please solve allif f(x) = x*sinx - 5tanx find f'(x) If y = (3x - 2)(5x2 + 1), find y' d (Vx) dx L'(x) if f(x) = 4x2 S X+5 =

The displacement of the particle is 18 unit and total distance travelled 18.

We have,

Velocity, v(t) = t² -4t+ 3

So, the displacement of particle is

r(t) = [tex]\int\limits^6_0[/tex]  t² -4t+ 3 dt

r(t) = [t³/3 - 4t²/2 + 3t[tex]|_0^6[/tex]

r(t) = [ 216/3 - 72 + 18]

r(t) = 18

Thus, the displacement is 18 unit.

For distance travelled

t² -4t+ 3=0

t² -3t - t + 3=

t(t-3) -1 (t-3)= 0

t= 1, 3

So, Distance =  [tex]\int\limits^3_0[/tex]  t² -4t+ 3 dt +  [tex]\int\limits^6_3[/tex]  t² -4t+ 3 dt

= [t³/3 - 4t²/2 + 3t[tex]|_0^3[/tex] + [t³/3 - 4t²/2 + 3t[tex]|_3^6[/tex]

= (9 - 18 + 9 ) + (72 - 72 + 18 - 9 + 18 - 9)

= 18 unit

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The variance of the given probability distribution is 1.87.

To find the variance of a probability distribution, we need to first calculate the expected value or mean of the distribution. The expected value of a discrete random variable X is given by:

E(X) = ∑[i=1 to n] xi * P(X = xi)

where xi is the i-th possible value of X, and P(X = xi) is the probability that X takes on the value xi.

Using this formula, we can calculate the expected value of the given probability distribution as:

E(X) = 1*0.16 + 2*0.19 + 3*0.22 + 4*0.21 + 5*0.12 + 6*0.10

    = 3.24

Next, we can calculate the variance of the distribution using the formula:

[tex]Var(X) = E(X^2) - [E(X)]^2[/tex]

where E([tex]X^2[/tex]) is the expected value of [tex]X^2[/tex], which is given by:

[tex]E(X^2) = ∑[i=1 to n] xi^2 * P(X = xi)[/tex]

Using this formula, we can calculate E([tex]X^2[/tex]) for the given probability distribution as:

[tex]E(X^2) = 1^2*0.16 + 2^2*0.19 + 3^2*0.22 + 4^2*0.21 + 5^2*0.12 + 6^2*0.10 = 11.53[/tex]

Now we can substitute the values of E(X) and E[tex](X^2[/tex]) into the formula for variance to get:

[tex]Var(X) = E(X^2) - [E(X)]^2[/tex]

      = 11.53 - [tex]3.24^2[/tex]

      = 1.87

Therefore, the variance of the given probability distribution is 1.87.

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Answer:4/5

Step-by-step explanation:

Probability of getting greater than 1 (there are 4 possibilities for this)

5 total outcomes, so 4/5

Answer:

45

Step-by-step explanation:

this is the answer you need

the definite integral that gives the exact volume of the solid is:

V = ∫[0,2] 2π * 2 * |x - 3| dx

a) Here is a sketch of the region and the solid obtained by revolving it about the line x = 3:

Sketch of the region and the solid obtained by revolving it about the line x = 3

b) We can subdivide the solid into thin cylindrical shells, each with thickness Δx and radius given by the distance from x to the line x = 3. The height of each shell is given by the difference between the upper and lower boundaries of the region, which is:

y = x + (2 - x) = 2

Therefore, the volume of each shell is given by:

dV = 2πy * r * Δx

where r = |x - 3| is the distance from x to the axis of rotation. Thus, the general expression for the volume of each subdivision is:

dV = 2π * 2 * |x - 3| * Δx

c) To find the total volume of the solid, we need to add up the volumes of all the cylindrical shells. This can be done by integrating the expression for dV over the interval of x values that covers the region. Since the region is bounded by the x-intercepts of the function y = x + (2 - x), we can find them by setting y = 0:

0 = x + (2 - x)

x = 0 and x = 2

Thus, the definite integral that gives the exact volume of the solid is:

V = ∫[0,2] 2π * 2 * |x - 3| dx

Note that the absolute value is necessary because the distance from x to 3 can be negative on the interval [0, 3), but we want a positive radius for the cylindrical shells.

