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The quotient rule is proved below.

In mathematics, the derivative of a function of a real variable measures the sensitivity to change of the function value (output value) with respect to a change in its argument (input value).

Given is the expression -

Q(x) = F(x)/G(x)

Quotient rule in calculus is a method used to find the derivative of any function given in the form of a quotient obtained from the result of the division of two differentiable functions. Mathematically, for a function f(x), we can write -

f'(x) = [u(x)/v(x)]' = [v(x) × u'(x) - u(x) × v'(x)]/[v(x)]²

Now, let f(x) = u(x) and g(x) = v(x).

Q(x) = u(x)/v(x)

Q(x) = u(x) v⁻¹(x)

Using the product rule -

Q'(x) = u'(x) v⁻¹(x) + u(x) {d/dx v⁻¹(x)}

Q'(x) = u'(x) v⁻¹(x) + u(x) (-1)(v(x)⁻²) v'(x)

Q'(x) = u'(x)/v(x) - u(x)v'(x)/v²(x)

Q'(x) = [v(x) × u'(x) - u(x) × v'(x)]/[v(x)]²

Therefore, the quotient rule is proved above.

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The location of Kris' signature on the $601.73 check Elmo wrote to Kris and which Kris deposited into her checking account is on the back of the check. The correct option is option C.

C. Only on the back of the check

A checking account is an account opened at a financial institution into which deposits and withdrawals can be made by a customer. A checking account can also be referred to as a transaction account, a credit unions share draft account, demand deposit account or current account.

The amount of check Elmo wrote Kris = $601.73

The amount Kris deposited into her checking account = The check Elmo issued to her

The location to sign before depositing a check into a checking account is at the back of the check, including informing the cashier that the specified  check is to be deposited into the checking account.

The correct option is therefore option C.

C. Only on the back of the check

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The y-coordinate of point P is 16/3, which rounded to two decimal places is approximately 5.33.

First, we need to find the coordinates of point P. Since point P partitions the directed segment AB in the part-to-part ratio of 2:4, we can use the following formula to find its coordinates:

P = (4A + 2B) / 6

Substituting the coordinates of A and B, we get:

P = (4(5,7) + 2(1,-6)) / 6

P = (20,28) + (2,-12) / 6

P = (22,16) / 6

P = (11/3, 16/3)

y-value = 5.33

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The value of JI in the given triangle is 11.

A triangle is a polygon with three sides, vertices and angles.

Given that, in a triangle DEF, J is the intersection of the three medians.

EI = 33, we are asked to find the value of JI,

Since, all the median is intersecting at J, therefore, J is the centroid.

We know that, the centroid divides each median into two parts, which are always in the ratio 2:1.

Therefore,

Let, EI = 3x,

JI = x,

Therefore, JI = 11

Hence, the value of JI in the given triangle is 11.

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The complete question is :

In Δ DEF, J is the intersection of the three medians. If EI = 33, find JI.

Figure is attached

Answer:

The correct answer is D. y = 3.35x + 73.96.

Step-by-step explanation:

D. y = 3.35x + 73.96

73 degrees is the equivalent meausure of m<A

From the given line geometry below, the sum of the adjacent angles is 180 degrees as shown:

m∠A + m∠B = 180

Given the following

∠A = 6x - 35

∠B = 3x + 53

Substitute the given parameters

6x - 35 + 3x + 53 = 180

9x + 18 = 180

9x = 180 - 18

9x = 162
x = 162/9

x = 18

Determine the measure of <A

<A = 6x - 35

<A = 6(18) - 35

<A = 108 - 35

<A = 73 degrees

Hence the measure of angle A from the given diagram is 73 degrees

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The doubling time for the investment as required to be determined is; 15.58 years.

It follows from the task content that the model function is;

A(T)=650 (1.0455) ^T

For doubling time;

2 = (1.0455)^T

ln (2) = T ln (1.0455)

T = ln 2 / ln (1.0455)

T = 15.58 years.

Ultimately, the doubling time of the investment as required is; 15.58 years.

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For product=6 and sum=7, the numbers or integers  are 1 and 6.

