show that y=x-2 is a tangent to the curve y=x^3-7x^2+17x-14 and
find the point of tangency
include explanation

The vertical distance between points A and B is 12 units.

The vertical distance between two points on a coordinate plane is found by subtracting the y-coordinates of the two points. In this case, point A has coordinates (8, -5) and point B has coordinates (8, 7).

To find the vertical distance between these two points, we subtract the y-coordinate of point A from the y-coordinate of point B.

Vertical distance = y-coordinate of point B - y-coordinate of point A

Vertical distance = 7 - (-5)
Vertical distance = 7 + 5
Vertical distance = 12

Therefore, the vertical distance between points A and B is 12 units.

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Given, a>0 and b be integers (b can be negative). We need to show that there is an integer k such that b + ka > 0.To prove this, we will use the well-ordering principle. Let S be the set of all positive integers that cannot be written in the form b + ka, where k is some integer. We need to prove that S is empty.

To do this, we assume that S is not empty. Then, by the well-ordering principle, S must have a smallest element, say n.This means that n cannot be written in the form b + ka, where k is some integer. Since a>0, we have a > -b/n. Thus, there exists an integer k such that k < -b/n < k + 1. Multiplying both sides of this inequality by n and adding b,

we get: bn/n - b < kna/n < bn/n + a - b/n,

which can be simplified to: b/n < kna/n - b/n < (b + a)/n.

Now, since k < -b/n + 1, we have k ≤ -b/n. Therefore, kna ≤ -ba/n.

Substituting this in the above inequality, we get: b/n < -ba/n - b/n < (b + a)/n,

which simplifies to: 1/n < (-b - a)/ba < 1/n + 1/b.

Both sides of this inequality are positive, since n is a positive integer and a > 0.

Thus, we have found a positive rational number between 1/n and 1/n + 1/b. This is a contradiction, since there are no positive rational numbers between 1/n and 1/n + 1/b.

Therefore, our assumption that S is not empty is false. Hence, S is empty.

Therefore, there exists an integer k such that b + ka > 0, for any positive value of a and any integer value of b.

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To determine if two vectors are orthogonal, we need to check if their dot product is equal to zero.

The dot product of two vectors A = (a₁, a₂, a₃, a₄) and B = (b₁, b₂, b₃, b₄) is given by:

A · B = a₁b₁ + a₂b₂ + a₃b₃ + a₄b₄

Let's calculate the dot product of the given vectors:

(1, 2, -1, 0) · (3, 1, 5, -10) = (1)(3) + (2)(1) + (-1)(5) + (0)(-10)

                            = 3 + 2 - 5 + 0

                            = 0

Since the dot product of the vectors is equal to zero, the vectors (1, 2, -1, 0) and (3, 1, 5, -10) are indeed orthogonal.

Therefore, the statement is true.

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Based on the given options, both 3,4,5,6 and 3,4,5,6i could be the complete list of roots for a fourth-degree polynomial. So option 1 and 2 are correct answer.

A fourth-degree polynomial function can have up to four distinct roots. The given options are:

Therefore, both options 1 and 2 could be the complete list of roots for a fourth-degree polynomial.

The question should be:

The polynomial function f(x) is a fourth degree polynomial. Which of the following could be the complete list of the roots of f(x)

1. 3,4,5,6

2. 3,4,5,6i

3. 3,4,4+i[tex]\sqrt{6}[/tex]

4. 3,4,5+i, 5+i, -5+i

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Bonang's monthly installment is R7 492,35 (rounded to the nearest cent).

In order to calculate the annual rate of depreciation using the reducing-balance method, we need to know the initial cost of the asset and the estimated salvage value.

However, we can calculate Bonang's monthly installment as follows:

Given that Bonang is granted a home loan of R650 000 to be repaid over a period of 15 years and the bank charges interest at 11,5% per annum compounded monthly.

In order to calculate Bonang's monthly installment,

we can use the formula for the present value of an annuity due, which is:

PMT = PV x (i / (1 - (1 + i)-n)) where:

PMT is the monthly installment

PV is the present value

i is the interest rate

n is the number of payments

If we assume that Bonang will repay the loan over 180 months (i.e. 15 years x 12 months),

then we can calculate the present value of the loan as follows:

PV = R650 000 = R650 000 x (1 + 0,115 / 12)-180 = R650 000 x 0,069380= R45 082,03

Therefore, the monthly installment that Bonang has to pay is:

PMT = R45 082,03 x (0,115 / 12) / (1 - (1 + 0,115 / 12)-180)= R7 492,35 (rounded to the nearest cent)

Therefore, Bonang's monthly installment is R7 492,35 (rounded to the nearest cent).