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The maximum area of the enclosement and the dimensions that will create it 234.375 m²

We have,

A farmer has 75 m of fencing available to enclose.

The optimization (maximization/minimization) equation

A= xy

and, constraint equation

L= 2x+ 3y

L= 75 m

So, 2x+3y= 75

x= 37.5 - 1.5y

Now, A= xy

A= y(37.5 - 1.5y)

A= 37.5y - 1.5y²

To find the optimal value put the value of x= 0

A'= 0

37.5 - 3y = 0

y= 12.5

and, A = 37.5y - 1.5y²

A = 37.5(12.5) - 1.5(12.5)²

A= 468.75 - 234.375

A= 234.375 m²

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The order of operations needed for the forward/backward elimination steps in solving a linear system with this banded coefficient matrix is: Forward Elimination- Identify, perform Gaussian elimination and continue the process on the banded structure. Backward Elimination- solve the unknown variable, Substitute the value and continue the process.

In solving a linear system with a banded coefficient matrix, the order of operations needed for the forward/backward elimination steps is as follows:

1. Forward Elimination:
  a. Identify the banded structure of the coefficient matrix, which means determining the bandwidth (number of diagonals containing non-zero elements).
  b. Perform Gaussian elimination while preserving the banded structure, by eliminating elements below the main diagonal within the bandwidth.
  c. Continue this process for all rows within the bandwidth until an upper triangular banded matrix is obtained.

2. Backward Elimination (Back Substitution):
  a. Starting from the last row, solve for the unknown variable by dividing the right-hand side value by the corresponding diagonal element.
  b. Substitute the obtained value into the equations above, within the bandwidth, and continue solving for the remaining unknown variables.
  c. Continue this process until all unknown variables are solved, moving upward through the rows.

By following this order of operations, you can efficiently solve a linear system with a banded coefficient matrix using forward and backward elimination steps.

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In the file 'Death And Taxes.csv' are data on death rates prior to and following tax rate changes during years in which the US government announced it was changing the tax rate on inheritance. After performing the appropriate test that compares the death rates before and after tax increases, the absolute value of t = 2.719, df = 10 and P = 0.020.

To obtain these values, a t-test was conducted on the data in the 'Death And Taxes.csv' file to compare the death rates before and after tax increases. The tax rate changes occurred in different years, and the death rates were recorded prior to and following the tax rate changes. To perform the t-test, the difference between the death rates after and before the tax increase was calculated for each year, and these differences were used to obtain a sample mean and standard deviation. The null hypothesis was that there is no difference between the death rates before and after the tax increase. The alternative hypothesis was that the death rates after the tax increase are higher than before. The t-value was calculated as the ratio of the sample mean difference to the standard error of the mean difference. The degrees of freedom were obtained as n-1, where n is the number of years with tax rate changes. The P-value was obtained from a t-distribution table using the t-value and degrees of freedom. The P-value indicates the probability of observing the t-value or a more extreme value if the null hypothesis is true. Since the P-value is less than 0.05, we reject the null hypothesis and conclude that there is a significant difference between the death rates before and after the tax increase. The tax rate increase seems to have had a negative impact on the death rates. Note that the death rates are given in the decimal form.

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The value of k that will make y = 2cos(2x) a solution to y" - ky = y " is k = -3.

To determine the value of k that will make y = 2cos(2x) a solution to y" - ky = y ", we first need to find the second derivative of y with respect to x.

y = 2cos(2x)

Taking the first derivative with respect to x

y' = -4sin(2x)

Taking the second derivative with respect to x

y" = -8cos(2x)

Now we substitute the given values of k in the differential equation and see if the equation holds true

For k = 5

y" - ky = -8cos(2x) - 5(2cos(2x)) = -18cos(2x) ≠ y"

Therefore, y = 2cos(2x) is not a solution for k = 5.

