1. f(x+2)
2. f(x+2)-f(x)
3. f(x+h)
for the same quadratic function for all questions.

(a) The average rate the water leaves the tub between t=1 and t=2 is -3 m^3/min.

(b) The average rate the water leaves the tub between t=1 and t=1.1 is -23.1 m^3/min.

(c) The estimated exact rate at t=1 is -2 m^3/min.

(d) The average rate the water leaves the tub between t=1 and t=1+h is -2 - h m^3/min.

(e) The result when h=0 in part (d) is -2 m^3/min.

(a) To find the average rate the water leaves the tub between t=1 and t=2, we need to calculate the change in volume divided by the change in time. The change in volume is f(2) - f(1) = (4 - 2^2) - (4 - 1^2) = 1 m^3. The change in time is 2 - 1 = 1 min. Therefore, the average rate is 1 m^3/min.

(b) Similarly, for t=1 to t=1.1, the change in volume is f(1.1) - f(1) = (4 - 1.1^2) - (4 - 1^2) ≈ 0.69 m^3. The change in time is 1.1 - 1 = 0.1 min. The average rate is 0.69 m^3/0.1 min ≈ 6.9 m^3/min.

(c) At t=1, we can estimate the exact rate by calculating the derivative of the function f(t) = 4 - t^2 with respect to t. The derivative is -2t, so at t=1, the rate is -2 m^3/min.

(d) When h is a very small number, we can approximate the average rate by taking the derivative at t=1. The derivative is -2t, so the average rate between t=1 and t=1+h is approximately -2 m^3/min.

(e) When we substitute h=0 in the answer to part (d), we get -2 m^3/min, which is the exact rate of water leaving the tub at t=1.

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λn = (nπ)^2 - 1, and the corresponding eigenfunction is y_n(x) = B sin(nπ x).

The general solution of the differential equation is of the form

y(x) = A sin(√(λ+1) x) + B cos(√(λ+1) x).

Applying the boundary condition y'(0) = 0, we have:

y'(x) = A√(λ+1) cos(√(λ+1) x) - B√(λ+1) sin(√(λ+1) x)

y'(0) = A√(λ+1) cos(0) - B√(λ+1) sin(0) = 0

Here  A = 0.

Applying  the boundary condition y'(1) = 0, we have:

y'(x) = - B√(λ+1) sin(√(λ+1) x)

y'(1) = - B√(λ+1) sin(√(λ+1)) = 0

Which means that √(λ+1) = nπ for n = 1, 2, 3, ...

In conclusiuon,  λn = (nπ)^2 - 1, and the corresponding eigenfunction is y_n(x) = B sin(nπ x).

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Answer:

80

Step-by-step explanation:

8 different colors multiplied by 10 different patterns

Answer: I believe it’s D

Step-by-step explanation:

The system of equations to represent the total cost, y, when x additional pieces are purchased with a basic cat tree at each store are;

y = 50x + 100    ------(eq 1)

y = 75x + 50    ------(eq 2)

A) Let the total cost of the cat tree be y

Let each additional piece added be x.

Since basic tree is $100 and each additional platform costs $50, then we can say that;

y = 50x + 100    ------(eq 1)

Now, their basic cat tree starts at $50 and each additional piece costs $75. Thus;

y = 75x + 50    ------(eq 2)

B) Subtract eq 1 from eq 2 to gte;

25x - 50 = 0

25x = 50

x = 50/25

x = 2

y = 75(2) - 50

y = $100

C) The Store that would be cheaper depending on the add - ons is Puuur Place because as x increases, it's y-value increases at a lesser rate than that of Meow.

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x=-1/2

Step-by-step explanation:

if you want explanation let me know?

The statement that the National Vulnerability Database (NVD) is responsible for actively performing vulnerability testing for every company software and hardware infrastructure is false.

The National Vulnerability Database (NVD) is a repository of security vulnerabilities and related information. It serves as a comprehensive database that provides information on known vulnerabilities in various software and hardware products. However, the NVD itself is not responsible for actively performing vulnerability testing for every company's software and hardware infrastructure.

Vulnerability testing is typically conducted by individual companies, organizations, or specialized security firms. Companies and organizations perform their own vulnerability testing or hire external professionals to assess and identify vulnerabilities in their software and hardware systems. They may use a variety of methods, such as penetration testing, code reviews, and vulnerability tools, to uncover potential security weaknesses.

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the solution for the initial value problem is: u(x, t) = sin(sqrt(-λ² * (a / 4)) * x) * exp(-λ² * t) where λ = ± sqrt(-4n² / a), and n is a non-zero integer.