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A. The solution set is (-∞, -87/4). The solution set for the inequality is x < -87/4.

To solve the inequality −3(4x−1) < −2[5+8(x+5)], we will simplify the expression step by step and solve for x.

First, let's simplify both sides of the inequality:

−3(4x−1) < −2[5+8(x+5)]

−12x + 3 < −2[5+8x+40]

−12x + 3 < −2[45+8x]

Next, distribute the −2 inside the brackets:

−12x + 3 < −90 − 16x

Combine like terms:

−12x + 3 < −90 − 16x

Now, let's isolate the x term by adding 16x to both sides and subtracting 3 from both sides:

4x < −87

Finally, divide both sides of the inequality by 4 (since the coefficient of x is 4 and we want to isolate x):

x < -87/4

So, the solution set for the given inequality is x < -87/4.

In interval notation, this can be expressed as:

A. The solution set is (-∞, -87/4).

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If f is continuous and f(x+y) = f(x) + f(y) for all real numbers x and y, then there exists exactly one real

number a ∈ R, such that f(x) = ax, where a is a real number.

Given that f(x + y) = f(x) + f(y) for all x, y ∈ R.

To show that there exists exactly one real number a ∈ R and f is continuous such that for all rational numbers x, show that f(x) = ax

Let us assume that there exist two real numbers a, b ∈ R such that f(x) = ax and f(x) = bx.

Then, f(1) = a and f(1) = b.

Hence, a = b.So, the function is well-defined.

Now, we will show that f is continuous.

Let ε > 0 be given.

We need to show that there exists a δ > 0 such that for all x, y ∈ R, |x − y| < δ implies |f(x) − f(y)| < ε.

Now, we have |f(x) − f(y)| = |f(x − y)| = |a(x − y)| = |a||x − y|.

So, we can take δ = ε/|a|.

Hence, f is a continuous function.

Now, we will show that f(x) = ax for all rational numbers x.

Let p/q be a rational number.

Then, f(p/q) = f(1/q + 1/q + ... + 1/q) = f(1/q) + f(1/q) + ... + f(1/q) (q times) = a/q + a/q + ... + a/q (q times) = pa/q.

Hence, f(x) = ax for all rational numbers x.

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A cat weighing 5 lb and fed at a rate of 40 calories/lb/day should be fed a certain number of cups of commercial cat food at each meal if fed twice a day. We need to calculate this based on the given information that the cat food has 120 kcal/cup.

To determine the amount of cat food to be fed at each meal, we can follow these steps:

1. Calculate the total daily caloric intake for the cat:

  Total Calories = Weight (lb) * Calories per lb per day

                 = 5 lb * 40 calories/lb/day

                 = 200 calories/day

2. Determine the caloric content per meal:

  Since the cat is fed twice a day, divide the total daily caloric intake by 2:

  Caloric Content per Meal = Total Calories / Number of Meals per Day

                          = 200 calories/day / 2 meals

                          = 100 calories/meal

3. Find the number of cups needed per meal:

  Caloric Content per Meal = Calories per Cup * Cups per Meal

  Cups per Meal = Caloric Content per Meal / Calories per Cup

                = 100 calories/meal / 120 calories/cup

                ≈ 0.833 cups/meal

Therefore, the cat should be fed approximately 0.833 cups of commercial cat food at each meal if fed twice a day.

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The given congruence to show that 38592041 is divisible by 529.

To factor the number 38592041 using the given congruence 159238479574729≡529(mod38592041), we can utilize the concept of modular arithmetic and the fact that a≡b(modn) implies that a−b is divisible by n.

Let's go step by step:

1. Start with the congruence 159238479574729≡529(mod38592041).

2. Subtract 529 from both sides: 159238479574729−529≡529−529(mod38592041).

3. Simplify: 159238479574200≡0(mod38592041).

4. Since 159238479574200 is divisible by 38592041, we can conclude that 38592041 is a factor of

159238479574200

5. Divide 159238479574200 by 38592041 to obtain the quotient, which will be another factor of 38592041.

By following these steps, we have used the given congruence to show that 38592041 is divisible by 529. Further steps are needed to fully factorize 38592041, but without additional information or using more advanced factorization techniques, it may be challenging to find all the prime factors.

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The critical value for the goodness-of-fit test needed to test the claim is approximately 11.07.

