Exercises 2.17 Interpolate a cubic spline between the three
points (0, 1), (2, 2) and (4, 0).

The value that verifies 5p (1)-(1)-500 is -124.04

To approximate the value of the function f at x = 1 using the Taylor polynomial Pₙ = (-10)^n/ n! about x = 0, we need to find the value of P₅(1).

First, let's compute the derivatives of f(x) = e^x up to the fifth derivative:

f'(x) = e^x

f''(x) = e^x

f'''(x) = e^x

f''''(x) = e^x

f⁽⁵⁾(x) = e^x

Now, let's evaluate these derivatives at x = 0:

f(0) = e^0 = 1

f'(0) = e^0 = 1

f''(0) = e^0 = 1

f'''(0) = e^0 = 1

f''''(0) = e^0 = 1

f⁽⁵⁾(0) = e^0 = 1

Using these values, we can compute the Taylor polynomial P₅(x):

P₅(x) = f(0) + f'(0)(x - 0) + f''(0)(x - 0)²/2! + f'''(0)(x - 0)³/3! + f''''(0)(x - 0)⁴/4! + f⁽⁵⁾(0)(x - 0)⁵/5!

P₅(x) = 1 + 1x + 1x²/2! + 1x³/3! + 1x⁴/4! + 1x⁵/5!

Now, let's evaluate P₅(1):

P₅(1) = 1 + 1(1) + 1(1)²/2! + 1(1)³/3! + 1(1)⁴/4! + 1(1)⁵/5!

P₅(1) = 1 + 1 + 1/2 + 1/6 + 1/24 + 1/120

P₅(1) = 227/120

Therefore, the value that verifies 5P₅(1) - (1) - 500 is:

5P₅(1) - (1) - 500 = 5 * (227/120) - 1 - 500

= 1135/120 - 1 - 500

= 1135/120 - 120/120 - 60000/120

= (1135 - 120 - 60000)/120

= -59485/120

= -124.04

So, the value that verifies 5P₅(1) - (1) - 500 is approximately -124.04.

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Answer:

Use the degree and the leading coefficient to determine the behavior.

Falls to the left and falls to the right

Step-by-step explanation:

Answer:

60°

Step-by-step explanation:

90°-30°…

according to your question

After considering the given data we conclude that the distance between stone and ground level [tex]h(t) = -4.9t^2 + 900[/tex], time taken for the stone to reach the ground 18.22 seconds,the velocity with which it strikes the ground 178.76 m/s, if thrown with a down ward speed of 3m/s then the duration needed is 18.47 seconds.

A stone is dropped from the upper observation deck of a tower, 900 m above the ground. We can use the kinematic equations of motion to answer the following questions:
a) The distance of the stone above ground level at time t can be found using the equation:
[tex]h(t) = -1/2gt^2 + v_0t + h_0[/tex]
where g is the acceleration due to gravity (9.8 m/s²), v0 is the initial velocity (0 m/s), h0 is the initial height (900 m), and t is the time elapsed. Plugging in the values, we get:
[tex]h(t) = -4.9t^2 + 900[/tex]
b) To find how long it takes for the stone to reach the ground, we need to find the time when h(t) = 0:
[tex]-4.9t^2 + 900 = 0[/tex]
Solving for t, we get:
[tex]t = \sqrt(900/4.9) = 18.22 seconds[/tex]
Therefore, it takes the stone 18.22 seconds to reach the ground.
c) To find the velocity with which the stone strikes the ground, we can use the equation:
[tex]v(t) = -gt + v_0[/tex]
where v(t) is the velocity at time t. Plugging in the values, we get:
[tex]v(t) = -9.8(18.22) + 0 = -178.76 m/s[/tex]
Therefore, the stone strikes the ground with a velocity of 178.76 m/s.
d) If the stone is thrown downward with a speed of 3 m/s, we can use the same equation [tex]v(t) = -9.8(18.22) + 0 = -178.76 m/s[/tex] to find how long it takes to reach the ground. This time, [tex]v_0[/tex] is -3 m/s (since it is thrown downward) and [tex]h_0[/tex] is still 900 m. Plugging in the values, we get:
[tex]-4.9t^2 - 3t + 900 = 0[/tex]
Solving for t, we get:
t = 18.47 seconds
Therefore, it takes the stone 18.47 seconds to reach the ground when thrown downward with a speed of 3 m/s.
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The solution to the given differential equation with the initial conditions y(0) = 1 and y'(0) = 1 is:

[tex]y(t) = -e^{-t}/10 + (11/10)t*e^{-t} + (1/10)cos(2t) + (3/10)sin(2t)[/tex]

To solve the given differential equation using Laplace transform, we will apply the Laplace transform to both sides of the equation and then solve for the transformed variable.

Let's denote the Laplace transform of y(t) as Y(s).

Taking the Laplace transform of both sides of the differential equation, we get:

[tex]s^2Y(s) + 2sY(s) + Y(s) = (s^2 + 2s + 1)/(s^2 + 4)[/tex]

Now, let's solve for Y(s):

[tex]Y(s)(s^2 + 2s + 1) = (s^2 + 2s + 1)/(s^2 + 4)\\Y(s) = (s^2 + 2s + 1)/(s^2 + 4)(s^2 + 2s + 1)[/tex]

Factoring the denominator:

[tex]Y(s) = (s^2 + 2s + 1)/((s + 1)^2(s^2 + 4))[/tex]

Now, we need to decompose the fraction into partial fractions. Let's express the numerator in terms of A, B, C, and D:

[tex]s^2 + 2s + 1 = A/(s + 1) + B/(s + 1)^2 + (Cs + D)/(s^2 + 4)[/tex]

To find the values of A, B, C, and D, we can equate the numerators:

[tex]s^2 + 2s + 1 = A(s + 1)(s^2 + 4) + B(s^2 + 4) + (Cs + D)(s + 1)^2[/tex]

Expanding and equating coefficients:

[tex]s^2 + 2s + 1 = A(s^3 + 5s^2 + 4s) + B(s^2 + 4) + (C(s^2 + 2s + 1) + D(s + 1)^2)[/tex]

Simplifying:

[tex]s^2 + 2s + 1 = (A + C)s^3 + (5A + C + D)s^2 + (4A + 2C + D)s + (4A + D)[/tex]

Equating coefficients:

A + C = 0 (coefficient of [tex]s^3[/tex])

5A + C + D = 1 (coefficient of [tex]s^2)[/tex]

4A + 2C + D = 2 (coefficient of s)

4A + D = 1 (constant term)

Solving these equations simultaneously, we find A = -1/10, B = 11/10, C = 1/10, and D = 3/10.

Now, substituting these values back into Y(s):

[tex]Y(s) = (-1/10)/(s + 1) + (11/10)/(s + 1)^2 + (1/10)(s + 3)/(s^2 + 4) + (3/10)/(s^2 + 4)[/tex]

To find y(t), we need to take the inverse Laplace transform of Y(s). Fortunately, we can use a Laplace transform table to find the inverse Laplace transform of each term.

The inverse Laplace transform of (-1/10)/(s + 1) is [tex]-e^{-t}/10.[/tex]

The inverse Laplace transform of (11/10)/(s + 1)² is (11/10)t*[tex]e^{-t}.[/tex]

The inverse Laplace transform of (1/10)(s + 3)/(s² + 4) is (1/10)cos(2t).

The inverse Laplace transform of (3/10)/(s² + 4) is (3/10)sin(2t).

Combining these results, the solution y(t) is:

[tex]y(t) = -e^{-t}/10 + (11/10)t*e^{-t} + (1/10)cos(2t) + (3/10)sin(2t)[/tex]

Therefore, the solution to the given differential equation with the initial conditions y(0) = 1 and y'(0) = 1 is:

[tex]y(t) = -e^{-t}/10 + (11/10)t*e^{-t} + (1/10)cos(2t) + (3/10)sin(2t)[/tex]

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The statement "W is a subspace of P2" is true because the set W, defined as W = {a + 2x + bx² ∈ P2: a, b ∈ R}, is a subspace of P2.

