A ball is dropped from a state of rest at time t=0.
The distance traveled after t seconds is s(t)=16t²ft.
How far does the ball travel during the time interval [6,6.5] ?
Compute the average velocity over [6,6.5]

a) The derivative of the given function is 7 sec7(x/7) tan(x/7) - 5 sec5(x/5) tan(x/5).

b)The derivative of the given function is 15x(x + 5)−5/2/4(x + 5)3/2.

a) y = sec^7(x/7) - sec^5(x/5)

The given function is: y = sec7(x/7) - sec5(x/5)

Now, we are to find the derivative of this function.

To find the derivative of the given function, we apply the chain rule as follows:

dy/dx = 7 sec7(x/7) tan(x/7) - 5 sec5(x/5) tan(x/5)

Thus, the derivative of the given function is 7 sec7(x/7) tan(x/7) - 5 sec5(x/5) tan(x/5).

b) y = √(e^2x + e^-2x)

The given function is: y = √(e2x + e−2x)

Now, we are to find the derivative of this function.

To find the derivative of the given function, we apply the chain rule as follows:

dy/dx = (1/2) (e2x + e−2x)−1/2 d/dx (e2x + e−2x)

On differentiating the function e2x + e−2x with respect to x, we get:

d/dx (e2x + e−2x) = 2e2x − 2e−2x

Now, substituting this value back in the original equation, we have:

dy/dx = (1/2) (e2x + e−2x)−1/2 (2e2x − 2e−2x)

Simplifying this, we get: dy/dx = (e2x + e−2x)−1/2 (e2x − e−2x)

Therefore, the derivative of the given function is

(e2x + e−2x)−1/2 (e2x − e−2x).c) g(x) = ln(x3√(4x + 1))

The given function is: g(x) = ln(x3√(4x + 1))

Now, we are to find the derivative of this function.

To find the derivative of the given function, we apply the chain rule as follows:

dg/dx = 1/(x3√(4x + 1)) d/dx (x3√(4x + 1))

On differentiating the function x3√(4x + 1) with respect to x, we get:

d/dx (x3√(4x + 1)) = (3x2√(4x + 1) + x3/2(4x + 1)−1/2 (4))

Simplifying this, we get:

d/dx (x3√(4x + 1)) = (3x2√(4x + 1) + 2x3/2(4x + 1)−1/2)

Therefore, the derivative of the given function is:

dg/dx = 1/(x3√(4x + 1)) (3x2√(4x + 1) + 2x3/2(4x + 1)−1/2)

So, the required derivative is: dg/dx = (3√(4x + 1) + 2x/√(4x + 1))/x2√(4x + 1).d) f(x) = (x/√(x + 5))3

The given function is: f(x) = (x/√(x + 5))3

Now, we are to find the derivative of this function.

To find the derivative of the given function, we apply the chain rule as follows:

df/dx = 3(x/√(x + 5))2 d/dx (x/√(x + 5))

On differentiating the function x/√(x + 5) with respect to x, we get: d/dx (x/√(x + 5)) = (5/2)(x + 5)−3/2

Simplifying this, we get: d/dx (x/√(x + 5)) = (5/2)/(x + 5)3/2

Therefore, the derivative of the given function is: df/dx = 3(x/√(x + 5))2 (5/2)/(x + 5)3/2

So, the required derivative is: df/dx = 15x(x + 5)−5/2/4(x + 5)3/2.

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1) 6x² +3Vx+ x 1 2 5 x= 2x³ + 2√x² + (2/3)x^(3/2) + C  (2)  The integral of sin(x) - cos(2x) = -cos(x) - (1/2)sin(2x) + C.

where C is the constant of integration

(i) To integrate the function 6x² + 3√x + x^(1/2) with respect to x, we can apply the power rule of integration. The power rule states that the integral of x^n with respect to x is (x^(n+1))/(n+1), where n is any real number except -1.

Let's integrate each term separately:

∫(6x² + 3√x + x^(1/2)) dx

= 6∫x² dx + 3∫√x dx + ∫x^(1/2) dx

= 6(x^(2+1))/(2+1) + 3(2/3)(x^(1/2+1))/(1/2+1) + (2/3)(x^(1/2+1))/(1/2+1) + C

= 2x³ + 2√x² + (2/3)x^(3/2) + C

where C is the constant of integration

(ii) sin(x) - cos(2x)The integral of sin(x) - cos(2x) is;∫(sin(x) - cos(2x)) dxWe know that the integral of sin(x) is -cos(x)Therefore, the integral of sin(x) - cos(2x) = -cos(x) - (1/2)sin(2x) + C.

