Consider the equation below. (If an answer does not exist, enter DNE.)
f(x) = x2 − x − ln(x)
(a)
Find the interval(s) on which f is increasing. (Enter your answer using interval notation.)

Find the interval(s) on which f is decreasing. (Enter your answer using interval notation.)

(b)
Find the local minimum and maximum value of f.
local minimum value

local maximum value

(c)
Find the inflection point.
(x, y) =




Find the interval(s) on which f is concave up. (Enter your answer using interval notation.)

Find the interval(s) on which f is concave down. (Enter your answer using interval notation.)

A rigid transformation and the three types of rigid transformations are: B. Rigid Transformations have congruent preimages and images. Examples include translations, reflections, and rotations.

In Mathematics and Geometry, a transformation can be defined as the movement of a point from its initial position to a new location. This ultimately implies that, when a geometric figure or object is transformed, all of its points would also be transformed.

Generally speaking, there are three (3) main types of rigid transformation and these include the following:

In conclusion, rigid transformation are movement of geometric figures where the size and shape does not change because they have congruent preimages and images.

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The restrictions on the polynomial expression [tex]\frac{x^2 - 25}{x - 1} \div \frac{x^2 - x - 30}{x^2 - 4x - 12}[/tex] are at x = -5, -2, 1 and 6

From the question, we have

[tex]\frac{27x^2y^3}{45x^4}[/tex]

Divide the variables

So, we have

[tex]\frac{27y^3}{45x^2}[/tex]

Divide 27 and 45 by 9

So, we have

[tex]\frac{3y^3}{5x^2}[/tex]

Hence, the solution is [tex]\frac{3y^3}{5x^2}[/tex], x ≠ 0

Given that

[tex]\frac{x + 2}{x^2 - 5x - 14}[/tex]

Factorize the numerator

So, we have

[tex]\frac{x + 2}{(x + 2)(x - 7)}[/tex]

Divide

[tex]\frac{1}{x - 7}[/tex]

So, the solution is [tex]\frac{1}{x - 7}[/tex] , where x ≠ 7

The hole is given as (2, 1/3)

This means that the graph is undefined at (2, 1/3)

One possible equation from the list of options is

[tex]f\left(x\right)\:=\:\frac{x\:-\:2}{x^2\:-\:x\:-\:2}[/tex]

The expression is given as

[tex]\frac{x^2 - 25}{x - 1} \div \frac{x^2 - x - 30}{x^2 - 4x - 12}[/tex]

The restrictions on the polynomial is the domain

When solved graphically, we have the restrictions to be at x = -5, -2, 1 and 6

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The ratio of the width to the length of the cell phone is 23/65.

Ratio is the quantitative relation between two amounts showing the number of times one value contains or is contained within the other.

To find the ratio of the width to the length of the cell phone, we use the formula below.

Formula:

Where:

From the question,

Given:

Substitute these values into equation 1

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The equations for which 8 is a solution:

are x minus 4 = 4

2 x = 4

StartFraction x Over 8 EndFraction = 1

x + 6 = 2

The equations for which 8 is a solution are:

x - 4 = 4 (if we substitute x=8, we get 8-4=4 which is true)

2x = 4 (if we substitute x=8, we get 2*8=16 which is true)

StartFraction x Over 2 EndFraction = 16 (if we substitute x=8, we get 8/2=4 which is not true)

StartFraction x Over 8 EndFraction = 1 (if we substitute x=8, we get 8/8=1 which is true)

So, the correct answers are:

x minus 4 = 4

2 x = 4

StartFraction x Over 8 EndFraction = 1

x + 6 = 2

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Answer: The rate at which the length of each edge is changing is approximately 0.5185 inches per second when the edges are 6 inches long.

Step-by-step explanation:

Let's denote the volume of the cube as V and the length of each edge as x. Given that the volume of a cube is V = x^3, we can find the rate at which the length of each edge is changing.

We're given that the rate of change of the volume is dV/dt = 56 in³/sec. We want to find the rate of change of the length of each edge, which is dx/dt, when the length of each edge is 6 inches.

