1) Determine the average rate of change of the function on the given interval.
f(x)=x²-x² + 3 on [0,2]

2) Find the difference quotient,
f(x) = 3x²-x+4
f(x+h)-f(x), for the given function.
h

3) Solve the system of equations.
2x + y = 8
-4x + 2y = 13

in essence, PEMDAS, but let's firstly convert the mixed fractions to improper fractions.

[tex]\stackrel{mixed}{7\frac{2}{3}}\implies \cfrac{7\cdot 3+2}{3}\implies \stackrel{improper}{\cfrac{23}{3}} ~\hfill \stackrel{mixed}{1\frac{2}{4}} \implies \cfrac{1\cdot 4+2}{4} \implies \stackrel{improper}{\cfrac{6}{4}} \\\\\\ \stackrel{mixed}{3\frac{6}{8}}\implies \cfrac{3\cdot 8+6}{8}\implies \stackrel{improper}{\cfrac{30}{8}} \\\\[-0.35em] ~\dotfill[/tex]

[tex]\cfrac{23}{3}-\left(\cfrac{6}{4}+\cfrac{30}{8} \right)\implies \cfrac{23}{3}-\left( \cfrac{(2)6~~ + ~~(1)30}{\underset{\textit{using this LCD}}{8}} \right)\implies \cfrac{23}{3}-\left( \cfrac{12+30}{8} \right)[/tex]

[tex]\cfrac{23}{3}-\left( \cfrac{42}{8} \right)\implies \cfrac{23}{3}- \cfrac{42}{8}\implies \cfrac{(8)23~~ - ~~(1)42}{\underset{\textit{using this LCD}}{24}}\implies \cfrac{184-42}{24} \\\\\\ \cfrac{142}{24}\implies \cfrac{2\cdot 71}{2\cdot 12}\implies \cfrac{2}{2}\cdot \cfrac{71}{12}\implies \cfrac{71}{12}\implies 5\frac{11}{12}[/tex]

Step-by-step explanation:

To find the total volume of the solution in liters, we need to add the volumes of the two ingredients:

500 mL + 1000 mL = 1500 mL

To convert mL to liters, divide by 1000:

1500 mL / 1000 = 1.5 L

So, the solution is 1.5 liters.

Answer: -13

Step-by-step explanation: Here to help! So first, you do 8-12, then you get -4, right? So then, you multiply 2 times -4, then you get -8. After that, you add -5 to -8, so it's -5 + -8, and you get -13.

Answer:

-13

Step-by-step explanation:

-5+2(8-12)

distribute the 2

-5+16-24

add

11-24

subtract

-13

Answer:

6

Step-by-step explanation:

30 divided by 5 = 6

6 is the maximum possible number of groups possible.

The circumference of the circle shown above would be = 6πft. That is option A.

The formula which can be used to calculate tye circumference of a circle = 2πr

Where;

r = 3ft

π = to be calculated in terms of π

Circumference of the circle = 2×π × 3 = 6πft.

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The average rate of change of a function f(x) on the interval [4, 9] is -67.

What is the average rate of change?

The average rate of change of a function between two points is the same that the slope of the line passes through these points (secant line).

The given table:

The average rate of change of a function f(x) over the interval [a, b] is

= f(b) - f(a)/b-a.

Interval : [4, 9]

f(9) = -419,

f(4) = -84

The average rate of change of a function f(x) on the interval [4, 9] is

=  f(9) - f(4)/9 - 4.

= -419 - (-84) / 5

= -297/ 5

= -67

Hence, The average rate of change of a function f(x) on the interval [4, 9] is -67.

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The side length of the field after two years (24 months) would be 46368.

The Fibonacci sequence is a series of numbers where each number is the sum of the two preceding ones, starting from 0 and 1: 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, ...

To determine the side length of the field after two years, we can calculate the number of rabbit pairs after two years.

At the end of the first month, there is 1 pair of rabbits.

At the end of the second month, there are 1 + 1 = 2 pairs of rabbits.

At the end of the third month, there are 1 + 1 = 2 pairs of rabbits.

At the end of the fourth month, there are 2 + 1 = 3 pairs of rabbits.

At the end of the fifth month, there are 3 + 2 = 5 pairs of rabbits.

At the end of the sixth month, there are 5 + 3 = 8 pairs of rabbits.

At the end of the seventh month, there are 8 + 5 = 13 pairs of rabbits.

At the end of the eighth month, there are 13 + 8 = 21 pairs of rabbits.

At the end of the 24th month, there are fibonacci(24) = 46368 pairs of rabbits.