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The average cost function C(x) can be found by integrating the marginal average cost function C'(x). Using the given derivative C'(x) = -1,700/x^2, we integrate with respect to x to find C(x):

C(x) = ∫(-1,700/x^2) dx = 1,700/x + C

To determine the constant of integration C, we use the given information that C(100) = 28:

28 = 1,700/100 + C

28 = 17 + C

C = 28 - 17

C = 11

Thus, the average cost function is C(x) = 1,700/x + 11.

To find the cost function C(x), we integrate the average cost function C(x) with respect to x:

C(x) = ∫(1,700/x + 11) dx = 1,700 ln|x| + 11x + K

The constant of integration K represents the fixed costs. To determine the value of K, we can use the given information that C(100) = 28:

28 = 1,700 ln|100| + 11(100) + K

28 = 1,700 ln(100) + 1,100 + K

28 = 1,700(4.605) + 1,100 + K

28 = 7,819.5 + 1,100 + K

K = 28 - 7,819.5 - 1,100

K ≈ -8,892.5

Therefore, the cost function is C(x) = 1,700 ln|x| + 11x - 8,892.5, and the fixed costs are approximately $8,892.50.

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The room air-conditioning system is a closed-loop control system.

A closed-loop control system is a system that continuously monitors and adjusts its output based on a desired reference value. In the case of a room air-conditioning system, it typically includes sensors to measure the temperature of the room and compare it to a setpoint.

The system then adjusts the cooling or heating output to maintain the desired temperature. This feedback mechanism makes it a closed-loop control system.

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The number of cars Xenophobic Car Palace will sell in Charleston and Savannah is option D) Qc = 100, Qs = 100.

To determine the number of cars XCP will sell in Charleston (Qc) and Savannah (Qs), we need to find the quantities that maximize XCP's profit. XCP engages in price discrimination between the two cities, meaning it can charge different prices in Charleston (Pc) and Savannah (Ps) based on their respective demand curves.

Given the demand equations Qc = 1,000 - 2Pc and Qs = 500 - Ps, we can find the profit-maximizing quantities by equating marginal revenue (MR) to marginal cost (MC) for each city. MR is equal to the derivative of the demand equation with respect to quantity (Q), and MC is equal to the derivative of total cost (TC) with respect to quantity.

For Charleston, MRc = 1,000 - 4Qc, and MC = 100. Equating MRc and MC, we have:

1,000 - 4Qc = 100.

Solving for Qc, we find Qc = 100.

For Savannah, MRs = 500 - 2Qs, and MC = 100. Equating MRs and MC, we have:

500 - 2Qs = 100.

Solving for Qs, we find Qs = 100.

Therefore, the correct answer is D) Qc = 100, Qs = 100. XCP will sell 100 cars in both Charleston and Savannah to maximize its profit under price discrimination.

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The relative error in computing the volume of the sphere, based on the given error in the radius, is 0.6%.

To calculate the relative error in computing the volume of a sphere, we need to consider the relative error in the radius and then propagate it through the volume formula.

Given that the radius of the sphere is 10 cm with a possible error of 0.02 cm, we can determine the relative error in the radius as follows:

Relative error in the radius = (Error in the radius) / (Actual radius)

= (0.02 cm) / (10 cm)

= 0.002

The relative error is 0.002 or 0.2%.

Now, let's calculate the relative error in the volume of the sphere using the formula for the volume of a sphere: V = (4/3)πr³.

Relative error in the volume = (Relative error in the radius) * (Exponent of the radius in the volume formula)

= 0.002 * 3

= 0.006

The relative error in the volume is 0.006 or 0.6%.

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1) the equation of the tangent line to the curve \(x^2 - xy - y^2 = 1\) at the point (2, 1) is \(y = \frac{1}{2}x - 1\).

2) the equation of the tangent line is \[y = -\frac{57}{25}x + \frac{171}{25}\].

1) To find the equation of the tangent line to the curve \(x^2 - xy - y^2 = 1\) at the point (2, 1), we'll use implicit differentiation.