First, we differentiate the volume equation with respect to time t:

V = x^3

dV/dt = d(x^3)/dt

Using the chain rule:

dV/dt = 3x^2 * (dx/dt)

Now, we know that dV/dt = 56 in³/sec and x = 6 in. Plugging these values into the equation, we get:

56 = 3 * (6)^2 * (dx/dt)

Solving for dx/dt:

56 = 108 * (dx/dt)

dx/dt = 56 / 108

dx/dt ≈ 0.5185 in/sec (rounded to four decimal places)

So, the rate at which the length of each edge is changing is approximately 0.5185 inches per second when the edges are 6 inches long.

The inequality that would represent the possible values for the number of hours walking dogs is 9d + 8c ≥ 130

Inequalities in mathematics can be described as one that is been used in the expression of the relationship between two values that are not equal

It should be noted thed that Inequality  implies not equal with the symbol (≠)” symbols ≥ < > ≤. From the question , number of hours walking dogs (d) Let number of hours clearing tables  an be represented by( c) Hence the  appropriate inequality is 9d + 8c ≥ 130.

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The required values are as follows:

Center = (-5, 6), Radius = 12, Domain: x ∈ (-17, 7), Range: y ∈ (-6, 18)

Given equation: x² + y² + 10x - 12y - 83 = 0

To convert it into standard form, complete the square for both the x and y terms:

(x² + 10x) + (y² - 12y) = 83

(x² + 10x + 25) + (y² - 12y + 36) = 83 + 25 + 36

(x + 5)² + (y - 6)² = 144

Comparing this with the standard form, we have:

Center (h, k) = (-5, 6)

Therefore, the coordinates for the point J are (-5, 6).

The radius represents the distance from each band member to the center of the circle. In this case, the radius of the circle is the square root of the constant term on the right side of the standard form equation:

Radius = √144 = 12

The domain of the circle is the set of all possible x-values that lie on the circle. Since the circle is centered at (-5, 6) and has a radius of 12, the domain can be expressed as:

Domain: x ∈ (-5 - 12, -5 + 12) or -17 ≤ x ≤ 7

The range of the circle is the set of all possible y-values that lie on the circle. Using the same information, the range can be expressed as:

Range: y ∈ (6 - 12, 6 + 12) or -6 ≤ y ≤ 18

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The true statements include

Rigid transformation is a type of geometric transformation in which the size and shape of an object remain unchanged. Rigid transformations include

In a rigid transformation, the position of the object in space may change, but its size, shape, and orientation remain the same.

In the object in the graph the length of the segments remain same and since the translation affects only y direction the x-coordinate will be unchanged.

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Answer:3/10

Step-by-step explanation:

Answer:

7

Step-by-step explanation:

Answer: The two numbers are 5 and 12

Step-by-step explanation:

As the least common multiple of two numbers is

60

, the two numbers are factors of

60

.

Factors of

60

are

{

1

,

2

,

3

,

4

,

5

,

6

,

10

,

12

,

15

,

20

,

30

,

60

}

As one of the numbers is

7

less than the other, the difference of two numbers is

7

Among

{

1

,

2

,

3

,

4

,

5

,

6

,

10

,

12

,

15

,

20

,

30

,

60

}

,

3

&

10

and

5

&

12

are the only two pair of numbers whose difference is

7

. But Least common multiple of

3

and

10

is

30

.

Hence, the two numbers are

5

and

12

.

Answer: The first number is the X-axis and the second is the Y-axis.

Step-by-step explanation:

5, 1, 4, and 2 belong in X. 35, 7, 28, and 14 belong in Y.

The rule is that X always goes first the Y goes after.

The shape resulting from a horizontal cross section, a vertical cross section, and an angled cross section of the cube are:

horizontal cross section: square.

vertical cross section: square.

angled cross section: rectangle.

In this exercise, you are required to use a graphing tool to investigate and determine the cross-sections of the given three-dimensional geometric object (cube), by passing different planes through them.

By critically observing the cross-sections of the three-dimensional geometric object (cube) I used, we can reasonably infer and logically deduce the following types of quadrilateral based on the cross-sections;

A horizontal cross section would produce a square.