To determine if the set W = {a + 2x + bx² ∈ P2: a, b ∈ R} is a subspace of P2, we need to check if it satisfies three conditions: closure under addition, closure under scalar multiplication, and contains the zero vector.

Closure under addition: For any two polynomials p(x) = a + 2x + bx² and q(x) = c + 2x + dx² in W, their sum p(x) + q(x) = (a + c) + 4x + (b + d)x² is also a polynomial in W. This shows that W is closed under addition.

Closure under scalar multiplication: For any polynomial p(x) = a + 2x + bx² in W and any scalar c, the scalar multiple c * p(x) = ca + 2cx + cbx² is also a polynomial in W. Therefore, W is closed under scalar multiplication.

Contains the zero vector: The zero vector in P2 is the polynomial 0x² + 0x + 0, which can be expressed as a + 2x + bx² with a = 0 and b = 0. Since this polynomial satisfies the conditions of W, W contains the zero vector.

Since W satisfies all three conditions, it is a subspace of P2.

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the average can be calculated by adding the scores together and then dividing by the number of scores.

we can set up an equation:

let x = test score needed on next test

(72+72+80+x)/4 = 71

multiply both sides by 4

72+72+80+x = 284

add like terms

224+x=284

subtract 224 from both sides

x=60

she will need a 60 for her average to be 71

She needs to score a 60 to have a average score of 71,

72+72+80+60=284

284/4=71

Answer:

17

Step-by-step explanation:

The lower quartile Q₁ is positioned at the left side of the box.

The value at the left is Q₁ is 17

The critical value (upper) for a 1% level of significance is 2.33. Therefore, we can reject the null hypothesis.

To determine whether the average revenue is significantly different from $8 million, we perform a hypothesis test using the sample data. The null hypothesis (H0) states that the average revenue is equal to $8 million, while the alternative hypothesis (Ha) states that the average revenue is significantly different.

Given a sample of 400 companies, the sample average revenue is found to be $8.30 million, and the population standard deviation is known to be 3. We can use a z-test since the population standard deviation is known and the sample size is large.

Next, we calculate the test statistic (z-score) using the formula: z = (sample mean - hypothesized mean) / (population standard deviation / √sample size).

Plugging in the values, we get z = (8.30 - 8) / (3 / √400) = 1.73.

To determine the critical value (upper) at a 1% level of significance, we look up the z-value from the standard normal distribution table, which is approximately 2.33.

Since the calculated z-score of 1.73 is less than the critical value of 2.33, we do not have enough evidence to reject the null hypothesis. Therefore, we cannot conclude that the average revenue is significantly different from $8 million at a 1% level of significance.

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The probability that at least 5 of the 8 adults say they get their financial advice from family members can be calculated using binomial probability distribution.

Formula for binomial probability distribution is:P(X=k) = nCk * pk * (1-p)n-kwhere, P(X=k) is the probability of k successes in n independent trials, p is the probability of success in one trial, q=1-p is the probability of failure in one trial, nCk is the combination of k successes in n independent trials.

In this case, the probability of success is p=0.4 as 40% of adults in the US say they get their financial advice from family members.