Differentiating the equation implicitly with respect to x, we get:

\[2x - y - x\frac{dy}{dx} - 2y\frac{dy}{dx} = 0\]

Next, we substitute the coordinates of the point (2, 1) into the equation. We have x = 2 and y = 1:

\[2(2) - 1 - 2(2)\frac{dy}{dx} - 2(1)\frac{dy}{dx} = 0\]

\[4 - 1 - 4\frac{dy}{dx} - 2\frac{dy}{dx} = 0\]

\[3 - 6\frac{dy}{dx} = 0\]

\[-6\frac{dy}{dx} = -3\]

\[\frac{dy}{dx} = \frac{1}{2}\]

So, the slope of the tangent line to the curve at the point (2, 1) is \(\frac{1}{2}\).

Using the point-slope form of a line, we can write the equation of the tangent line:

\[y - 1 = \frac{1}{2}(x - 2)\]

\[y = \frac{1}{2}x - 1\]

Therefore, the equation of the tangent line to the curve \(x^2 - xy - y^2 = 1\) at the point (2, 1) is \(y = \frac{1}{2}x - 1\).

2) To find the equation of the tangent line to the curve \(2(x^2+y^2)^2 = 25(x^2-y^2)\) at the point (3, 1), we'll again use implicit differentiation.

Differentiating the equation implicitly with respect to x, we get:

\[8x(x^2+y^2) + 8y^2x - 25(2x - 2y\frac{dy}{dx}) = 0\]

Next, we substitute the coordinates of the point (3, 1) into the equation. We have x = 3 and y = 1:

\[8(3)(3^2 + 1^2) + 8(1^2)(3) - 25(2(3) - 2(1)\frac{dy}{dx}) = 0\]

\[8(3)(10) + 8(3) - 25(6 - 2\frac{dy}{dx}) = 0\]

\[240 + 24 - 150 + 50\frac{dy}{dx} = 0\]

\[264 - 150 + 50\frac{dy}{dx} = 0\]

\[50\frac{dy}{dx} = -114\]

\[\frac{dy}{dx} = -\frac{114}{50} = -\frac{57}{25}\]

So, the slope of the tangent line to the curve at the point (3, 1) is \(-\frac{57}{25}\).

Using the point-slope form of a line, we can write the equation of the tangent line:

\[y - 1 = -\frac{57}{25}(x - 3)\]

\[y = -\frac{57}{25}x + \frac{171}{25}\]

Therefore, the equation of the tangent line is \[y = -\frac{57}{25}x + \frac{171}{25}\].

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The value of x = 1 does not match any of the mathematics values given (10, 12, 22, 31, 42, 55).

The given set of values for x is 10, 12, 22, 31, 42, and 55. However, none of these values equal 1. Therefore, the value of x = 1 is not present in the given set.

In mathematics and programming, the equal sign (=) is used for assignment, not for equality. So when we say "x = 1," we are assigning the value 1 to the variable x. However, in the given set, x takes the values 10, 12, 22, 31, 42, and 55, which means x can only have those specific values, not 1.

It's important to distinguish between assignment and equality. In this case, the assignment statement "x = 1" does not match any of the values in the given set. If we were looking for a value of x that equals 1, we would need to search for it in a different context or equation.

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The limit of the expression limx→6 [f(x) + 2g(x) + h(x)²] is 14.

To evaluate this limit, we can use the limit laws step by step. Let's break down the process:

First, we use the limit law for addition: limx→a [f(x) + g(x)] = limx→a f(x) + limx→a g(x). Applying this law, we have limx→6 [f(x) + 2g(x)] = limx→6 f(x) + limx→6 (2g(x)).

Since we know limx→6 f(x) = 3 and limx→6 g(x) = 5, we substitute these values into the equation: limx→6 [f(x) + 2g(x)] = 3 + 2 * 5 = 13.

Next, we use the limit law for multiplication: limx→a (c * f(x)) = c * limx→a f(x), where c is a constant. Applying this law to the term h(x)², we have limx→6 (h(x)²) = (limx→6 h(x))².

Given that limx→6 h(x) = -1, we substitute this value into the equation: (limx→6 h(x))² = (-1)² = 1.

Now, we can combine all the parts of the expression: limx→6 [f(x) + 2g(x) + h(x)²] = limx→6 [f(x) + 2g(x)] + limx→6 (h(x)²) = 13 + 1 = 14.

Therefore, the limit of the given expression limx→6 [f(x) + 2g(x) + h(x)²] is equal to 14.

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The distance between the given pair of points ( -1,1 ) and (4,-3) is √41.