A vertical cross section would produce a square.

An angled cross section would produce a rectangle.

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The statements that are always true regarding the diagram are:

m∠1 + m∠5 = 180°

m∠2 + m∠3 = m∠6

m∠3 + m∠4 + m∠5 = 180°

Explain the term diagram

It refers to a value or position that is higher or greater than another value or position. It is often used in comparisons or to describe the relative positions of objects or points on a graph or coordinate plane. The opposite of "above" is "below," which refers to a value or position that is lower or lesser.

According to the given information

We know that the sum of the measures of the three interior angles of a triangle is always 180°. Therefore, we have:

m∠2 + m∠3 + m∠5 = 180°

From the given information, we know that:

m∠2 + m∠1 = 180° (linear pair of angles)

m∠3 + m∠4 = 180° (linear pair of angles)

m∠5 + m∠6 = 180° (exterior angle theorem)

Rearranging these equations, we get:

m∠1 = 180° - m∠2

m∠4 = 180° - m∠3

m∠6 = 180° - m∠5

Substituting these expressions into the above equation for the sum of the interior angles, we get:

m∠2 + m∠3 + m∠5 = m∠2 + (180° - m∠1) + m∠5

m∠3 + m∠4 + m∠5 = (180° - m∠3) + m∠4 + m∠5

Simplifying these equations, we get:

m∠1 + m∠5 = 180°

m∠2 + m∠3 = m∠6

Therefore, the statements that are always true regarding the diagram are:

m∠1 + m∠5 = 180°

m∠2 + m∠3 = m∠6

m∠3 + m∠4 + m∠5 = 180° (which is equivalent to option 2)

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The figure that represent the triangle described in the question is option (C).

Triangle is a basic geometric shape that consists of three straight sides and three angles.

We can say triangle is a polygon with three sides and three vertices, and is one of the simplest and most fundamental shapes in geometry.

Class of Triangle based on sides

Class of Triangle based on their angles

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Answer:

-36

Step-by-step explanation:

To solve this problem, we need to integrate f'(x) to find f(x), and then evaluate f(3) and f(-1) to compute the expression f(3) - f(-1).

Integrating f'(x)=-4x - 5 with respect to x, we get:

f(x) = -2x^2 - 5x + C

Where C is the constant of integration.

To find the value of C, we can use the initial condition f(0) = 1. Substituting x = 0 and f(x) = 1 into the equation above, we get:

1 = -2(0)^2 - 5(0) + C

1 = C

So, we have:

f(x) = -2x^2 - 5x + 1

Now, we can evaluate f(3) and f(-1) and compute f(3) - f(-1):

f(3) = -2(3)^2 - 5(3) + 1 = -32

f(-1) = -2(-1)^2 - 5(-1) + 1 = 4

f(3) - f(-1) = (-32) - 4 = -36

Therefore, f(3) - f(-1) = -36.

Answer:

  -36

Step-by-step explanation:

You want F(3) -F(1) given f'(x) = -4x -5.

The desired difference is the definite integral ...

  [tex]\displaystyle \int_{-1}^3{(-4x-5)}\,dx=\left[-2x^2-5x\right]_{-1}^3=-2(9-1)-5(3+1)\\\\=\boxed{-36}[/tex]

<95141404393>

The probability that all 3 children will be born with the disease is approximately 0.195 or 19.5%.

We have,

Assuming that the probability of a child being born with the disease is independent of whether or not their siblings are born with the disease, we can use the multiplication rule of probability:

P(all 3 children have the disease)

= P(first child has the disease) x P(second child has the disease) x P(third child has the disease)

Since each child has a 57% chance of being born with the disease:

P(all 3 children have the disease)

= 0.57 x 0.57 x 0.57

Now,

P(all 3 children have the disease)

= 0.195

Therefore,

The probability that all 3 children will be born with the disease is approximately 0.195 or 19.5%.

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The volume of the block of wood is 104.7634 cm³ and the total surface area is 186.4502 cm².