Therefore, the probability of failure is q=1-0.4=0.6.So, P(X ≥ 5) = P(X = 5) + P(X = 6) + P(X = 7) + P(X = 8)P(X=k) = nCk * pk * (1-p)n-kwhere, n=8, p=0.4, q=0.6For k = 5, nCk = 8C5P(X = 5) = 8C5 * (0.4)5 * (0.6)3= 0.2787

For k = 6, nCk = 8C6P(X = 6) = 8C6 * (0.4)6 * (0.6)2= 0.1960For k = 7, nCk = 8C7P(X = 7) = 8C7 * (0.4)7 * (0.6)1= 0.0575For k = 8, nCk = 8C8P(X = 8) = 8C8 * (0.4)8 * (0.6)0= 0.0030Therefore,P(X ≥ 5) = P(X = 5) + P(X = 6) + P(X = 7) + P(X = 8) = 0.2787 + 0.1960 + 0.0575 + 0.0030 = 0.5352

Hence, the probability that at least 5 of the 8 adults say they get their financial advice from family members is 0.5352.

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Given information that 40% of adults in the US say they get their financial advice from family members. We have to find the probability of at least 5 of the 8 say they get their financial advice from family members.

Hence, the required probability is 0.7530.

The probability of getting financial advice from family members is 40%. Let X be the number of people out of 8, who get their financial advice from family members. Here, X follows a binomial distribution with parameters n = 8 and p = 0.4. The probability of getting atleast 5 people getting their financial advice from family members is

P(X ≥ 5) = P(X = 5) + P(X = 6) + P(X = 7) + P(X = 8)

Using binomial distribution formula, we get

P(X = x) = ${n\choose x}p^xq^{n-x}$

Where, n = 8, p = 0.4, q = 0.6

The probability of getting exactly 5 people out of 8 getting their financial advice from family members

P(X = 5) = ${8\choose 5} 0.4^5 (0.6)^{8-5}$

= 0.27869

The probability of getting exactly 6 people out of 8 getting their financial advice from family members

P(X = 6) = ${8\choose 6} 0.4^6 (0.6)^{8-6}$

= 0.29360

The probability of getting exactly 7 people out of 8 getting their financial advice from family members

P(X = 7) = ${8\choose 7} 0.4^7 (0.6)^{8-7}$

= 0.16493

The probability of getting exactly 8 people out of 8 getting their financial advice from family members

P(X = 8) = ${8\choose 8} 0.4^8 (0.6)^{8-8}$

= 0.01678

Therefore, the probability of getting at least 5 people getting their financial advice from family members is

P(X ≥ 5) = 0.27869 + 0.29360 + 0.16493 + 0.01678

= 0.7530

Hence, the required probability is 0.7530.

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Answer:

$0.24 per orange

Step-by-step explanation:

$3.84 / 16 = 0.24

To make cookies for 48 people, the recipe requires 6 cups of flour.

The oatmeal raisin cookie recipe serves 36 people and requires 2 eggs and 2 cups of flour. Since we need to make cookies for 48 people, we can calculate the amount of flour required as follows:

36 people → 2 cups of flour

1 person → (2 cups of flour) / (36 people) = (1/18) cups of flour

To make cookies for 48 people:

48 people × (1/18) cups of flour = 2.67 cups of flour

Therefore, to make cookies for 48 people, we need approximately 2.67 cups of flour.

Note: Since the result is a fraction of a cup, it is advisable to round up to the nearest whole number, so in this case, 3 cups of flour would be needed.

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Answer:

[tex]\dfrac{7}{10}[/tex]

Step-by-step explanation:

We need to find a rational number between 3/5 and 4/5.

We can find a rational number between two fractions as :

[tex]\dfrac{a+b}{2}[/tex]

We have, a = 3/5 and b = 4/5

So,

[tex]\dfrac{\dfrac{3}{5}+\dfrac{4}{5}}{2}\\\\=\dfrac{\dfrac{7}{5}}{2}\\\\=\dfrac{7}{10}[/tex]

So, a rational number between 3/5 and 4/5 is equal to 7/10.

1. Using Gauss Divergence theorem, we have to calculate S. (5xi + ay3 - 23k). ndA over the sphere S: 1+ y + x2 = 9. We have the following information: S.(5xi + ay3 - 23k).ndA over the sphere S: 1+ y + x2 = 9.