The distance formula used in finding the distance between two points is expressed as;

[tex]d = \sqrt{(x_2 - x_1)^2+(y_2 - y_1)^2}[/tex]

Given the data in the question;

Point 1( -1,1 )

x₁ = -1

y₁ = 1

Point 2( 4,-3 )

x₂ = 4

y₂ = -3

Plug the given values into the distance formula and simplify.

[tex]d = \sqrt{(x_2 - x_1)^2+(y_2 - y_1)^2}\\\\d = \sqrt{(4 - (-1))^2+(-3 - 1)^2}\\\\d = \sqrt{(4 + 1)^2+(-3 - 1)^2}\\\\d = \sqrt{(5)^2+(-4)^2}\\\\d = \sqrt{25+16}\\\\d = \sqrt{41}\\\\d = \sqrt{41}[/tex]

Therefore, the distance is radical form is √41.

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The area of the given regular octagon is 222.848 square inches.

apothem of length a = 8.2 in.

sides of length s = 6.8 in.

The area of a regular octagon can be calculated using the formula:

A=1/2 aP

Where, a is the apothem and P is the perimeter of the octagon.

A regular octagon is an eight-sided polygon, where all sides are of equal length and the angles are of equal measure. It is divided into eight congruent triangles, and the area of each triangle can be found out to find the total area of the octagon.

Area of each triangle:

Area of the triangle = 1/2 × apothem × side

Apothem (a) = 8.2 in

Side (s) = 6.8 in

Area of the triangle = 1/2 × 8.2 × 6.8

Area of the triangle = 27.856 in²

Area of the octagon:

Total area of octagon = 8 × (Area of the triangle)

[As there are 8 congruent triangles in the octagon]

Total area of octagon = 8 × 27.856

Total area of octagon = 222.848 in²

Therefore, the area of the given regular octagon is 222.848 square inches.

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The width of the rectangle is 7 inches.

To find the width of a rectangle given its length and area, we can use the formula for the area of a rectangle:

Area = Length × Width

In this case, we are given that the length of the rectangle is 28 inches and the area is 196 square inches. Let's substitute these values into the formula:

196 = 28 × Width

To find the width, we divide both sides of the equation by 28:

Width = 196 / 28

Simplifying the division:

Width = 7

Therefore, the width of the rectangle is 7 inches.

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Two points are given, and we are to find the equation of the sphere such that these two points are on opposite sides of the sphere.

1. A sphere with center at (a,b,c) and radius r has equation[tex](x-a)² + (y-b)² + (z-c)² = r².[/tex]

Thus, the equation of the sphere is[tex](x - 3)² + (y - 1)² + (z - 2)² = 14. 2. For the function f(x, y) = x+y x²+y²[/tex]

2. To be continuous at the origin, it must be defined at the origin, that is, f(0, 0) must exist.

Hence, we have:

f(0,0) = 0 + 0 0² + 0² = 0Hence, f(x, y) can be defined at the origin such that it is continuous. The limit at the origin can be shown to be zero, thus we have:[tex]lim (x, y)→(0,0) (x+y) x²+y² = lim (x, y)→(0,0) (x+y) (x²+y²) = lim (x, y)→(0,0) x³+y³ + x²y + xy² = 0[/tex]

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If x denotes one of the sides of the rectangle, then the adjacent side must also be x to form a rectangle. The perimeter of this rectangle is given by P(x) = 2x + 2x = 4x.

To minimize the perimeter P(x), we need to find the critical points by setting the derivative P'(x) equal to zero.

Taking the derivative of P(x) = 4x with respect to x, we have:

P'(x) = 4

Setting P'(x) equal to zero, we find:

4 = 0

Since 4 is a nonzero constant, there are no values of x that satisfy P'(x) = 0. Therefore, there are no critical points.

Since the interval is (0, ∞) and there are no critical points or endpoints, we need to analyze the behavior of P(x) as x approaches the boundaries of the interval.

As x approaches 0, the perimeter P(x) approaches 4(0) = 0.

As x approaches ∞, the perimeter P(x) approaches 4(∞) = ∞.

Since the perimeter P(x) approaches 0 as x approaches 0 and approaches ∞ as x approaches ∞, there is no local maximum or minimum. The function P(x) does not have any extrema.

Regarding the second derivative P''(x), since P(x) is a linear function with a constant derivative of 4, the second derivative P''(x) is zero.

Therefore, the brief summary is as follows:

The adjacent side of the rectangle must also be x.

The perimeter of the rectangle is given by P(x) = 4x.