To determine the critical value for the goodness-of-fit test, we need to use the chi-square distribution with (k - 1) degrees of freedom, where k is the number of categories or color options in this case.

In this scenario, there are 6 color categories, so k = 6.

To find the critical value, we need to consider the significance level, which is given as 0.05.

Since we want to test the claim, we perform a goodness-of-fit test to compare the observed frequencies with the expected frequencies based on the claimed distribution. The chi-square test statistic measures the difference between the observed and expected frequencies.

The critical value is the value in the chi-square distribution that corresponds to the chosen significance level and the degrees of freedom.

Using a chi-square distribution table or statistical software, we can find the critical value for the given degrees of freedom and significance level. For a chi-square distribution with 5 degrees of freedom and a significance level of 0.05, the critical value is approximately 11.07.

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a. The surface area of the band cut from the paraboloid is approximately 314.16 square units.

b.  We have:

S = ∫[-π/4,π/4]∫[-π/4,π/4] √(tan^2 θ/2 + 1) sec^2 θ/2 dθ dφ

a) To find the surface area of the band cut from the paraboloid x^2 + y^2 - z = 0 by planes z = 2 and z = 6, we can use the formula for the surface area of a parametric surface:

S = ∫∫ ||r_u × r_v|| du dv

where r(u,v) is the vector-valued function that describes the surface, and r_u and r_v are the partial derivatives of r with respect to u and v.

In this case, we can parameterize the surface as:

r(u, v) = (u cos v, u sin v, u^2)

where 0 ≤ u ≤ 2 and 0 ≤ v ≤ 2π.

To find the partial derivatives, we have:

r_u = (cos v, sin v, 2u)

r_v = (-u sin v, u cos v, 0)

Then, we can calculate the cross product:

r_u × r_v = (2u^2 cos v, 2u^2 sin v, -u)

and its magnitude:

||r_u × r_v|| = √(4u^4 + u^2)

Therefore, the surface area of the band is:

S = ∫∫ √(4u^4 + u^2) du dv

We can evaluate this integral using polar coordinates:

S = ∫[0,2π]∫[2,6] √(4u^4 + u^2) du dv

= 2π ∫[2,6] u √(4u^2 + 1) du

This integral can be evaluated using the substitution u^2 = (1/4)(4u^2 + 1) - 1/4, which gives:

S = 2π ∫[1/2,25/2] (√(u^2 + 1/4))^3 du

= π/2 [((25/2)^2 + 1/4)^{3/2} - ((1/2)^2 + 1/4)^{3/2}]

≈ 314.16

Therefore, the surface area of the band cut from the paraboloid is approximately 314.16 square units.

b) To find the surface area of the upper portion of the cylinder x^2 + z^2 = 1 that lies between the planes x = ±1/2 and y = ±1/2, we can also use the formula for the surface area of a parametric surface:

S = ∫∫ ||r_u × r_v|| du dv

where r(u,v) is the vector-valued function that describes the surface, and r_u and r_v are the partial derivatives of r with respect to u and v.

In this case, we can parameterize the surface as:

r(u, v) = (x(u, v), y(u, v), z(u, v))

where x(u,v) = u, y(u,v) = v, and z(u,v) = √(1 - u^2).

Then, we can find the partial derivatives:

r_u = (1, 0, -u/√(1 - u^2))

r_v = (0, 1, 0)

And calculate the cross product:

r_u × r_v = (u/√(1 - u^2), 0, 1)

The magnitude of this cross product is:

||r_u × r_v|| = √(u^2/(1 - u^2) + 1)

Therefore, the surface area of the upper portion of the cylinder is:

S = ∫∫ √(u^2/(1 - u^2) + 1) du dv

We can evaluate the inner integral using trig substitution:

u = tan θ/2, du = (1/2) sec^2 θ/2 dθ

Then, the limits of integration become θ = atan(-1/2) to θ = atan(1/2), since the curve u = ±1/2 corresponds to the planes x = ±1/2.

Therefore, we have:

S = ∫[-π/4,π/4]∫[-π/4,π/4] √(tan^2 θ/2 + 1) sec^2 θ/2 dθ dφ

This integral can be evaluated using a combination of trig substitutions and algebraic manipulations, but it does not have a closed form solution in terms of elementary functions. We can approximate the value numerically using a numerical integration method such as Simpson's rule or Monte Carlo integration.

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a. CCA's current liabilities are approximately $21,000. b. CCA's inventory is approximately $10,500.

To find the current liabilities and inventory of Credit Card of America (CCA), we can use the current ratio and quick ratio along with the given information.