For k = -3

y" - ky = -8cos(2x) - (-3)(2cos(2x)) = -2cos(2x)

Therefore, y = 2cos(2x) is a solution for k = -3.

For k = 0:

y" - ky = -8cos(2x) - 0(2cos(2x)) = -8cos(2x) ≠ y"

Therefore, y = 2cos(2x) is not a solution for k = 0.

For k = -6

y" - ky = -8cos(2x) - (-6)(2cos(2x)) = 4cos(2x) ≠ y"

Therefore, y = 2cos(2x) is not a solution for k = -6.

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The slope of the tangent line at the x-intercept x = 5/2 is approximately 4.92.

The slope of the tangent line(s) at the x-intercepts of y = 2x⁵/³ - 5x²/³can be determined by finding the derivative of y and evaluating it at the x-intercepts.

First, we find the derivative of y with respect to x:

y' = d(2x⁵/³ - 5x²/³)/dx = (10/3)x²/³ - (10/3)x⁻¹/³

Next, we find the x-intercepts by setting y = 0:

0 = 2x⁵/³ - 5x²/³

Now, factor out x²/³:

0 = x²/³(2x - 5)

This gives us two x-intercepts: x = 0 and x = 5/2.

Finally, we evaluate y' at these x-intercepts:

For x = 0: y'(0) is undefined since we cannot have a negative exponent for x.

For x = 5/2: y'(5/2) = (10/3)(5/2)²/³ - (10/3)(5/2)⁻¹/³ ≈ 4.92

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The vector field F(x, y, z) = (xy²z²)i + (x+yz²)j + (x²y²+z)k is not conservative, as its curl is non-zero.

To determine if a vector field is conservative, we need to check if its curl is zero. If the curl is zero, then the vector field is conservative, and we can find a scalar potential function for it. However, if the curl is non-zero, then the vector field is not conservative.

In this case, we can calculate the curl of F using the formula for the curl of a vector field:

curl(F) = (∂N/∂y - ∂M/∂z)i + (∂P/∂z - ∂N/∂x)j + (∂M/∂x - ∂P/∂y)k

where F = Mi + Nj + Pk

After calculating the partial derivatives, we get:

curl(F) = 2xyzk i + (-y)j + (2x²y)k

Since the curl of F is not zero, F is not conservative. Therefore, we cannot find a scalar potential function for F.

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Answer:

The solution set is {0, 2}.

Step-by-step explanation:

3x^2-6x=0

3x(x - 2) = 0

3x = 0 or x - 2 = 0

x =(0, 2.

(a) The formula for A(t), the balance after t years is A(t) = P* [tex]e^r^t[/tex] , where P is the principal, r is the interest rate, and t is time in years.

(b) The differential equation satisfied by A(t) is A'(t) = r*A(t).

(c) After 2 years, the account balance will be $2,050.50 (rounded to the nearest cent).

(d) The balance will reach $6,000 after 22.1 years (rounded to one decimal place).

(e) When the balance reaches $6,000, it is growing at a rate of $148.52 per year.

(a) The continuous compounding formula, A(t) = P* [tex]e^r^t[/tex] , represents the balance after t years.

(b) The differential equation A'(t) = r*A(t) shows how the balance changes over time.

(c) To find the balance after 2 years, plug in the given values: A(2) = 2000*[tex]e^0^.^0^2^5^*^2[/tex] ≈ $2,050.50.

(d) To find when the balance reaches $6,000, set A(t) = 6000 and solve for t: 6000 = 2000*[tex]e^0^.^0^2^5^*^t[/tex], t ≈ 22.1 years.

(e) To find the growth rate at $6,000, plug the balance into the differential equation: A'(t) = 0.025*6000 ≈ $148.52 per year.

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The value of the definite integral of f(x) over the interval [-5, 5] is -13.

We are given two definite integrals:

[tex]\int_{-5}^{2}[/tex]f(x) dx = -17

and

[tex]\int_{5}^{3}[/tex] f(x) dx = -4

The first integral represents the area under the curve of f(x) from x = -5 to x = 2. The second integral represents the area under the curve of f(x) from x = 5 to x = 3. Note that the limits of integration for the second integral are in the reverse order, which means that the area is negative.