The derivative of P(x) is P'(x) = 4, which does not have any critical points.

There is no local maximum or minimum.

The second derivative of P(x), P''(x), is zero.

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The converse of this statement would be: If two lines are cut by a transversal and the lines are parallel, then the corresponding angles formed are congruent.

Without specific information or equations, it is not possible to determine which lines are parallel.

However, to determine if lines are parallel, we can use the converse of the corresponding angles postulate. If two lines are cut by a transversal and the corresponding angles formed are congruent, then the lines are parallel.

The converse of this statement would be: If two lines are cut by a transversal and the lines are parallel, then the corresponding angles formed are congruent.

This converse can be used to justify the parallelism of lines when the corresponding angles are congruent.

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the work done by the force is 3974.7 foot pounds.

Force (F) = 87.9

lbAngle (θ) = 87.9°

Horizontal displacement (d) = 48.3 ftTo find: Work (W)

Formula to calculate work done by a force is:

W = Fdcosθ

Where,θ = 87.9°d = 48.3 ftF = 87.9 lb

We know that the angle is given in degrees,

so we need to convert it into radians because the unit of angle in the formula is radians.θ (radians) = (87.9° * π) / 180= 1.534 radian

Work done W = Fdcosθ= 87.9 * 48.3 * cos 1.534W = 3974.7 ft lb

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The Fourier Transform of f(t) = x(t - 1/2)cos(pt) is given by 1/2[X(f - p)exp(-j2πf(1/2)) + X(f + p)exp(-j2πf(1/2))], where X(f) is the Fourier Transform of x(t).

To find the Fourier Transform of the signal f(t) = x(t - 1/2)cos(pt), where x(t) is an arbitrary function, we can apply the time-shifting property and the modulation property of the Fourier Transform.

Let's denote F{ } as the Fourier Transform operator.

Using the time-shifting property, we have x(t - 1/2) = X(f)exp(-j2πf(1/2)), where X(f) is the Fourier Transform of x(t).

Applying the modulation property, we know that F{cos(2πft)} = 1/2[δ(f - f0) + δ(f + f0)], where δ is the Dirac delta function.

Combining these two properties, we get the Fourier Transform of f(t) as follows:

F{f(t)} = F{x(t - 1/2)cos(pt)} = X(f)exp(-j2πf(1/2)) * 1/2[δ(f - p) + δ(f + p)] = 1/2[X(f - p)exp(-j2πf(1/2)) + X(f + p)exp(-j2πf(1/2))].

In summary, the Fourier Transform of f(t) = x(t - 1/2)cos(pt) is given by 1/2[X(f - p)exp(-j2πf(1/2)) + X(f + p)exp(-j2πf(1/2))], where X(f) is the Fourier Transform of x(t).

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a. T1(N) + T2(N) = O(f(N)): True b. T1(N) - T2(N) = o(f(N)): False c. T2(N) * T1(N) = O(1): True d. T1(N) = O(T2(N)): False

a. T1(N) + T2(N) = O(f(N)): True

To justify this, we can use the definition of big O notation. If T1(N) = O(f(N)) and T2(N) = O(f(N)), it means that there exist positive constants c1 and c2, and a positive integer N0, such that for all N ≥ N0:

|T1(N)| ≤ c1 * |f(N)|

|T2(N)| ≤ c2 * |f(N)|

Now, let's consider the sum T1(N) + T2(N):

|T1(N) + T2(N)| ≤ |T1(N)| + |T2(N)| ≤ c1 * |f(N)| + c2 * |f(N)|

We can rewrite the above inequality as:

|T1(N) + T2(N)| ≤ (c1 + c2) * |f(N)|

Therefore, T1(N) + T2(N) = O(f(N)).

b. T1(N) - T2(N) = o(f(N)): False

To prove this statement false, we need to provide a counterexample. Consider the case where T1(N) = 2N and T2(N) = N. In this case, T1(N) = O(f(N)) and T2(N) = O(f(N)), where f(N) = N.

However, if we subtract T2(N) from T1(N):

T1(N) - T2(N) = 2N - N = N

Now, let's examine the relationship between N and f(N):

N = f(N)

Since the difference between T1(N) - T2(N) is equal to f(N), we can say that T1(N) - T2(N) is not strictly smaller than f(N) (o(f(N))). Hence, the statement T1(N) - T2(N) = o(f(N)) is not true in this case.

c. T2(N) * T1(N) = O(1): True

Multiplying two functions that are both bounded by O(f(N)) will result in a function that is bounded by O(f(N) * f(N)), which simplifies to O(f(N)^2).

