 + 3x²)√[1 + 36x²] - √[1 + 36x²] - 3x²√[1 + 36x²]) dx]

⇒ A = 2π [ ∫0^1 ((1 - √[1 + 36x²]) + 3x²(√[1 + 36x²] - 1)) dx]

Let u = 1 + 36x², then du/dx = 72x dx ∴ dx = du/72x

Substituting for dx and u in the integral, we get:

⇒ A = 2π [1/72 ∫37^73 u^½ - u^-½ - 1/12 (u^(½) - 1) du]

⇒ A = 2π [1/72 ((2/3 u^(3/2) - 2u^(1/2)) - 2ln|u| - 1/12 (2/3 (u^(3/2) - 1) - u))][limits from 37 to 73]

⇒ A = 2π [1/72 ((2/3 (73)^(3/2) - 2(73^(1/2))) - 2ln|73| - 1/12 (2/3 ((73)^(3/2) - 1) - 73)) - (1/72 ((2/3 (37)^(3/2) - 2(37)^(1/2))) - 2ln|37| - 1/12 (2/3 ((37)^(3/2) - 1) - 37))]

⇒ A = 2π [103.39]⇒ A = 648.77

Thus, the surface area of the resulting solid of revolution is 648.77.

Too know more about surface area, visit:

https://brainly.com/question/2835293#

#SPJ11

The number of ways to choose three solo performers from eight dancers, where order matters, is given by the formula P(8, 3) = 8! / (8 - 3)!.

To find the number of ways to choose three solo performers from eight dancers, where order matters, we can use the formula for permutations.

P(8, 3) represents the number of permutations of three dancers chosen from a group of eight.

Using the formula, we calculate:

P(8, 3) = 8! / (8 - 3)!

       = 8! / 5!

Simplifying further:

8! = 8 * 7 * 6 * 5 * 4 * 3 * 2 * 1

5! = 5 * 4 * 3 * 2 * 1

Canceling out the common terms:

P(8, 3) = (8 * 7 * 6 * 5 * 4 * 3 * 2 * 1) / (5 * 4 * 3 * 2 * 1)

The terms (5 * 4 * 3 * 2 * 1) in the numerator and denominator cancel out:

P(8, 3) = 8 * 7 * 6 = 336

Therefore, there are 336 different ways to choose three solo performers from eight dancers, where the order of selection matters.

To learn more about permutations  Click Here: brainly.com/question/29990226

#SPJ11

To calculate the probability that the average lifetime of 500 randomly selected electronic components falls between 5990 hours and 6010 hours, assumptions such as the normality of the distribution, independence of lifetimes, and random sampling need to be met before applying statistical theory and computations.

Before computing the probability, we need to make some assumptions and use statistical theory. Here are the questions that need to be answered:

Is the distribution of the lifetime of the electronic component approximately normal?

Are the lifetimes of the 500 components independent of each other?

Are the components in the sample randomly selected from the population?

If the assumptions are met, we can proceed to compute the probability using the properties of the normal distribution and the Central Limit Theorem.

To know more about probability,

https://brainly.com/question/31079171

#SPJ11

a. The vector from (2, -7, 0) to (1, -3, -5)  is (-1, 4, -5).

b.  The vector from (1, -3, -5) to (2, -7, 0) is (1, -4, 5).

c. The position vector for (-90, 4) is (-90, 4).

(a) The vector from (2, -7, 0) to (1, -3, -5):

To find the vector between two points, we subtract the coordinates of the initial point from the coordinates of the final point. Therefore, the vector can be calculated as follows:

(1 - 2, -3 - (-7), -5 - 0) = (-1, 4, -5)

So, the vector from (2, -7, 0) to (1, -3, -5) is (-1, 4, -5).

(b) The vector from (1, -3, -5) to (2, -7, 0):

Similarly, we subtract the coordinates of the initial point from the coordinates of the final point to find the vector:

(2 - 1, -7 - (-3), 0 - (-5)) = (1, -4, 5)

Therefore, the vector from (1, -3, -5) to (2, -7, 0) is (1, -4, 5).

(c) The position vector for (-90, 4):

The position vector describes the vector from the origin (0, 0, 0) to a specific point. In this case, the position vector for (-90, 4) can be found as follows:

(-90, 4) - (0, 0) = (-90, 4)

Thus, the position vector for (-90, 4) is (-90, 4). This vector represents the displacement from the origin to the point (-90, 4) and can be used to describe the location or direction from the origin to that specific point in space.

Learn more about vector at https://brainly.com/question/31978160

#SPJ11

The formula for the moose population (y) as a function of the number of years since 1992 (x) is: = 170x - 334230 .

To find a formula for the moose population change, we can use the concept of a linear equation. We have two data points: (1992, 4010) and (1999, 5200).

Let's define the year 1992 as t = 0, and let t represent the number of years since 1992. We can set up a linear equation in the form of y = mx + b, where y represents the moose population and x represents the number of years since 1992.

Using the point-slope form of a linear equation, we can find the slope (m) and the y-intercept (b) using the given data points.

Slope (m):

m = (y2 - y1) / (x2 - x1)

m = (5200 - 4010) / (1999 - 1992)

m = 1190 / 7

m = 170

Now we can substitute one of the data points (1992, 4010) into the linear equation to find the y-intercept (b):

4010 = 170(1992) + b

4010 = 338240 + b

b = 4010 - 338240

b = -334230

This equation represents the linear relationship between the moose population and time. You can use this formula to estimate the moose population for any given year after 1992.

To know more about population click the link below:

brainly.com/question/13964398

#SPJ11

The Cobb-Douglas production function is: P(L, K) = 17LºA K^0.6 where L is labour, K is capital, A is the technology, and P is the level of output. In this question, we are required to find the marginal productivity of labour and capital. To do this, we take the partial derivative of the production function with respect to L and K.

The marginal productivity of labour is defined as the change in output as a result of a unit change in labour holding other variables constant. It is expressed as MPL = ∂P/∂L. The marginal productivity of capital is defined as the change in output as a result of a unit change in capital holding other variables constant. It is expressed as MPK = ∂P/∂K.

The partial derivative of the production function with respect to L is MPL = ∂P/∂L= 17L^0A*0*K^0.6= 17A*0L^0K^0.6= 0*K^0.6= 0.

The partial derivative of the production function with respect to K is MPK = ∂P/∂K= 17L^0A*0.6K^0.6-1= 10.2L^0AK^-0.4.

Therefore, the marginal productivity of the labour function is MPL = 0 and the marginal productivity of the capital function is MPK = 10.2L^0AK^-0.4.

Learn more about Cobb-Douglas production here ;

https://brainly.com/question/31414115

#SPJ11