(a) Current liabilities:

The current ratio is calculated as the ratio of current assets to current liabilities. In this case, the current ratio is 3.5, which means that for every dollar of current liabilities, CCA has $3.5 of current assets.

Let's assume the current liabilities as 'x'. We can set up the following equation based on the given information:

3.5 = $73,500 / x

Solving for 'x', we find:

x = $73,500 / 3.5 ≈ $21,000

Therefore, CCA's current liabilities are approximately $21,000.

(b) Inventory:

The quick ratio is calculated as the ratio of current assets minus inventory to current liabilities. In this case, the quick ratio is 3.0, which means that for every dollar of current liabilities, CCA has $3.0 of current assets excluding inventory.

Using the given information, we can set up the following equation:

3.0 = ($73,500 - Inventory) / $21,000

Solving for 'Inventory', we find:

Inventory = $73,500 - (3.0 * $21,000)

Inventory ≈ $73,500 - $63,000

Inventory ≈ $10,500

Therefore, CCA's inventory is approximately $10,500.

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The solutions to the equation x² + 3x - 28 = 0 are x = -7 and x = 4.

1. Solve x^(1/4) = 3x^(1/8):

To solve this equation, we can raise both sides to the power of 8 to eliminate the fractional exponent:

(x^(1/4))⁸ = (3x^(1/8))⁸

x² = 3⁸ * x

x² = 6561x

Now, we'll rearrange the equation and solve for x:

x² - 6561x = 0

x(x - 6561) = 0

From the zero-factor property, we set each factor equal to zero and solve for x:

x = 0 or x - 6561 = 0

x = 0 or x = 6561

So the solutions to the equation x^(1/4) = 3x^(1/8) are x = 0 and x = 6561.

2. Solve |4x - 8| = |2x + 8|:

To solve this equation, we'll consider two cases based on the absolute value.

Case 1: 4x - 8 = 2x + 8

Solving for x:

4x - 2x = 8 + 8

2x = 16

x = 8

Case 2: 4x - 8 = -(2x + 8)

Solving for x:

4x - 8 = -2x - 8

4x + 2x = -8 + 8

6x = 0

x = 0

Therefore, the solutions to the equation |4x - 8| = |2x + 8| are x = 0 and x = 8.

3. Solve using the zero-factor property x² + 3x - 28 = 0:

To solve this equation, we can factor it:

(x + 7)(x - 4) = 0

Setting each factor equal to zero and solving for x:

x + 7 = 0 or x - 4 = 0

x = -7 or x = 4

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Lombardi House won 8 soccer games and 4 football games, found by following system of equations.

Let's assume Lombardi House won x soccer games and y football games. From the given information, we have the following system of equations:

x + y = 12 (total number of wins)

2x + 4y = 32 (total points earned)

Simplifying the first equation, we have x = 12 - y. Substituting this into the second equation, we get 2(12 - y) + 4y = 32. Solving this equation, we find y = 4. Substituting the value of y back into the first equation, we get x = 8.

Therefore, Lombardi House won 8 soccer games and 4 football games.

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The cubic polynomial interpolation function for the given data using different methods is as follows:

Polynomial Coefficient Interpolation Method: p(x) = -1x³ + 4x² - 2x - 8

Newton Interpolation Method: p(x) = -8 + 6(x+1) - 4(x+1)(x-0) + 2(x+1)(x-0)(x-2)

Lagrange Interpolation Method: p(x) = -4((x-0)(x-2)(x-3))/((-1-0)(-1-2)(-1-3)) - 8((x+1)(x-2)(x-3))/((0-(-1))(0-2)(0-3)) + 2((x+1)(x-0)(x-3))/((2-(-1))(2-0)(2-3)) + 28((x+1)(x-0)(x-2))/((3-(-1))(3-0)(3-2))

Polynomial Coefficient Interpolation Method: In this method, we find the coefficients of the polynomial directly. By substituting the given data points into the polynomial equation, we can solve for the coefficients. Using this method, the cubic polynomial interpolation function is p(x) = -1x³ + 4x² - 2x - 8.

Newton Interpolation Method: This method involves constructing a divided difference table to determine the coefficients of the polynomial. The divided differences are calculated based on the given data points. Using this method, the cubic polynomial interpolation function is p(x) = -8 + 6(x+1) - 4(x+1)(x-0) + 2(x+1)(x-0)(x-2).