Since f(N) can be any function, including a constant function, it is valid to say that T2(N) * T1(N) = O(1).

d. T1(N) = O(T2(N)): False

To disprove this statement, we need to provide a counterexample. Consider the case where T1(N) = 2N and T2(N) = N. In this case, T1(N) = O(T2(N)), as T1(N) = O(N), but T1(N) is not equal to O(T2(N)), since T2(N) = O(N) but not O(2N).

Hence, the statement T1(N) = O(T2(N)) is false in this case.

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The complete question is:

You must justify your answer. You will not earn any point if you simply say True or False (even the answer is correct). In case your answer is false, a counterexample must be given. Note: True means always true, so a valid justification is needed (such as using a rule or using a definition).

False means not always true, so you should be able to show at least once case it is not hold. So in case you think the answer should be false, you must provide a counterexample; i.e., you should show particular functions T1 and T2, such as T1 = 3N2 and T2 = 4N + 2.

Suppose T 1 (N)=O(f(N)) and T 2 (N)=O(f(N)). Which of the following are true? a. T 1 (N)+T 2(N)=O(f(N)) b. T 1(N)−T 2(N)=o(f(N)) c.T 2(N)T 1(N)​=0(1) d. T 1(N)=O(T 2 (N))

The antiderivative of [tex](-2+x^3[/tex]) with respect to x is[tex](-2x + (1/4)x^4) + C[/tex], where C is the constant of integration.

To find the antiderivative  [tex](-2+x^3)[/tex] with respect to x, we need to apply the power rule for integration and the constant multiple rules. The power rule states that the antiderivative of xⁿ with respect to x is [tex](1/(n+1))x^(n+1) + C[/tex], where C is the constant of integration.

Applying the power rule to the term x³, we get:

[tex]\int\limits^_[/tex][tex]x^3 dx = (1/(3+1))x^(3+1) + C = (1/4)x^4 + C[/tex]

Now, we must consider the antiderivative of the constant term (-2). The antiderivative of a constant multiplied by x is simply the constant multiplied by x. Thus, the antiderivative of -2 with respect to x is -2x.

Putting it all together, the antiderivative of[tex](-2+x^3)[/tex] with respect to x is [tex](-2x + (1/4)x^4) + C[/tex], where C is the constant of integration.

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The alternative that should be selected is alternative A.

To determine which alternative to choose, we need to compare their annual cash flows using an annual cash flow analysis and a 10% interest rate. Let's analyze each alternative:

Alternative A: It has an initial cost of $50,000 and generates an annual cash inflow of $15,000 for 10 years. The annual cash inflow remains constant throughout the life of the project.

Alternative B: It has an initial cost of $80,000 and generates an annual cash inflow of $20,000 for 5 years. At the end of the 5-year period, the unit is replaced with an identical one.

Alternative C: It has an initial cost of $100,000 and generates an annual cash inflow of $30,000 for 4 years. At the end of the 4-year period, the unit is replaced with an identical one.

To determine the present value of each alternative, we need to discount the annual cash flows at a 10% interest rate. By calculating the present value of each alternative, we can compare their net present values (NPVs). The alternative with the highest NPV should be selected.

Performing the calculations, we find that the net present value (NPV) of alternative A is the highest, indicating that it is the most financially favorable option. Therefore, alternative A should be selected.

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Therefore, each pendant cost $13.20.

To find the cost of each pendant, we can subtract the cost of the beads from the total cost of the necklaces.

Total cost of the necklaces = $24.40

Cost of the beads = $11.20

Cost of each pendant = Total cost of the necklaces - Cost of the beads

= $24.40 - $11.20

= $13.20

Therefore, each pendant cost $13.20.

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Yes it can be written like that.

Answer: yes it can

Step-by-step explanation: 2.0 is the same as 2/10

The value exceeds $1 million for the first time on the 21st square of the checkerboard.

The value of each square follows a doubling pattern:

$1, $2, $4, $8, $16, and so on.

We can express this pattern as [tex]2^{(n-1)}[/tex], where n represents the square number.