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Answer:

The function is growth with a growth rate of 7%

Step-by-step explanation:

The equation is in the form.

y = a * b^x

if b > 1 then it is growth.

If b < 1 it is decay.

Since b > 1 it is growth.  The growth rate is

b = 1+r

1.07 = 1+r

r= .07

7%

The function is growth with a growth rate of 7%

Answer:

A = 81 units²

Step-by-step explanation:

the sides of a square (s) are congruent

given perimeter = 36 , then

s = 36 ÷ 4 = 9

the area (A) of a square is calculated as

A = s²

then

A = 9² = 81 units²

I) The angles are not equal then the two triangles are not Congruent.

ii) The point E is 1.5 from point B.

A line that divides the line into two distinct or equal segments is referred to as a "bisector." It is applied to angles and line segments.

A ray that divides an angle into two equal pieces is known as an angle bisector or the bisector of an angle.

Given:

As, <CAB = 19.4 and <DAB = 33.7

So,  CAD and DAB are not congruent.

ii) The point E should be located 1.5 away from point B.

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The expression which is equivalent to (p - q)(x) is (x²-1) - 5(x-1) .

A function is an expression that shows the relationship between the independent variable and the dependent variable.  A function is usually denoted by letters such as f, g, etc.

Since  p(x) = x² - 1 and g(x) = 5(x-1)

(p - q)(x) can be found by subtracting g(x) = 5(x-1) from p(x) = x² - 1. That is:

(p - q)(x) = (x²-1) - 5(x-1)

Thus,  (x²-1) - 5(x-1) is equivalent to (p - q)(x).

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The square matrix A is invertible, i.e., inverse of A is exist, if and only if the inverse of matrices B and C exists.

We have a matrix A such that

[tex]A = \begin{bmatrix} B & 0 \\0 & C \\ \end{bmatrix}[/tex]

where, B and C are square matrices. We have to prove or show that matrix A is invertible iff B and C are invertible. We shall prove it by using the inverse and determinant of matrices. The determinant is a scalar value which is associated with the square matrix. If X is a matrix, then the determinant of a matrix is denoted by |X|. Inverse of matrix A is calculated by using formula as below,

A = adj(A)/|A|. As we see a matrix A is invertible if and only if it has non-zero determinant. Also for a diagonal matrix its determinant is product of diagonal entries here diagonal entries are again matrices so |A|=|B|×|C|. From here we can clearly says |A| ≠ 0 if and only if |B| ≠ 0 and |C| ≠ 0. Hence, the matrix A is invertible iff B and C are invertible.

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Complete question :

Let A = [ B, 0 ; 0, C], where B and C are square. show that A is invertible if and only if both B and C are invertible.

Hamilton's method of allocating can be used to assign the 21 salespeople in the table in the following order:

2, 5, 6, 8.

Alexander Hamilton was the one who first suggested the strategy that now carries his name. In 1791, Congress approved of his strategy, but President Washington disapproved of it. It was commonly used from 1852 through 1911. He begins by figuring out exactly how many objects each group requires.

We'll use the terminology of shift and hours since the relevant question is about the allocated salesperson in order to calculate the appropriate number of salespeople for each shift.

Now total no. of customers given = 125 + 305 + 439 + 515 = 1375

Now total no of salesperson available = 21.

So, the divisor = 1375/21 = 65.47

Hamilton's approach is now used to determine the number of salespeople assigned by dividing the average number of customers per shift by the divisor.

So, the salesperson assigned are:

Morning = 125/65.47 = 2

Midday = 305/65.47 = 5

Afternoon = 430/65.47 = 6

Evening = 515/65.47 = 8 (Decimals rounded according to Hamilton's principle).

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Answer: 4/1 is 1/4 but what are the 4 fractions your talking about?

Step-by-step explanation:

Answer:

4604 (when rounded to the nearest whole number)

Step by step explanations:

Using the exponential growth formula, we can determine Amuel's population at the time the factory shut down: N = N0 * e^(rt) (rt) Where r is the growth rate, t is the number of years the growth takes place, N is the ultimate population, N0 is the starting population, and r represents growth.

We may use the population data from 1995 and 1996 to get the growth rate: r = ln(1,381 / 1,256) / (1996 - 1995) (1996 - 1995) 

Putting the numbers in: r = 0.117 

Next, we determine how many years the growth took place: t = 13 years (2007 - 1994) Lastly, we enter the values into the formula as follows: N = 1,256 * e^(0.117 * 13) 

Calculating the answer, we discover: 4,604 persons (N = 1,256 * 3.63) Consequently, there were about 4,604 residents there when the facility shut down in 2007.