Lagrange Interpolation Method: This method uses the Lagrange basis polynomials to construct the interpolation function. Each basis polynomial is multiplied by its corresponding function value and summed to obtain the final interpolation function. The Lagrange basis polynomials are calculated based on the given data points. Using this method, the cubic polynomial interpolation function is p(x) = -4((x-0)(x-2)(x-3))/((-1-0)(-1-2)(-1-3)) - 8((x+1)(x-2)(x-3))/((0-(-1))(0-2)(0-3)) + 2((x+1)(x-0)(x-3))/((2-(-1))(2-0)(2-3)) + 28((x+1)(x-0)(x-2))/((3-(-1))(3-0)(3-2)).

These interpolation methods provide different ways to approximate a function based on a limited set of data points. The resulting polynomial functions can be used to estimate function values at intermediate points within the given data range.

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Here are the solutions to the given problems :

1. 0.12 + 143 = 143.12 (The answer is 143.12)

2. 0.00843 + 0.0144 = 0.02283 (The answer is 0.02283)

3. 1.2 × 10^(-3) + 27 = 27.0012 (The answer is 27.0012)

4. 1.2 × 10^(-3) + 1.2 × 10^(-4) = 0.00132 (The answer is 0.00132)

5. 2473.86 + 123.4 = 2597.26 (The answer is 2597.26)

Hence, we can say that these are the answers of the given problems.

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V= (3,4,0) can be expressed as a linear combination of the basis vectors {(1, 0, 0), (1, 1, 0), (1, 1, 1)} as v = (-1, 0, 0) + 4 * (1, 1, 0).

To express vector v = (3, 4, 0) as a linear combination of the basis vectors {(1, 0, 0), (1, 1, 0), (1, 1, 1)}, we need to find the coefficients that satisfy the equation:

v = c₁ * (1, 0, 0) + c₂ * (1, 1, 0) + c₃ * (1, 1, 1),

where c₁, c₂, and c₃ are the coefficients we want to determine.

Setting up the equation for each component:

3 = c₁ * 1 + c₂ * 1 + c₃ * 1,

4 = c₂ * 1 + c₃ * 1,

0 = c₃ * 1.

From the third equation, we can directly see that c₃ = 0. Substituting this value into the second equation, we have:

4 = c₂ * 1 + 0,

4 = c₂.

Now, substituting c₃ = 0 and c₂ = 4 into the first equation, we get:

3 = c₁ * 1 + 4 * 1 + 0,

3 = c₁ + 4,

c₁ = 3 - 4,

c₁ = -1.

Therefore, the linear combination of the basis vectors that expresses v is:

v = -1 * (1, 0, 0) + 4 * (1, 1, 0) + 0 * (1, 1, 1).

So, v = (-1, 0, 0) + (4, 4, 0) + (0, 0, 0).

v = (3, 4, 0).

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Therefore, the coops will extend straight out from the barn approximately 23.12 feet.

To find how far the coops will extend straight out from the barn, we need to determine the size of each coop and divide it by 2.

The area of the barn is 26 feet * 36 feet = 936 square feet.

To have the coops cover half of this area, each coop should have an area of 936 square feet / 7 coops:

= 133.71 square feet.

Since the coops are rectangular, we can find the width and depth of each coop by taking the square root of the area:

Width of each coop = √(133.71 square feet)

≈ 11.56 feet

Depth of each coop = √(133.71 square feet)

≈ 11.56 feet

Since the coops are placed along two sides of the barn, the total extension will be twice the width of each coop:

Total extension = 2 * 11.56 feet

= 23.12 feet.

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The functions sin(x) and cos(x) are periodic functions that represent the sine and cosine of an angle, respectively. When plotted on the interval 0≤x≤2π, the graph of sin(x) starts at the origin, reaches its maximum at π/2, returns to the origin at π, reaches its minimum at 3π/2, and returns to the origin at 2π. The graph of cos(x) starts at its maximum value of 1, reaches its minimum at π, returns to 1 at 2π, and continues in a repeating pattern.

The function sin(x) represents the ratio of the length of the side opposite to an angle in a right triangle to the length of the hypotenuse. When plotted on the interval 0≤x≤2π, the graph of sin(x) starts at the origin (0,0) and oscillates between -1 and 1 as x increases. It reaches its maximum value of 1 at π/2, returns to the origin at π, reaches its minimum value of -1 at 3π/2, and returns to the origin at 2π.

The function cos(x) represents the ratio of the length of the side adjacent to an angle in a right triangle to the length of the hypotenuse. When plotted on the interval 0≤x≤2π, the graph of cos(x) starts at its maximum value of 1 and decreases as x increases. It reaches its minimum value of -1 at π, returns to 1 at 2π, and continues in a repeating pattern.