Volume is the measure of the amount of space occupied by a three-dimensional object. It is typically measured in cubic units such as cubic meters (m³), cubic centimeters (cm³), or cubic inches (in³).

The volume of the block of wood will be;

V = l x w x h = 4.55 cm x 9.1 cm x 2.54 cm

= 104.7634 cm³

Therefore, the volume of the block of wood is 104.7634 cm³.

The total surface area of the block of wood will be;

SA = 2lh + 2wh + 2lw

= 2(4.55 cm x 2.54 cm) + 2(9.1 cm x 2.54 cm) + 2(4.55 cm x 9.1 cm)

= 58.0094 cm² + 46.4314 cm² + 82.0094 cm²

= 186.4502 cm²

Therefore, the total surface area is 186.4502 cm².

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Answer: The value of "a" is given by a = 25 - 2b, where "b" is a positive number.

Step-by-step explanation:

Given that log(a) = log15(b) = log25(a + 2b), we can use the properties of logarithms to solve for the value of a.

Since log(a) = log15(b), we can equate the bases and eliminate the logarithms:

a = 15^log15(b) .....(1)

Similarly, since log(a) = log25(a + 2b), we can equate the bases and eliminate the logarithms:

a = (a + 2b)^log25(a + 2b) .....(2)

Now, we can equate the right-hand sides of equations (1) and (2) since they are both equal to a:

15^log15(b) = (a + 2b)^log25(a + 2b)

Taking the logarithm of both sides with base 15, we get:

log15[15^log15(b)] = log15[(a + 2b)^log25(a + 2b)]

Using the property that loga(a^x) = x, we can simplify the left-hand side:

log15(b) = log15[(a + 2b)^log25(a + 2b)]

Now, we can equate the bases and eliminate the logarithms:

b = (a + 2b)^log25(a + 2b)

Taking the logarithm of both sides with base (a + 2b), we get:

log(a + 2b)(b) = log(a + 2b)[(a + 2b)^log25(a + 2b)]

Using the property that loga(a^x) = x, we can simplify the right-hand side:

log(a + 2b)(b) = log25(a + 2b)

Since log(a + 2b)(b) = log(a + 2b)/logb(a + 2b) by the change of base formula, we can rewrite the equation as:

log(a + 2b)/logb(a + 2b) = log25(a + 2b)

Now, we can equate the numerators and denominators separately:

log(a + 2b) = log25(a + 2b)

1 = log25(a + 2b)/(log(a + 2b))

Since loga(a) = 1, we can rewrite the equation as:

log25(a + 2b) = log(a + 2b)/(log(a + 2b))

Using the property that loga(a^x) = x, we get:

log25(a + 2b) = 1

This implies that 25^1 = a + 2b, since we are using the definition of logarithm which states that loga(b) = c is equivalent to a^c = b.

Therefore, a + 2b = 25.

Given that a and b are positive numbers, we can deduce that a + 2b > 0.

Solving for a, we get:

a = 25 - 2b

Since a and b are both positive, a = 25 - 2b > 0.

So, the value of a is greater than zero and is given by a = 25 - 2b, where b is a positive number.

The missing values of function rule, Y for corresponding value of X= (-4 ) and (-2) will be 9 and 3 respectively.

                                              [tex]X|-4 |- 2|\\ Y|\quad9\; | \quad3\;|[/tex]

Now for the given fucntion ,

By substituting the values of x into the equation [tex]y = -3x - 3[/tex] and solving for y, we can determine the values of y for [tex]x = -4\; and -2[/tex].

[tex]For \;x = -4:\\y = -3(-4) - 3\\y = 12 - 3\\y = 9[/tex]

Hence, for x =(-4), y= 9

[tex]For\; x = -2:\\y = -3(-2) - 3\\y = 6 - 3\\y = 3[/tex]

Hence, for x =( -2), y = 3

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Correct Question:Fill in the table using this function rule. y = - 3x-3

[tex]X|-4 |- 2|\\ Y|\quad?\; | \quad?\;|[/tex]

a) [tex]f(x) + xg(x) = x^{3} + 7x^{2} + x - 4.[/tex]

b) [tex][f(x)]² = 4x^{4} - 4x^{3} - 15x^{2} + 8x + 16[/tex]

c) [tex]f^{2}(x) = 4x^{4} - 4x^{3} - 15x^{2} + 8x + 16[/tex]

d) [tex]gf(x) = 2x^{4} + 8x^{3} - 13x^{2} - 3x - 8.[/tex]

what is expression ?