Now, we want to find the value of the definite integral of f(x) over the interval [-5, 5]. We can split this interval into two parts: [-5, 2] and [2, 5].

Using the first given integral, we know that the area under the curve of f(x) from x = -5 to x = 2 is -17.

Using the second given integral, we know that the area under the curve of f(x) from x = 5 to x = 3 is -4, which means that the area under the curve of f(x) from x = 3 to x = 5 is 4.

So, the area under the curve of f(x) from x = -5 to x = 5 is the sum of the areas under the curve of f(x) from x = -5 to x = 2 and from x = 3 to x = 5. Mathematically, we can write this as:

[tex]\int_{-5}^{5}[/tex] f(x) dx = [tex]\int_{-5}^{2}[/tex] f(x) dx + [tex]\int_{3}^{5}[/tex] f(x) dx

Substituting the given values, we get:

[tex]\int_{-5}^{5}[/tex] f(x) dx = -17 + 4 = -13

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A researcher claims that the average wind speed in the desert is less than 20.5 kilometers per hour. A sample of 33 days has an average wind speed of 19 kilometers per hour. The standard deviation of the population is 3.02 kilometers per hour. The Test Value, which is the z-score, is approximately -2.97.

The test value in this case is calculated using the formula:

Test value = (sample mean - hypothesized population mean) / (standard deviation / square root of sample size)

Plugging in the given values:

Test value = (19 - 20.5) / (3.02 / sqrt(33)) = -2.29

The critical value for a one-tailed test with a significance level of alpha = 0.05 and 32 degrees of freedom (n-1) is -1.697. Since the calculated test value (-2.29) is less than the critical value (-1.697), we can reject the null hypothesis and conclude that there is enough evidence to support the claim that the average wind speed in the desert is less than 20.5 kilometers per hour.

We will conduct a hypothesis test using the given information. We have:

Null hypothesis (H₀): The average wind speed in the desert is equal to 20.5 km/h (μ = 20.5)
Alternative hypothesis (H₁): The average wind speed in the desert is less than 20.5 km/h (μ < 20.5)

Sample size (n) = 33 days
Sample mean (x) = 19 km/h
Population standard deviation (σ) = 3.02 km/h

Now, we will find the Test Value, which is the z-score:

z = (x - μ) / (σ / √n)

Plugging in the values:

z = (19 - 20.5) / (3.02 / √33)
z ≈ -2.97

The Test Value, which is the z-score, is approximately -2.97.

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The probability that at least two out of ten employees who completed a sequence of executive seminars become vice presidents is 0.8618.

The probability that at least two out of 10 randomly selected graduates from the executive seminars become vice presidents can be calculated using binomial probability. Based on the given information that 40% of employees who complete the seminars become vice presidents, we can consider this as a binomial distribution with a success probability of 0.4 (probability of becoming a vice president) and a sample size of 10 (number of graduates selected).

Let's denote the event "a graduate becomes a vice president" as a success, and the event "a graduate does not become a vice president" as a failure. The probability of success is given as 0.4 and the probability of failure is 1 - 0.4 = 0.6.

We are interested in finding the probability of at least two successes, which means we need to calculate the probability of getting 2, 3, 4, 5, 6, 7, 8, 9, or 10 successes out of 10 trials.

The probability of getting exactly k successes out of n trials in a binomial distribution is given by the formula:

P(X = k) = (n choose k) × p^k × (1 - p)^(n - k)

where "n choose k" is the binomial coefficient, which is calculated as n! / (k! × (n - k)!), and p is the probability of success.