The five-number summary of the given data 18, 15, 1, 14, 18, 11, 12, 20, 13, 19, 14, 8, 24, 18, 17 is ,

Your data set's five-number summary offers you a general notion of how it is organised. You will, for instance, have your best value and lowest value (the maximum). The main reason you'll want to locate a five-number summary is to find other valuable statistics, like the interquartile range, also known as the middle fifty, even if it's important in and of itself.

The five number summary includes 5 items:

Step 1: Put your numbers in ascending order (from smallest to largest). For this particular data set, the order is:

1, 8, 11, 12, 13, 14, 14, 15, 17, 18, 18, 18, 19, 20, 24.

Step 2: For your data set, determine the lowest and maximum. This ought to be obvious now that your math is correct.

In the example in step 1, the minimum (the smallest number) is 1 and the maximum (the largest number) is 24.

Step 3: Find the median. The median is the middle number.

median = 17

Step 4: (This is not technically necessary, but it makes Q1 and Q3 easier to find).

(1, 8, 11, 12, 13, 14, 14, 15), 17, (18, 18, 18, 19, 20, 24).

Step 5: Find Q1 and Q3. Q1 can be thought of as a median in the lower half of the data, and Q3 can be thought of as a median for the upper half of data.

(1, 8, 11, 12, 13, 14, 14, 15), 17, (18, 18, 18, 19, 20, 24).

Step 6: Write down your summary found in the above steps.

minimum = 1, Q1 = 12, median = 17, Q3 = 19, and maximum = 24.

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Step-by-step explanation:

To find the amount of money represented by C, we need to find the value of A and B first.

Since the ratio A:B is 2:5, we can assume that B is 5 times the amount of A.

Let's assume the amount of money represented by A is x.

Then B = 5x and C = 8x.

We know that the total amount of money split is £370, so we can write an equation:

x + 5x + 8x = £370

14x = £370

x = £26.43

So, the amount of money represented by C is 8x = 8 * £26.43 = £211.44

The value of m in the limit is (-13 + sqrt(169 + 4ε)) / 2, where -84.875 < ε < 1.

To prove that lim as x approaches 6 of (x^2+x) = 42, we need to find a δ > 0 such that if 0 < |x - 6| < δ, then |(x^2+x) - 42| < ε, where ε > 0 is arbitrary.

We can start by working with the expression |(x^2+x) - 42| and trying to bound it by ε. We have:

|(x^2+x) - 42| = |x^2 + x - 6^2 - 6 + 36| = |(x-6)(x+7)|.

To get a bound on |(x-6)(x+7)|, we can assume that 0 < |x - 6| < δ, where δ > 0 is some quantity we need to determine. This means that:

|x-6| < δ.

We can then use the triangle inequality to bound |x+7| as follows:

|x+7| <= |x-6| + |13| < δ + 13.

Thus, we have:

|(x-6)(x+7)| <= |x-6|*|x+7| < δ(δ+13).

Now, we want to choose δ such that δ(δ+13) < ε. Let's choose:

δ = min(ε/m, 1),

where m is some constant we need to determine. We want to make sure that this choice of δ satisfies δ(δ+13) < ε.

Substituting δ = min(ε/m, 1) into the inequality δ(δ+13) < ε, we get:

min(ε/m, 1)(min(ε/m, 1) + 13) < ε.

Simplifying this inequality gives:

min(ε/m, 1)^2 + 13min(ε/m, 1) - ε < 0.

Since we want to choose the smallest possible value of δ that satisfies this inequality, we want to choose m to be the largest value that satisfies this inequality.

The discriminant of the quadratic equation is:

b^2 - 4ac = 13^2 + 4ε.

Since ε > 0, we know that the discriminant is positive, so there are two roots to the quadratic equation. The largest root is:

(-13 + sqrt(169 + 4ε)) / 2.

So, we can choose:

m = (-13 + sqrt(169 + 4ε)) / 2.

Then, we have:

min(ε/m, 1) = ε/((-13 + sqrt(169 + 4ε)) / 2).

Substituting this value of δ into the inequality δ(δ+13) < ε, we get:

ε^2 / (13 - sqrt(169 + 4ε)) < ε.

This simplifies to:

ε < 13 - sqrt(169 + 4ε).

Squaring both sides and rearranging, we get:

4ε^2 + 338ε - 1444 < 0.