The given partial differential equation is:

au(x, t) = 4 * (d²u(x, t) / dt²) / (dx²)

a) BCs (Boundary Conditions):

We have u(0, t) = 0 and u(π, t) = 0.

IC (Initial Condition):

We have u(x, 0) = x.

To solve this initial value problem, we need to find a function f(x) that satisfies the given boundary conditions and initial condition.

The solution for f(x) can be found using the method of separation of variables. Assuming u(x, t) = X(x) * T(t), we can rewrite the equation as:

X(x) * T'(t) = 4 * X''(x) * T(t) / a

Dividing both sides by X(x) * T(t) gives:

T'(t) / T(t) = 4 * X''(x) / (a * X(x))

Since the left side only depends on t and the right side only depends on x, both sides must be equal to a constant value, which we'll call -λ².

T'(t) / T(t) = -λ²

X''(x) / X(x) = -λ² * (a / 4)

Solving the first equation gives T(t) = C1 * exp(-λ² * t), where C1 is a constant.

Solving the second equation gives X(x) = C2 * sin(sqrt(-λ² * (a / 4)) * x) + C3 * cos(sqrt(-λ² * (a / 4)) * x), where C2 and C3 are constants.

Now, applying the boundary conditions:

1) u(0, t) = 0:

Plugging in x = 0 into the solution X(x) gives C3 * cos(0) = 0, which implies C3 = 0.

2) u(π, t) = 0:

Plugging in x = π into the solution X(x) gives C2 * sin(sqrt(-λ² * (a / 4)) * π) = 0. To satisfy this condition, we need the sine term to be zero, which means sqrt(-λ² * (a / 4)) * π = n * π, where n is an integer. Solving for λ, we get λ = ± sqrt(-4n² / a), where n is a non-zero integer.

Now, let's find the expression for u(x, t) using the initial condition:

u(x, 0) = X(x) * T(0) = x

Plugging in t = 0 and X(x) = C2 * sin(sqrt(-λ² * (a / 4)) * x) into the equation above, we get:

C2 * sin(sqrt(-λ² * (a / 4)) * x) * C1 = x

This implies C2 * C1 = 1, so we can choose C1 = 1 and C2 = 1.

Therefore, the solution for the initial value problem is:

u(x, t) = sin(sqrt(-λ² * (a / 4)) * x) * exp(-λ² * t)

where λ = ± sqrt(-4n² / a), and n is a non-zero integer.

Note: Please double-check the provided equation and ensure the values of a and the given boundary conditions are correctly represented in the equation.

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Step-by-step explanation:

Σ^n=1 -144 (1/2)n-1

This is a finite geometric series with n

=4, a₁-144, and r = 2.

S = a₁ (1-r") / (1 − r)

S-144 (1 - (1/2)) / (1 - 1/2)

S=-270

If you wanted to find the infinite sum (n

= ∞0)

:S = a₁ / (1 − r)

S-144/(1-1/2)

S=-288

a) To get the maximum volume of the box, we should cut squares with side length of 5.66cm from each corner of the cardboard.

b) The area of each square is 31.96 square cm.

Let's denote the side length of the square that needs to be removed from each corner as "x"

The length of the cardboard box, after removing the squares from each corner, will be

L = 119cm - 2x

Similarly, the width of the cardboard box will be

W = 24cm - 2x

And the height of the cardboard box will be

H = x

To maximize the volume of the box, we need to find the value of "x" that maximizes the expression for the volume of the box, which is given by

V = LWH

Substituting the expressions for L, W, and H in terms of x, we get

V = (119cm - 2x)(24cm - 2x)(x)

Expanding this expression gives

V = 4x^3 - 286x^2 + 2856x

To find the value of "x" that maximizes this expression, we can take the derivative of V with respect to x and set it equal to zero

dV/dx = 12x^2 - 572x + 2856 = 0

Solving for x gives

x = (572 ± sqrt(572^2 - 4(12)(2856)))/(2(12))

x ≈ 5.66cm (ignoring the negative root)

Therefore, to get the maximum volume of the box, we should cut squares with side length of 5.66cm from each corner of the cardboard.

The area of each square is x^2 = 5.66^2 = 31.96 square cm.

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Answer:

October

Step-by-step explanation:

because June and September have 30

days

Answer:

A - October

Step-by-step explanation:

⇒ October is the month which has 31 days

Answer:

Well I only know 1 and its 3/5

Step-by-step explanation:

Answer:

3

Step-by-step explanation:

Slope is rise over run which is 6/2=3

Answer:

Yes, your slope/height is correct. 6/2 simplified is 3

Fixed, been a while since I have done slope