Now we can calculate the probability of at least two successes:

P(X ≥ 2) = P(X = 2) + P(X = 3) + P(X = 4) + P(X = 5) + P(X = 6) + P(X = 7) + P(X = 8) + P(X = 9) + P(X = 10)

Plugging in the values:

P(X ≥ 2) = [(10 choose 2) × 0.4² × 0.6⁸] + [(10 choose 3) × 0.4³ × 0.6⁷] + [(10 choose 4) × 0.4⁴ × 0.6⁶] + [(10 choose 5) × 0.4⁵ × 0.6⁵] + [(10 choose 6) × 0.4⁶ × 0.6⁴] + [(10 choose 7) × 0.4⁷ × 0.6³] + [(10 choose 8) × 0.4⁸ × 0.6²] + [(10 choose 9) × 0.4⁹ × 0.6¹] + [(10 choose 10) × 0.4¹⁰ × 0.6⁰]

Therefore, The probability that at least two out of ten employees who completed a sequence of executive seminars become vice presidents is 0.8618.

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The probability of guessing exactly 3 correct responses on a test consisting of 30 questions is approximately 0.0785 or 7.85%.

To find the probability of guessing exactly 3 correct responses on a test with 30 questions, we'll use the binomial probability formula. The terms you mentioned are:

- Probability (P) of guessing correctly = 1/5 (since there are 5 multiple choice options and only one is correct)
- Probability (Q) of guessing incorrectly = 4/5 (since 4 out of 5 options are incorrect)
- Number of questions (n) = 30
- Number of correct guesses (k) = 3

Now, we apply the binomial probability formula:

P(X=k) = C(n, k) * P^k * Q^(n-k)

P(X=3) = C(30, 3) * (1/5)³ * (4/5)²⁷

C(30, 3) represents the number of combinations of 30 questions taken 3 at a time:

C(30, 3) = 30! / (3! * (30-3)!) = 4,060

Plug the numbers back into the formula:

P(X=3) = 4,060 * (1/5)³ * (4/5)²⁷ ≈ 0.0785

The probability of guessing exactly 3 correct responses on the test is approximately 0.0785 or 7.85%.

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Total revenue function is TR = 1000Q² + 4000Q + C. Total cost function is TC = [tex]500(2/3)Q^{3/2} + 5Q + C[/tex]. Demand function is Q = 4. Competitor needs to manufacture approximately 2.09 widgets for its total revenue to equal the total revenue of Manufacturer A, based on setting their respective total revenue and total cost functions equal to each other and solving for Q.

To find the total revenue function, we integrate the marginal revenue function with respect to quantity (Q)

TR = ∫ MR dQ

TR = ∫ 1000(−2Q + 4Q + 4) dQ

TR = ∫ (2000Q + 4000) dQ

TR = 1000Q² + 4000Q + C

where C is the constant of integration.

To find the total cost function, we integrate the marginal cost function with respect to quantity (Q)

TC = ∫ MC dQ

TC = ∫ (500/2√3 + 5) dQ

TC =[tex]500(2/3)Q^{3/2} + 5Q + C[/tex]

where C is the constant of integration.

The demand function can be found by setting the marginal revenue equal to zero, since this occurs at the quantity where revenue is maximized

MR = 0

1000(-2Q + 4 + 4) = 0

-2000Q + 8000 = 0

Q = 4

Therefore, the demand function is Q = 4.

To find the quantity of widgets that Manufacturer B needs to manufacture for its total revenue to equal the total revenue of Manufacturer A, we set their respective total revenue functions equal to each other and solve for Q:

1000Q² + 4000Q + C = 1000(-2Q + 2 + 7)² + C'

where C and C' are constants of integration.

Simplifying, we get

1000Q² + 4000Q + C = 1000(9 - 28Q + 16Q²) + C'

1000Q² + 4000Q + C = 9000 - 28000Q + 16000Q² + C'

Since the two manufacturers are producing the same widgets, their marginal costs are the same, so we can set their total costs equal to each other

[tex]500(2/3)Q^{3/2} + 5Q + C = 500(2/3)(9 - 28Q + 16Q^2){3/2} + 5(9 - 28Q + 16Q^2) + C'[/tex]

Simplifying and solving for Q, we get

Q = 2.09 or Q = 5.13

Since the quantity of widgets produced must be non-zero, the competitor needs to manufacture approximately 2.09 widgets for its total revenue to equal the total revenue of Manufacturer A.

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The sample space of the possible outcomes of tossing a coin twice is:

{HH, HT, TH, TT}

Each result represents a possible combination of two coin tosses. The primary letter represents the result of the primary hurl, and the moment letter speaks to the result of his moment hurl.  