Both sin(x) and cos(x) are periodic functions with a period of 2π, meaning that their graphs repeat after every 2π.

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The rectangular coordinates of the point (4,6π/7,7) are (4cos(6π/7), 4sin(6π/7), 7).

Now, For the first problem, we need to convert the given rectangular coordinates (0,3,5) into cylindrical coordinates (r,θ,z).

We know that:

r = √(x² + y²)

θ = tan⁻¹(y/x)

z = z

Substituting the given coordinates, we get:

r = √(0² + 3²) = 3

θ = tan⁻¹(3/0) = π/2

(since x = 0)

z = 5

Therefore, the cylindrical coordinates of the point (0,3,5) are (3,π/2,5).

For the second problem, we need to convert the given cylindrical coordinates (4, 6π/7, 7) into rectangular coordinates (x,y,z).

We know that:

x = r cos(θ)

y = r sin(θ)

z = z

Substituting the given coordinates, we get:

x = 4 cos(6π/7)

y = 4 sin(6π/7)

z = 7

Therefore, the rectangular coordinates of the point (4,6π/7,7) are (4cos(6π/7), 4sin(6π/7), 7).

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The given equation is `f(x) = 9cos²(x) - 18sin(x), 0 ≤ x ≤ 2π`.We can find the maximum value of `f(x)` between `0` and `2π` by using differentiation.

We get,`f′(x)

= -18cos(x)sin(x) - 18cos(x)sin(x)

= -36cos(x)sin(x)`We equate `f′(x)

= 0` to find the critical points.`-36cos(x)sin(x)

= 0``=> cos(x)

= 0 or sin(x)

= 0``=> x = nπ + π/2 or nπ`where `n` is an integer. To determine the nature of the critical points, we use the second derivative test.`f″(x)

= -36(sin²(x) - cos²(x))``

=> f″(nπ) = -36`

`=> f″(nπ + π/2)

= 36`For `x

= nπ`, `f(x)` attains its maximum value since `f″(x) < 0`. For `x

= nπ + π/2`, `f(x)` attains its minimum value since `f″(x) > 0`.Therefore, the maximum value of `f(x)` between `0` and `2π` is `f(nπ)

= 9cos²(nπ) - 18sin(nπ)

= 9`. The minimum value of `f(x)` between `0` and `2π` is `f(nπ + π/2)

= 9cos²(nπ + π/2) - 18sin(nπ + π/2)

= -18`.Thus, the maximum value of the function `f(x)

= 9cos²(x) - 18sin(x)` on the interval `[0, 2π]` is `9` and the minimum value is `-18`.

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Given information is as follows:Mr. Silverstone invested some amount of money in 3 different investment products. We need to determine the linear equation that represents the interest earned from each investment if the total was $950.

To solve this problem, we will write the equation representing the sum of all interest as per the given interest rates for all three investments.

Let the amount invested in annuity with 4% interest be 'a', the amount invested in annuity with 6% interest be 'b' and the amount invested in bond with 5% interest be 'c'. The linear equation that shows interest earned from each investment if the total was $950 is given by : 0.04a + 0.06b + 0.05c = $950

We need to determine the linear equation that represents the interest earned from each investment if the total was $950.Let the amount invested in annuity with 4% interest be 'a', the amount invested in annuity with 6% interest be 'b' and the amount invested in bond with 5% interest be 'c'. The total interest earned from all the investments is given as $950. To form an equation based on given information, we need to sum up the interest earned from all the investments as per the given interest rates.

The linear equation that shows interest earned from each investment if the total was $950 is given by: 0.04a + 0.06b + 0.05c = $950
The linear equation that represents the interest earned from each investment if the total was $950 is 0.04a + 0.06b + 0.05c = $950.

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The algebraic expression in u for CSC(COS⁻¹(u)) is 1/√(1 - u²).

Here, we have,

To write the trigonometric expression CSC(COS⁻¹(u)) as an algebraic expression in u,

we can use the reciprocal identities of trigonometric functions.

CSC(theta) is the reciprocal of SIN(theta), so CSC(COS⁻¹(u)) can be rewritten as 1/SIN(COS⁻¹(u)).

Now, let's use the definition of inverse trigonometric functions to rewrite the expression:

COS⁻¹(u) = theta

COS(theta) = u

From the right triangle definition of cosine, we have:

Adjacent side / Hypotenuse = u

Adjacent side = u * Hypotenuse

Now, consider the right triangle formed by the angle theta and the sides adjacent, opposite, and hypotenuse.