We need to find the value of n for which [tex]2^{(n-1)}[/tex] exceeds $1 million:

[tex]2^{(n-1)} > 1,000,000[/tex]

Taking the logarithm base 2 of both sides, we get:

[tex](n-1) > log_2(1,000,000)[/tex]

Using a calculator, we can determine the logarithm:

[tex]log_2(1,000,000) = 19.93[/tex]

Now, solving for n:

n-1 > 19.93

n > 20.93

Since n represents the square number, it must be a whole number. Therefore, we need to round up to the nearest whole number, giving us:

n = 21

Therefore, the value exceeds $1 million for the first time on the 21st square of the checkerboard.

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The complete question is as follows:

A father put a dollar on the first square of an 8x8 checkerboard. On the second square, the father doubled $2, on the third $4, on the fourth $8, and so on. At what square would the value be more than $1 million for the first time?

The remaining side PR = 33mm. The angle at P = 60 degrees. The angle at R = 60 degrees.



Given: PQ = 48 mm, QR = 39 mm, angle Q = 60 degrees.

Step 1: Draw a rough sketch of the triangle.

Step 2: Use the law of cosines to find the length of PR.

PR^2 = PQ^2 + QR^2 - 2(PQ)(QR)cosQ
PR^2 = (48)^2 + (39)^2 - 2(48)(39)cos60
PR^2 = 2304 + 1521 - 1872
PR^2 = 1953
PR = sqrt(1953)
PR = 44.19 mm (rounded to two decimal places)

Step 3: Use the law of sines to find the remaining angles.

sinP / PQ = sinQ / PR
sinP / 48 = sin60 / 44.19
sinP = (48)(sin60) / 44.19
sinP = 0.8295
P = sin^-1(0.8295)
P = 56.56 degrees (rounded to two decimal places)

Angle R = 180 - 60 - 56.56
Angle R = 63.44 degrees (rounded to two decimal places)

Therefore, the remaining side PR = 44.19 mm, the angle at P = 56.56 degrees, and the angle at R = 63.44 degrees.


In this question, we need to construct a triangle PQR such that PQ = 48mm, QR = 39mm, and the angle at Q = 60 degrees. We are asked to measure the remaining side PR and angles.

The length of the remaining side PR can be found using the law of cosines. The law of cosines states that the square of one side of a triangle is equal to the sum of the squares of the other two sides minus twice their product and the cosine of the angle between them.

Using this formula, we can find that the length of PR is 44.19mm.

We can then use the law of sines to find the remaining angles. The law of sines states that the ratio of the length of a side of a triangle to the sine of the angle opposite that side is the same for all sides and angles in the triangle.

Using this formula, we can find that the angle at P is 56.56 degrees and the angle at R is 63.44 degrees.

Therefore, the remaining side PR is 44.19mm, the angle at P is 56.56 degrees, and the angle at R is 63.44 degrees.

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The unknown current is 0 Amps (I = 0 A).

To determine the unknown current in the given network, we need to use Ohm's Law and apply Kirchhoff's laws.

Let's assume the unknown current as I. According to Kirchhoff's current law (KCL), the sum of currents entering and leaving a junction is zero.

At the junction between R1, R2, and R3, we have:

I - (I1 + I2) = 0

Applying Ohm's Law, we can express the currents in terms of resistances and the unknown current:

I - (V1/R1 + V2/R2) = 0

Now, we know that V1 = I * R1 and V2 = I * R2. Substituting these values:

I - (I * R1 / R1 + I * R2 / R2) = 0

Simplifying further:

I - (I + I) = 0

I - 2I = 0

-I = 0

Therefore, the unknown current is 0 Amps (I = 0 A).

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Without additional information, it is not possible to determine the specific value of (c) in this case.

To find the function (f(x)) and the number (c) such that

[tex]$\(\lim_{x\to 25}\frac{8x-40}{x-25} = f'(c)\),[/tex]

we can start by simplifying the expression inside the limit.

[tex]$\lim_{x\to 25}\frac{8x-40}{x-25} &= \lim_{x\to 25}\frac{8(x-5)}{x-25}\\[/tex]

[tex]$= \lim_{x\to 25}\frac{8(x-5)}{x-25}\cdot\frac{(x-25)}{(x-25)}\\[/tex]

[tex]$= \lim_{x\to 25}\frac{8(x-5)(x-25)}{(x-25)^2}\\[/tex]

[tex]$= \lim_{x\to 25}\frac{8(x-5)(x-25)}{(x-25)(x-25)}\\[/tex]

[tex]$= \lim_{x\to 25}\frac{8(x-5)}{(x-25)}[/tex]

Now, we can see that the limit expression simplifies to

[tex]$\(\lim_{x\to 25}8 = 8\)[/tex]

Therefore, (f'(c) = 8).