Gauss Divergence Theorem states that, The surface integral of a vector field F over a closed surface S equals the volume integral of the divergence of F over the enclosed volume V. To calculate the surface integral S.(5xi + ay3 - 23k).ndA, we need to first calculate the volume integral of the divergence of the vector field over the enclosed volume V which in this case is a sphere.

The divergence of the given vector field can be calculated as, div(F) = ∂P/∂x + ∂Q/∂y + ∂R/∂z = 5 + 3ay + 0 = 5 + 3ay

Thus, the volume integral of the divergence of F over the sphere S: 1+ y + x2 = 9 is given as ∭V div(F) dV = ∭V (5 + 3ay) dV.

The volume integral can be calculated using spherical coordinates.

We have the equation of the sphere as 1 + y + x2 = 9, substituting x = r sin θ cos φ, y = r sin θ sin φ, z = r cos θ, and simplifying, we get the limits of integration as 0 ≤ r ≤ 2, 0 ≤ θ ≤ π, and 0 ≤ φ ≤ 2π.

Therefore, the volume integral becomes:∭V div(F) dV = ∭V (5 + 3ay) dV = ∫0^2 ∫0^π ∫0^2π (5 + 3a(r cos θ sin φ)) r2 sin θ dr dθ dφ = 60πa

The surface integral of F over the sphere S can be calculated using the Gauss Divergence Theorem, which states that the surface integral of F over a closed surface S is equal to the volume integral of the divergence of F over the enclosed volume V.

Thus, S.(5xi + ay3 - 23k).ndA = ∭V div(F) dV = 60πa

Answer: S.(5xi + ay3 - 23k).ndA = 60πa2.

We are to calculate the surface integral ∬S G(r) da, where G = (12 + 12 + 36)/2 = 30.

The surface is given by r(u, v) = (3u, 2v, u), 0 ≤ u ≤ 1 and 0 ≤ v ≤ 2.

The surface area element da can be calculated as, da = |r/∂u x r/∂v| dudv = |(6, 0, 3) x (0, 2, 1)| dudv = |(-6, -3, 0)| dudv = 3 dudv

Hence, the surface integral ∬S G(r) da becomes ∬S G(r) da = ∫0^2 ∫0^1 G(r(u, v)) da = ∫0^2 ∫0^1 30 * 3 dudv = 180

Answer: ∬S G(r) da = 180

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Given system of differential equation is(D − 1)x+ (D² + 1)y = 1 ...

(i)(D² − 1)x+ (D + 1)y = 2 ...(ii)By using systematic elimination method, we have(D²+1)(D²−1)x+(D+1)(D−1)y=D²+1×1-(D+1)×1=0Simplifying the above equation, we get(D⁴-1)x=-(D-1)y...(iii)Applying D on both sides of (iii), we get D(D⁴-1)x=-(D-1 )DyD⁵x- Dx=-(Dy-y)or D⁵x+Dy=y ... (iv)Now applying D on (i), we get(D−1)Dx+(D²+1)Dy=0or D(D²+1)y=(1-D)x ...(v)Now applying D on (ii), we get(D²−1)Dx+(D+1)Dy=0or D(D+1)x=(1+D)y ...(vi)Now, substituting the value of x and y from equations (v) and (vi) in equation (iv), we getD⁵x+(1+D)Dx=(1-D)Dy D⁵x+(1+D)Dx=-(1-D)x ...(vii) Simplifying the above equation, we getD⁶x+2D⁴x+D²x+x=0or D²(D⁴+1)x+D²x=-x ...(viii)or D²(D⁴+2)x=-xor D⁴x+2x=-xor D⁴x=-3xNow using D on both sides, we get D⁵x=-3Dxor D⁶x=-3D²x