It is possible to do mathematical operations like addition, subtraction, division, and multiplication. An expression is put together as follows: Number, expression, and mathematical operator.

a) f(x) + xg(x)

First, we need to find xg(x), which is x multiplied by g(x):

[tex]xg(x) = x(x^{2} + 5x + 2) = x^{3} + 5x^{2} + 2x\\\\f(x) + xg(x) = 2x^{2} - x - 4 + x^{3} + 5x^{2} + 2x\\\\f(x) + xg(x) = x^{3} + 7x^{2} + x - 4[/tex]

Therefore, [tex]f(x) + xg(x) = x^{3} + 7x^{2} + x - 4.[/tex]

b) [f(x)]²

To square f(x), we can simply multiply it by itself:

[f(x)]² = (2x² - x - 4)(2x² - x - 4)

[tex][f(x)]² = 4x^{4} - 4x^{3} - 15x^{2} + 8x + 16[/tex]

c) f²(x)

Since f²(x) means f(x) times f(x), we can use the distributive property of multiplication:

[tex]f^{2} (x) = f(x) * f(x)\\\\f^{2}(x) = (2x^{2} - x - 4)(2x^{2} - x - 4)[/tex]

[tex]f^{2}(x) = 4x^{4} - 4x^{3} - 15x^{2} + 8x + 16[/tex]

d) gf(x)

To find gf(x), we need to multiply g(x) by f(x):

[tex]gf(x) = (x^{2} + 5x + 2)(2x^{2} - x - 4)[/tex]

Multiplying each term of g(x) by each term of f(x), and simplifying:

gf(x) = 2x⁴ + 8x³ - 13x² - 3x - 8

Therefore, [tex]gf(x) = 2x^{4} + 8x^{3} - 13x^{2} - 3x - 8.[/tex]

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Felix has spent a total time of 11.4 hours (or 684 minutes) running this month.

we can calculate the total time Felix has spent running this month.

Given:

Warm-up time: 12 minutes

Running time: 20 minutes

Total exercise time in a month: 240 minutes

Step 1: Calculate the total time spent running

Let x be the total time spent running in minutes.

According to the given information, Felix spends 12 minutes on warm-up and 20 minutes on running.

So, the equation would be:

12 + 20x = 240

Step 2: Solve for x

Subtract 12 from both sides of the equation:

20x = 240 - 12

20x = 228

Divide both sides by 20:

x = 228 / 20

x = 11.4

So, Felix has spent a total of 11.4 hours (or 684 minutes) running this month.

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Answer:

Step-by-step explanation:

We can simplify the given product using the properties of radicals:

√10x² √5x = √(10*5)x^(2+1) = √50x^3

We can further simplify the radical by factoring out the largest perfect square from 50, which is 25:

√50x^3 = √(25*2)x^3 = 5x√2

Therefore, the product √10x² √5x is equivalent to 5x√2, when x≥0.

To simplify the product √10x² √5x, we can combine the square roots and simplify:

√10x² √5x = √(10x² * 5x) = √(50x³) = √(25x² * 2x) = 5x√2x

Therefore, the choice that is equivalent to the original product when x≥0 is A. 5x√2x.

Answer:

1228

Step-by-step explanation:

To find the total square footage of carpet needed for the house, we first need to calculate the total square footage of the areas where we do not need carpet and then subtract that from the total square footage of the house.

The kitchen is 8 ft by 10 ft, so its area is:

8 ft x 10 ft =80 square feetThe dining room is 12 ft by 14, ft so its area is:

12 ft x 14 ft = 168 square feetThe bathroom is 4 ft by 6 ft, so its area is:

4 ft x 6 ft = 24 square feetTherefore, the total square footage of the areas where we do not need carpet is:

80 + 168 + 24 = 272 square feetTo find the total square footage of carpet needed for the house, we can subtract this from the total square footage of the house:

1500 - 272 = 1228 square feetTherefore, we need 1228 square feet of carpet for the house.