Since (f'(c) = 8), the function (f(x)) must be the antiderivative of 8, which is (f(x) = 8x + k), where (k) is a constant.

To find the value of (c), we need more information about the function \(f(x)) or the original limit expression. Without additional information, it is not possible to determine the specific value of (c) in this case.

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To find the polar equation of a hyperbola with eccentricity 3 and the directrix (x = 1), we can start by defining the standard polar equation for a hyperbola.

Like this :

[r = frac{ed}{1 - e\cos(theta)}]

where (r) is the distance from the origin, (e) is the eccentricity, (d) is the distance from the origin to the directrix, and \(\theta\) is the angle from the positive x-axis.

In this case, the eccentricity is given as 3 and the directrix is (x = 1). The distance from the origin to the directrix is the absolute value of 1, which is simply 1.

Substituting these values into the polar equation, we get:

[r = frac{3}{1 - 3\cos(theta)}]

Therefore, the polar equation of the hyperbola with eccentricity 3 and the directrix (x = 1) is \(r = frac{3}{1 - 3\cos(theta)}).

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(a) The derivative of the vector function r(t) = 8cos(t)i + 8sin(t)j is r'(t) = -8sin(t)i + 8cos(t)j. (b) The second derivative of the vector function r(t) = 8cos(t)i + 8sin(t)j is r''(t) = -8cos(t)i - 8sin(t)j. (c) The dot product of r'(t) and r''(t) is r'(t)⋅r''(t) = 64sin^2(t) + 64cos^2(t) = 64.

(a) To find the derivative of the vector function r(t) = 8cos(t)i + 8sin(t)j, we differentiate each component with respect to t:

r'(t) = d/dt (8cos(t)i) + d/dt (8sin(t)j)

= -8sin(t)i + 8cos(t)j

Therefore, r'(t) = -8sin(t)i + 8cos(t)j.

(b) To find the second derivative of r(t), we differentiate each component of r'(t) with respect to t:

r''(t) = d/dt (-8sin(t)i) + d/dt (8cos(t)j)

= -8cos(t)i - 8sin(t)j

So, r''(t) = -8cos(t)i - 8sin(t)j.

(c) To find r'(t)⋅r''(t), we take the dot product of r'(t) and r''(t):

r'(t)⋅r''(t) = (-8sin(t)i + 8cos(t)j)⋅(-8cos(t)i - 8sin(t)j)

= 64sin^2(t) + 64cos^2(t)

= 64

Hence, r'(t)⋅r''(t) = 64. The dot product of the first derivative r'(t) and the second derivative r''(t) is a constant value of 64.

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Answer:

4K+6K =10+5

10K=15

K=25

The answer is:

k = -5/2

Work/explanation:

Our equation is:

[tex]\sf{4k+5=6k+10}[/tex]

Subtract 4k from each side

[tex]\sf{5=2k+10}[/tex]

[tex]\sf{2k+10=5}[/tex]

Subtract 10 from each side

[tex]\sf{2k=-5}[/tex]

[tex]\sf{k=-\dfrac{5}{2}}[/tex]

The equation of motion is x(t) = 3 sin(2t) meters.

To find the equation of motion, we need to determine the angular frequency (ω) and the coefficients A and B. The angular frequency can be calculated using the formula ω = √(k/m), where k is the spring constant and m is the mass attached to the spring.

Given that the force of 640 newtons stretches the spring by 4 meters, we can use Hooke's Law to determine the spring constant: F = kx. Thus, k = F/x = 640 N / 4 m = 160 N/m.

Now, we can calculate the angular frequency: ω = √(k/m) = √(160 N/m / 40 kg) = 2 rad/s.

To determine the coefficients A and B, we need to consider the initial conditions. Since the mass is initially released from the equilibrium position with an upward velocity of 6 m/s, the displacement at t = 0 is zero (x(0) = 0) and the velocity at t = 0 is 6 m/s (x'(0) = 6 m/s).

Substituting these initial conditions into the equation of motion, we can solve for A and B. Since x(0) = A cos(0) + B sin(0) = A, we have A = 0. And x'(0) = -ωA sin(0) + ωB cos(0) = ωB, so B = x'(0)/ω = 6 m/s / 2 rad/s = 3 m.

Therefore, the equation of motion is x(t) = 3 sin(2t) meters.

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