Now, substituting the value of D²x from equation (iii) in equation (i), we get(D-1)x+(D²+1)y=1 ...(i)⇒ (D-1)x+y=1 ...(ix)Now, substituting the value of D²x from equation (iii) in equation (ii), we get(D²-1)x+(D+1)y=2 ...(ii)⇒ -(D+1)x+y=0or (D+1)x-y=0 ...(x)From equation (ix) and (x), we have2x=1or x=1/2Now, substituting the value of x in equation (ix), we have D(1/2)+y=1or y=1-1/2=1/2Thus, the solution of the given system of differential equation is(x(t), y(t))=(e^(-t/2))[(-5/3)cos((sqrt(47)/2)t)-(125/3)sin((sqrt(47)/2)t)]+(20/3)cos(3t)+(20/3)sin(3t), (1/2)

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Answer:

h = 7.832

Step-by-step explanation:

This is a right angled triangle so, taking 50 as reference angle,

hypotenuse = ?

perpendicular = 6

The ratio for p and h is given by

Sin 50 = p/h

Sin 50 = 6 /h

h = 6 / Sin 50

h = 7.832

Answer:

I feel like the answers are 10 and -2

Step-by-step explanation:

Hope this helps! Have a great day and good luck! :)

Answer:B

Step-by-step explanation:if it's a negative 3/4x then in be there is a -4 and there is a -7 so we can do -7 -4 but 4 is a negative so it turns into a positive so it's _

-7+4 and it become a smaller negative so -7+4 = -3. So it has to be B. Hope this gets brainliest

Answer:

B and D

Explanation:

Answer:

√49

Step-by-step explanation:

It's a perfect square root.

Please brainliest ;)

Answer:

x = 20

Step-by-step explanation:

The two angles are verticals angles and vertical angles are equal

7x -99 = 2x+1

Subtract 2x from each side

7x-99-2x =2x+1-2x

5x-99 =1

Add 99 to each side

5x-99+99 = 1+99

5x =100

Divide each side by 5

5x/5 =100/5

x = 20

Answer:

x=20

set them equal to eac hother

7x-99=2x+1

-2x+99-2x+99

---------------------

5x=100

---- ------

5 5

x=20

First. Finding the x-intercepts of  

Let  be the change in water level. So to find the x-intercepts of this function we can use The Rational Zero Test that states:

To find the zeros of the polynomial:

We use the Trial-and-Error Method which states that a factor of the constant term:

can be a zero of a polynomial (the x-intercepts).

So let's use an example: Suppose you have the following polynomial:

where the constant term is . The possible zeros are the factors of this term, that is:

.

Thus:

From the foregoing, we can affirm that  are zeros of the polynomial.

Second. Construction a rough graph of  

Given that this is a polynomial, then the function is continuous. To graph it we set the roots on the coordinate system. We take the interval:

and compute  where  is a real number between -2 and -1. If , the curve start rising, if not, the curve start falling. For instance:

Therefore the curve start falling and it goes up and down until  and from this point it rises without a bound as shown in the figure below

Answer:

s = 3

Step-by-step explanation:

The volume formula for a cube is V = s^3, where “s” is the edge length. Since we know the volume and need to find “s,” we just do the inverse operation for an exponent, which is a radical. The cubed root of 27 is 3, so there’s your answer! Hope this is helpful & accurate. Best wishes.

Answer:

y = 15x

Step-by-step explanation:

For every flower pot purchased (y), the quantitity of the price (x) will go up by $15.

Answer:

y = 15x

Step-by-step explanation:

Answer:

385=80x+45

x=4.25 hours

Answer:

d = 20

Step-by-step explanation:

90-19 = 71

(4d-9) = 71

4d = 80

d = 20

Answer:

d = 20

Step-by-step explanation:

Complementary angles are two angles that add up to 90°.

We know that one angle is 19° and the other is (4d − 9)°. So, we can set up the equation:

19 + (4d − 9) = 90.

Solving for d, we get:

19 + (4d − 9) = 90

19 + 4d − 9 = 90

4d + 10 = 90

4d = 80

d = 20

Therefore, the value of d is 20.

Answer:

Sadly no but also brainly isn't for this

Step-by-step explanation:

(Ima get so much hate rn from y'all)