For example, "HH" means the coin landed heads twice in a row, and "HT" means the first toss was headed and the second was tails. 

In probability theory, sampling space refers to the set of all possible outcomes of an experiment or random event.

Before analyzing the probability of a particular event it is important to define the sampling space. This is because the sampling space determines the possible events and their probabilities. 

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There is proof that the equation; a²+b²+c²-ab-bc-ca = ((a - b)/√2)²  ((b - c)/√2)²  ((c - a)/√2)²

We can start by expanding the right-hand side of the equation:

((a - b)/√2)² ((b - c)/√2)² ((c - a)/√2)²

= (a² - 2ab + b²)/2 * (b² - 2bc + c²)/2 * (c² - 2ac + a²)/2

= (a⁴ - 2a³b + 3a²b² - 2ab³ + b⁴)/8 * (b⁴ - 2b³c + 3b²c² - 2bc³ + c⁴)/8 * (c⁴ - 2a³c + 3a²c² - 2ac³ + a⁴)/8

Multiplying out the terms, obtain:

(a⁴b⁴ - 2a³b⁵ + 3a²b⁶ - 2ab⁷ + b⁸)/512

(b⁴c⁴ - 2b³c⁵ + 3b²c⁶ - 2bc⁷ + c⁸)/512

(a⁴c⁴ - 2a³c⁵ + 3a²c⁶ - 2ac⁷ + c⁸)/512

Now, we can simplify the left-hand side of the equation by using the identity (a-b)² = a² - 2ab + b²:

a² + b² + c² - ab - bc - ca

= (a² - 2ab + b²) + (b² - 2bc + c²) + (c² - 2ca + a²)

= 2(a² - ab - ca) + 2(b² - bc - ab) + 2(c² - ca - bc)

= 2(a - b)(a - c) + 2(b - c)(b - a) + 2(c - a)(c - b)

= 2[(a - b)(c - a) + (b - c)(a - b) + (c - a)(b - c)]

= 2[(a² - ab - ca - ac + b² - bc + ba - cb + c² - ca - cb)]

= 2[(a² + b² + c² - ab - bc - ca)]

Substituting this back into the original equation:

a² + b² + c² - ab - bc - ca = ((a - b)/√2)² ((b - c)/√2)² ((c - a)/√2)²

Therefore, the equation is proved.

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Answer:

aprox. 5537.84

Step-by-step explanation:

The distance D between the two airplanes is increasing at a rate of approximately 119 knots.

This can be found by using the Pythagorean theorem to determine the distance between the two airplanes at the given distances from the airport: D² = 75² + (76/1.15)², where 1.15 is the conversion factor from knots to miles per hour.

Taking the derivative of both sides with respect to time, we get 2D(dD/dt) = 2(75)(0) + 2(76/1.15)(148/1.15) + 2D(0)(dD/dt).

Simplifying and solving for dD/dt, we get approximately 119 knots. This means that the distance between the two airplanes is increasing at a rate of 119 nautical miles per hour.

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Answer:

Step-by-step explanation:

I think the question is missing something. There is no x in the equation for h(x) so for this h(x)=0 always.

According to the recursive formula, the value of f(6) is 243.

A mathematical rule that takes one or more inputs, runs a series of operations on them, and produces a single output is known as a function. To put it another way, a function is a mapping between an input and an output in which there is only one output for each input.

Using the given recursive formula, we can find the value of f(6) by repeatedly applying the formula until we reach the desired value.

f(1) = 1 and

f(n)=3f(n−1)           (given in the question)

f(2) = 3f(1) = 3(1) = 3

f(3) = 3f(2) = 3(3) = 9

f(4) = 3f(3) = 3(9) = 27

f(5) = 3f(4) = 3(27) = 81

f(6) = 3f(5) = 3(81) = 243

Therefore, the value of f(6) is 243.

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Question-

If f(1) = 1 and f(n) = 3f(n−1) then find the value of  f(6).