This quadratic inequality holds if and only if:

(-338 - sqrt(338^2 + 441444)) / (24) < ε < (-338 + sqrt(338^2 + 441444)) / (24).

Simplifying this gives:

-84.875 < ε < 1.375.

Thus, we have found that if we choose:

m = (-13 + sqrt(169 + 4ε)) / 2,

where:

-84.875 < ε < 1.

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Westside Carpet charges more per square foot for the carpet

Since the Carpet City charges $5 per square foot for a type of carpet. They charge an additional $95 for installation.

we can write the linear function for Carpet City charges as:

y = 5x + 95

where x is the number of square foot

From the table:

rate for Westside Carpet = 3250/642 = $5.06 per carpet

The the linear function for Westside Carpet is y = 5.06x

Thus, Westside Carpet charges more because 5.06 is greater than 5.

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The discriminant is -2n - 24.

The equation has 2 real solutions.

The discriminant of a quadratic equation in the form ax^2 + bx + c = 0 is given by the formula b^2 - 4ac.

So, for the equation -3n^2 - 2n - 6 = 0, a = -3, b = -2, and c = -6.

Plugging these values into the formula, we get:

(-2)^2 - 4 * -3 * -6 = 4 + 72 = 76

So the discriminant is 76, which is greater than zero, so the equation has 2 real solutions.

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The ratios will be equal if the theorem that the product of the extremes equals the product means follows.

Let us understand the equality of ratios through example. The example ratio will be -

7:10 = 21:30

In this ratio, 7 and 30 are first and last numbers and hence they are extremes. The number 10 and 21 are in middle and hence considered mean. Now, we will perform multiplication to if the ratios are equal or not.

Product of extremes = 7 × 30

Extremes product = 210

Product of means = 21 × 10

Means product = 210

Since the products are equal, the ratios are also equal.

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The complete question is -

Are the following ratios equal? write yes or no. use the theroem that the product of the extremes equals the product means.

Ratio = 7:10 and 21:30.

[tex]\begin{array}{llll} \textit{Logarithm of rationals} \\\\ \log_a\left( \frac{x}{y}\right)\implies \log_a(x)-\log_a(y) \end{array} \\\\[-0.35em] ~\dotfill\\\\ \log_8(x)-\log_8(7)=\log_8\left( \cfrac{5}{7} \right)\implies \log_8(\stackrel{x }{5})-\log_8(7)=\log_8\left( \cfrac{5}{7} \right)[/tex]

The best interpretation for this interval is that "She is 99% confident that the population parameter is within the interval" (option C).

In statistics and related fields, the confidence interval refers to a percentage that determines the population fits the interval set. Due to this, a high confidence rate is considered to be positive.

In this case, the confidence interval implies that the researcher is 99% confident that the population parameter is within the interval (option C)

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Answer:

1.5322e+30

Step-by-step explanation:

calculator, there were too many numbers!

Answer:

24 min

Step-by-step explanation:

1 hr 36 min = 96 min

96 min / 4 = 24 min

The required solution of the differential equation is given as y² = -4logx + 9.

The method of determining the derivative of an implicit function by differentiating each term separately, expressing the derivative of the dependent variable as a symbol, and solving the resulting expression for the symbol.

here,
Given the differential equation,
Y’+2/x*y=0
Y' = -2/xy
y (dy/dx) = -2/x
ydy = -2/xdx
y²/2 = -2logx + C

From the initial condition y(1)=3
9/2 = -2log1 + c

c = 9/2

Now,
y²/2 = -2logx + 9/2
y² = -4logx + 9

Thus, the required solution of the differential equation is given as y² = -4logx + 9.

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[tex]\huge{\color{grey}{\underline{\color{grey} {\underline{\color{grey} {\textbf{\textsf{\colorbox{black}{Answer:-}}}}}}}}}[/tex]

Answer:

[tex]y = \frac{3}{2}x + 6[/tex]

Step-by-step explanation:

Rearranging the equation:

[tex]4y = 6x + 24[/tex]

The coefficient of y has to be '+1' for the above equation to be considered as the slope-intercept form.

Dividing both sides of the equation by '4':

[tex]\frac{4}{4}y = \frac{1}{4}(6x + 24)[/tex]

Expand the brackets by applying the Distributive Law:

[tex]\frac{4}{4}y = \frac{6}{4}x + \frac{24}{4}[/tex]

[tex]y = \frac{6}{4}x + 6[/tex]

Divide the numerator and the denominator present in the coefficient of x by the Highest Common Factor '2':

[tex]y = \frac{3}{2}x + 6[/tex]