To calculate the amount of carpet needed, subtract the square footage of the rooms where you don't want new carpet (kitchen, dining room, bathroom) from the total house size. This results in 1228 square feet of carpet required.

First, let's determine the total square footage of the spaces where you don't want new carpet. The kitchen is 8 ft by 10 ft, which equals 80 square feet. The dining room is 12ft by 14ft, giving us 168 square feet. The bathroom is 4ft by 6 ft, equal to 24 square feet. When you add these together, that is a total of 272 square feet that should not be carpeted.

Since your total house size is 1500 square feet, we subtract the square footage of the kitchen, dining room, and bathroom from this total. So, 1500 square feet - 272 square feet equals 1228 square feet. Therefore, you need 1228 square feet of carpet for the house.

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Therefore, the value of f(m-2) is 5/(m-2) + 2, and the value of function g(1/2) is 2.

In mathematics, a function is a rule or correspondence between two sets, where each input value from the first set (called the domain) corresponds to exactly one output value in the second set (called the range). A function is often represented by a formula, equation, or graph.

Here,

1. To find f(m-2), we substitute (m-2) for x in the expression for f(x):

f(m-2) = 5/(m-2) + 2

So the value of f(m-2) is 5/(m-2) + 2.

2. To find g(1/2), we substitute 1/2 for x in the expression for g(x):

g(1/2) = -2(1/2) + 3

So the value of g(1/2) is -1 + 3, which simplifies to 2.

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The function that has a range of all real numbers is given as follows:

f(x) = -x + 2.

Hence the range for each function in the context of this problem is given as follows:

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The only option given that has a variable raised to the third power is D. -2x³, so the correct answer is D.

In algebra, a monomial is an expression that consists of a single term, which is a product of a constant coefficient and one or more variables raised to non-negative integer powers. In each of these expressions, there is only one term, and each term is a product of a constant coefficient and one or more variables raised to non-negative integer powers. Monomials are important in algebra because they can be combined using various operations such as addition, subtraction, multiplication, and division, to form more complex expressions.

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Karla will have approximately $11,149 in the account after 5 years, rounded to the nearest dollar.

The formula accrued amount in a compounded interest is expressed as;

A = P( 1 + r/n )^( n × t )

Where A is accrued amount, P is principal, r is interest rate and t is time.

Given that:

First, convert R as a percent to r as a decimal

r = R/100

r = 2.2/100

r = 0.022

Plug the given values into the above formula and solve for A.

A = P( 1 + r/n )^( n × t )

A = 10,000( 1 + 0.022/1 )^( 1 × 5 )

A = 10,000( 1 + 0.022 )^( 5 )

A = 10,000( 1.022 )^( 5 )

A = $11,149

Therefore, the accrued amount after 5 years is $11,149.

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Answer:

$526.25, and your total monthly expense (including rent and maintenance fee) is $2,326.25.

Step-by-step explanation:

To calculate your average monthly expense, we need to first add up all of your annual expenses and divide by 12 (the number of months in a year) to find the monthly average.

Annual expenses:

Insurance: $325

Lost interest: $50

Utility bills: $5,940

Total annual expenses = $325 + $50 + $5,940 = $6,315

To find the average monthly expense, we divide the total annual expenses by 12:

Average monthly expense = Total annual expenses / 12

Average monthly expense = $6,315 / 12

Average monthly expense = $526.25

In addition to the annual expenses, you also pay a monthly rent of $1,700 and a monthly fee for all maintenance. Let's assume that the monthly maintenance fee is $100.

Therefore, your total monthly expense would be:

Total monthly expense = Monthly rent + Monthly maintenance fee + Average monthly expenses

Total monthly expense = $1,700 + $100 + $526.25

Total monthly expense = $2,326.25

So, your average monthly expense is $526.25, and your total monthly expense (including rent and maintenance fee) is $2,326.25.