The slope of the tangent at theta = Phi/2 is: dy/dx = (dy/dtheta) / (dx/dtheta) = (4/0) = undefined

To find the slope of the tangent to the curve r = -8 + 4 cos theta at the value Theta = Phi/2, we need to take the derivative of the polar equation with respect to theta, then plug in theta = Phi/2 and evaluate.

The derivative of r with respect to theta is given by:

dr/dtheta = -4 sin theta

At theta = Phi/2, we have:

dr/dtheta|theta=Phi/2 = -4 sin(Phi/2)

Now we need to find the slope of the tangent, which is given by:

dy/dx = (dy/dtheta) / (dx/dtheta)

To find dy/dtheta and dx/dtheta, we use the formulas:

x = r cos theta
y = r sin theta

dx/dtheta = -r sin theta + dr/dtheta cos theta
dy/dtheta = r cos theta + dr/dtheta sin theta

Plugging in r = -8 + 4 cos theta and dr/dtheta = -4 sin theta, we get:

dx/dtheta = -(4 cos theta)(-4 sin Phi/2) + (-8 + 4 cos theta)(-sin theta)
dy/dtheta = (4 cos theta)(cos Phi/2) + (-8 + 4 cos theta)(cos theta)

At theta = Phi/2, we have:

dx/dtheta|theta=Phi/2 = 0
dy/dtheta|theta=Phi/2 = 4

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The linearization of the function f(x,y) = √(x² + 16y²) is L(x,y) = 5 + 3(x-3)/5 + 16(x-1)/5 and the approximation of f(2.9.1.1) is 5.26.

The linearization of a function f(x,y) at the point (a,b) is given by,

L(x,y) = f(a,b) + fₓ(a,b)*(x - a) + fᵧ(a,b)*(y - b)

where fₓ and fᵧ are the partial derivatives of function 'f' with respect to 'x' and 'y' respectively.

Given the function is,

f(x,y) = √(x² + 16y²)

Now partially differentiate the above function firstly with respect to 'x' and then by 'y' we get,

fₓ(x,y) = (1/(2√(x² + 16y²)))*(2x) = x/√(x² + 16y²)

fᵧ(x,y) = (1/(2√(x² + 16y²)))*(32y) = 16y/√(x² + 16y²)

Given the point (a,b) = (3,1).

So substituting we get,

fₓ(3,1) = 3/√(9+16) = 3/√25 = 3/5

fᵧ(3,1) = 16/√(9+16) = 16/√25 = 16/5

f(3,1) = √(9 + 16) = √25 = 5

Then the linearization of the function f(x,y) = √(x² + 16y²) at the point (3,1) we get,

L(x,y) = f(a,b) + fₓ(a,b)*(x - a) + fᵧ(a,b)*(y - b)

L(x,y) = 5 + 3(x-3)/5 + 16(x-1)/5

Now approximating by this linear equation we get,

L(2.9, 1.1) = 5 + 3(2.9 - 3)/5 + 16(1.1 - 1)/5 = 5 - 0.3/5 + 1.6/5 = (25-0.3+1.6)/5 = 26.3/5 = 5.26

And

f(2.9, 1.1) = √((2.9)² + 16(1.1)²) = 5.27 (Rounding up to 2 decimal places)

So we can approximate using the linear function.

Hence, the linearization of the function is L(x,y) = 5 + 3(x-3)/5 + 16(x-1)/5 and the approximation of f(2.9.1.1) is 5.26.

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The volume of the solid is 99 cubic units.

In mathematics (particularly multivariable calculus), a volume integral (∭) refers to an integral over a 3-dimensional domain; that is, it is a special case of multiple integrals. Volume integrals are especially important in physics for many applications, for example, to calculate flux densities.

To find the volume of the solid, we need to integrate the cross-sectional area function over the length of the solid.

So, the volume (V) of the solid is given by:

[tex]V = \int_0^3 Al(x) dx[/tex]

where Al(x) = 2x + 34 is the cross-sectional area function.

Integrating Al(x) with respect to x, we get:

[tex]V = \int_0^3 (2x + 34) dx V = [x^2 + 34x]_0^3 \\V = (3^2 + 34(3)) - (0^2 + 34(0))[/tex]

V = 99 cubic units

Therefore, the volume of the solid is 99 cubic units.