Since COS(theta) = u, we have:

Adjacent side = u

Hypotenuse = 1

Using the Pythagorean theorem, we can find the opposite side:

Opposite side = √(Hypotenuse² - Adjacent side²)

Opposite side = √(1² - u²)

Opposite side =√(1 - u²)

Now, we can rewrite the expression CSC(COS^(-1)(u)) as:

CSC(COS⁻¹(u)) = 1/SIN(COS⁻¹(u))

CSC(COS⁻¹)(u)) = 1/(Opposite side)

CSC(COS⁻¹)(u)) = 1/√(1 - u²)

Therefore, the algebraic expression in u for CSC(COS⁻¹(u)) is 1/√(1 - u²).

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An approximation for the area f(x)=3eˣ. is 489.2158.

Given:

f(x)=3eˣ.

Here, a = 3 b = 5 and n = 4.

h = (b - a) / n =(5 - 3)/4 = 0.5.

Now, [tex]f (3.5) = 3e^{3.5}.[/tex]

[tex]f(4) = 3e^{4}[/tex]

[tex]f(4.5) = 3e^{4.5}[/tex]

[tex]f(5) = 3e^5.[/tex]

Area = h [f(3.5) + f(4) + f(4.5) + f(5)]

[tex]= 0.5 [3e^{3.5} + e^4 + e^{4.5} + e^5][/tex]

[tex]= 1.5 (e^{3.5} + e^4 + e^{4.5} + e^5)[/tex]

Area = 489.2158.

Therefore, an approximation for the area f(x)=3eˣ. is 489.2158.

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(a) The vector u such that it has the same direction as v and one-half its length is u = (4, 4, 5)

(b) The vector u such that it has the direction opposite that of v and one-fourth its length is u = (-2, -2, -2.5)

(c) The vector u such that it has the direction opposite that of v and twice its length is u = (-16, -16, -20)

To obtain vector u with specific conditions, we can manipulate the components of vector v accordingly:

(a) The vector u has the same direction as v and one-half its length.

To achieve this, we need to scale down the magnitude of vector v by multiplying it by 1/2 while keeping the same direction. Therefore:

u = (1/2) * v

  = (1/2) * (8, 8, 10)

  = (4, 4, 5)

So, vector u has the same direction as v and one-half its length.

(b) The vector u has the direction opposite that of v and one-fourth its length.

To obtain a vector with the opposite direction, we change the sign of each component of vector v. Then, we scale down its magnitude by multiplying it by 1/4. Thus:

u = (-1/4) * v

  = (-1/4) * (8, 8, 10)

  = (-2, -2, -2.5)

Therefore, vector u has the direction opposite to that of v and one-fourth its length.

(c) The vector u has the direction opposite that of v and twice its length.

We change the sign of each component of vector v to obtain a vector with the opposite direction. Then, we scale up its magnitude by multiplying it by 2. Hence:

u = 2 * (-v)

  = 2 * (-1) * v

  = -2 * v

  = -2 * (8, 8, 10)

  = (-16, -16, -20)

Thus, vector u has the direction opposite to that of v and twice its length.

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Identities are given as cos(a + B) = cosa cosß-sina sinß, cos²p+ sin²p=1,(a) cos(a+B) =cosa cosß + sina sinß (b)  (27 - (a− p)) = cos((2-a) + B)=cos(2-a + B) (c) sin(7/12)cos(7/12)= (√6+√2)/4

Part (a)To prove the identity for cos(a-B) = cosa cosß+ sina sinß, we start from the identity

cos(a+B) = cosa cosß-sina sinß, and replace ß with -ß,

thus we getcos(a-B) = cosa cos(-ß)-sina sin(-ß) = cosa cosß + sina sinß

To prove the identity for sin(a-B)=sina-cosß-cosa sinß, we first replace ß with -ß in the identity sin(a+B) = sina cosß+cosa sinß,

thus we get sin(a-B) = sin(a+(-B))=sin a cos(-ß) + cos a sin(-ß)=-sin a cosß+cos a sinß=sina-cosß-cosa sinß

Part (b)To prove that as (27 - (a− p)) = cos((2-a) + B),

we use the identity cos²p+sin²p=1cos(27-(a-p)) = cos a sin p + sin a cos p= cos a cos 2-a + sin a sin 2-a = cos(2-a + B)

Part (c)Given cos²a= 1+cos2a 2 , sin² a= 1-cos2a 2We are required to calculate cos(7/12) and sin(7/12)cos(7/12) = cos(π/2 - π/12)=sin (π/12) = √[(1-cos(π/6))/2]

= √[(1-√3/2)/2]

= (2-√3)/2sin (7/12)

=sin(π/4 + π/6)

=sin(π/4)cos(π/6) + cos(π/4) sin(π/6)

= √2/2*√3/2 + √2/2*√1/2

= (√6+√2)/4

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The answer is y=(x+4)3−5. The equation defines the graph of y=x3 after it is shifted vertically 5 units down and horizontally 4 units left.Final Answer: y=(x+4)3−5.