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The variance is Var(Y) = [tex]σ^2[/tex] - μ/σ, which matches the second answer choice based on random variable.

EY=0, VarY=1

To see why, we can use the properties of expected value and variance.

First, we can find E(Y) as follows:

E(Y) = E(X - μ/σ)
    = E(X) - μ/σ       (since E(aX + b) = aE(X) + b for constants a and b)
    = μ - μ/σ          (since E(X) = μ)
    = 0                (simplifying)

Therefore, E(Y) = 0.

Next, we can find Var(Y) as follows:

Var(Y) = Var(X - μ/σ)
      = Var(X)             (since Var(aX + b) = a^2Var(X) for constants a and b)
      = [tex]σ^2[/tex]        (since Var(X) = σ^2)

However, we can also find Var(Y) by using the formula for variance in terms of expected value:

[tex]Var(Y) = E(Y^2) - [E(Y)]^2[/tex]

To find E(Y^2), we can use the fact that:

Y^2 = (X - μ/σ)^2
   = X^2 - 2X(μ/σ) + (μ/σ)^2

Therefore, we have:

[tex]E(Y^2) = E(X^2) - 2(μ/σ)E(X) + (μ/σ)^2\\       = σ^2 + μ^2/σ^2 - 2μ/σ * μ + μ^2/σ^2\\       = σ^2 + μ^2/σ^2 - 2μ^2/σ^2 + μ^2/σ^2\\       = σ^2 - μ^2/σ^2[/tex]

Using this, we can now find Var(Y):

Var(Y) = [tex]E(Y^2) - [E(Y)]^2\\       = (σ^2 - μ^2/σ^2) - 0^2\\       = σ^2 - μ/σ[/tex]

Therefore, Var(Y) = [tex]σ^2[/tex] - μ/σ, which matches the second answer choice.

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a) The instantaneous rate of change of the function h(x) when x = -1 is -32.

b) The derivative of the function H(x) at x = 2 is -17/(2√19).

To find the instantaneous rate of change of a function at a specific point, we need to take the derivative of the function at that point. In this case, we are given the function h(x) = (x² – 2x – 1) (3x³ + 2), and we need to find its derivative when x = -1.

So, applying the product rule, we get:

h'(x) = (x² – 2x – 1) x (9x²) + (3x³ + 2) x (2x – 2)

To find the instantaneous rate of change at x = -1, we substitute -1 for x in the above equation, and we get:

h'(-1) = (-1² – 2(-1) – 1) x (9(-1)²) + (3(-1)³ + 2) x (2(-1) – 2)

h'(-1) = (-4) x 9 + (-1) x (-4)

h'(-1) = -36 + 4

h'(-1) = -32

Moving on to the second question, we are asked to find the derivative of the function H(x) = √x³ – 5x² – 7x + 1 using the chain rule. The chain rule is used when we have a function within a function, or a composite function.

In this case, we can express H(x) as a composite function of g(f(x)), where f(x) = x³ – 5x² – 7x + 1, and g(x) = √x. So, H(x) = g(f(x)) = √(x³ – 5x² – 7x + 1).

To find the derivative of H(x), we use the chain rule, which states that the derivative of a composite function is equal to the derivative of the outer function evaluated at the inner function, multiplied by the derivative of the inner function.

So, applying the chain rule, we get:

H'(x) = g'(f(x)) x f'(x)

where g'(x) = 1/(2√x), and f'(x) = 3x² – 10x – 7.

Substituting these values, we get:

H'(x) = 1/(2√(x³ – 5x² – 7x + 1)) x (3x² – 10x – 7)

To find the derivative of H(x) at a particular point, we substitute that point for x in the above equation. For example, to find the derivative of H(x) at x = 2, we substitute 2 for x in the above equation, and we get:

H'(2) = 1/(2√(5(2)² - 7(2) + 1) x (3(2)² - 10(2) - 7)

H'(2) = 1/(2√(-19)) x (-17)

H'(2) = -17/(2√19)

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