The original equation of the graph is y = x^3. We need to determine the equation of the graph after it is shifted five units down and four units left. When a graph is moved, it's called a shift.The shifts on a graph can be vertical (up or down) or horizontal (left or right).When a graph is moved vertically or horizontally, the equation of the graph changes. The changes in the equation depend on the number of units moved.

To shift a graph horizontally, you add or subtract the number of units moved to x. For example, if the graph is shifted 4 units left, we subtract 4 from x.To shift a graph vertically, you add or subtract the number of units moved to y. For example, if the graph is shifted 5 units down, we subtract 5 from y.To shift a graph five units down and four units left, we substitute x+4 for x and y-5 for y in the original equation of the graph y = x^3.y = (x+4)^3 - 5Therefore, the answer is y=(x+4)3−5. The equation defines the graph of y=x3 after it is shifted vertically 5 units down and horizontally 4 units left.Final Answer: y=(x+4)3−5.

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The true statements regarding the function f(x) = b∗ are that the range of the exponential function is f(x) > 0 or y > 0, and the domain of the exponential function is x > 0.

The range of the exponential function f(x) = b∗ is indeed f(x) > 0 or y > 0. Since the base b is positive, raising it to any power will always result in a positive value.

Therefore, the range of the function is all positive real numbers.

Similarly, the domain of the exponential function f(x) = b∗ is x > 0. Exponential functions are defined for positive values of x, as raising a positive base to any power remains valid.

Consequently, the domain of f(x) is all positive real numbers.

However, the other statements provided are not true for the given function. The horizontal asymptote of the function f(x) = b∗ is not the line y = 0.

It does not have a horizontal asymptote since the function's value continues to grow or decay exponentially as x approaches positive or negative infinity.

Additionally, the horizontal asymptote is not the line x = 0. The function does not have a vertical asymptote because it is defined for all positive values of x.

Lastly, the horizontal asymptote is not the point (0, b). As mentioned earlier, the function does not have a horizontal asymptote.

In conclusion, the true statements regarding the function f(x) = b∗ are that the range of the exponential function is f(x) > 0 or y > 0, and the domain of the exponential function is x > 0.

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The approximate average rate at which the area decreases as the rectangle's length goes from 13 cm to 16 cm is equal to the width (w) of the rectangle.

To determine the approximate average rate at which the area decreases as the rectangle's length goes from 13 cm to 16 cm, we need to calculate the change in area and divide it by the change in length.

Let's denote the length of the rectangle as L (in cm) and the corresponding area as A (in square cm).

Given that the initial length is 13 cm and the final length is 16 cm, we can calculate the change in length as follows:

Change in length = Final length - Initial length

= 16 cm - 13 cm

= 3 cm

Now, let's consider the formula for the area of a rectangle:

A = Length × Width

Since we are interested in the rate at which the area decreases, we can consider the width as a constant. Let's assume the width is w cm.

The initial area (A1) when the length is 13 cm is:

A1 = 13 cm × w

Similarly, the final area (A2) when the length is 16 cm is:

A2 = 16 cm × w

The change in area can be calculated as:

Change in area = A2 - A1

= (16 cm × w) - (13 cm × w)

= 3 cm × w

Finally, to find the approximate average rate at which the area decreases, we divide the change in area by the change in length:

Average rate of area decrease = Change in area / Change in length

= (3 cm × w) / 3 cm

= w

Therefore, the approximate average rate at which the area decreases as the rectangle's length goes from 13 cm to 16 cm is equal to the width (w) of the rectangle.

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The points of intersection of the given parabolic equations y = x² and y = x² + 18x are (0, 0).

Thus, the solution is obtained.

The given parabolic equations are:

y = x² ..............(1)y = x² + 18x ........(2)

The points of intersection can be found by substituting (1) in (2).

Then, [tex]x² = x² + 18x[/tex]

⇒ 18x = 0

⇒ x = 0

Since x = 0,

substitute this value in (1),y = (0)² = 0

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