What exactly are integers?

In mathematics, integers are the collection of positive and negative numbers. Integers, like whole numbers, do not include the fractional element. Thus, integers are numbers that can be positive, negative, or zero, but not fractions. On integers, we can do all arithmetic operations such as addition, subtraction, multiplication, and division. Integers include numbers such as 1, 2, 5, 8, -9, -12, and so on. "Z" is the symbol for integers.

now,

As given product=6 and sum=7  

make factors for 6 i.e 1,2,3,6

from these choose the numbers whose sum=7

and they are 1,6.

Hence,

           For product=6 and sum=7, the numbers are 1 and 6.

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Answer:

240 cm, 320 cm, 400 cm

Step-by-step explanation:

The drawing length multiplied by the scale factor gives the real length.

3 cm × 80 = 240 cm

4 cm × 80 = 320 cm

5 cm × 80 = 400 cm

Answer: 240 cm, 320 cm, 400 cm

The probability of choosing a preschooler who is a girl and prefers the color blue is 0.3 or 30%.

The total number of preschoolers who were asked about their color preference is 70. Out of these, 35 girls chose pink, 15 girls chose blue, 5 boys chose pink, and 15 boys chose blue.

To find the probability of choosing a girl who prefers blue, we need to use the formula:

Probability = (Number of favorable outcomes) / (Total number of outcomes)

Here, the favorable outcome is choosing a girl who prefers blue. We can see that 15 girls chose blue as their preferred color. Out of these, we need to choose a girl, so the total number of favorable outcomes is 15.

The total number of outcomes is the total number of preschoolers, which is 70. Out of these, the number of girls is 35+15=50. Therefore, the total number of outcomes where a girl can be chosen is 50.

Putting these values in the formula, we get:

Probability = 15/50 = 0.3

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Complete Question;

70 preschoolers were asked whether they preferred blue or pink.

35 Girls chose pink as preferred color, 15 girls chose blue as preferred color, 5 boys chose pink as preferred color, and 15 boys chose blue as preferred color.

Assuming one of the preschoolers is chosen at random determine the probability of choosing a preschooler that is a girl that prefers the color blue. Round answer to 2 decimal places.

[tex]\begin{array}{ccll} words&minutes\\ \cline{1-2} 90 & 4\\ x& 10 \end{array} \implies \cfrac{90}{x}~~=~~\cfrac{4}{10} \\\\\\ \cfrac{ 90 }{ x } ~~=~~ \cfrac{ 2 }{ 5 }\implies 450=2x\implies \cfrac{450}{2}=x\implies 225=x[/tex]

Answer:

Percentage of adults = 80%

Number of adults = 2000

Step-by-step explanation:

Percentage of men = 2/5 x 100 = 40%

Percentage of women = 2/5 x 100 = 40%

Percentage of adults = 40+ 40= 80%

If there are 2500 people, then number of adults
= 2500 x 80%

= 2500 x 80/100

= 2000 adults

No, the marked price of $73.25 is not correct.

To calculate the sale price, we start by multiplying the original price by the discount percentage:

$95.00 * 0.25 = $23.75

This means that the original price of $95.00 should be reduced by $23.75 to find the sale price:

$95.00 - $23.75 = $71.25

Since the marked price is $73.25, this means that the store made a mistake and overpriced the shoes by $2.00. The correct sale price should have been $71.25.

The values of sec a - cosec a/sec a+cosec a is 4/25.

The trigonometric ratios of any acute angle in a right-angled triangle are the ratios of its sides to that angle.

In a right-angled triangle,

Here we have

tan A = 21/29

As we know Tan θ = Opposite side/Adjacent side  

=> Opposite side = 21

=> Adjacent side = 29

In a right-angle triangle,

=> Hypotenuse² = Opposite side² + Adjacent side²  

=> Hypotenuse² = 21² +29²    

=> Hypotenuse² = 441 + 841    

=> Hypotenuse² = (35.8)²  

Cos a = Adjacent side /Hyopenuse = 29/ 35.8

Sin a = Opposite side/Hypotenuse = 21/35.8

The given expression can be solved as follows

=> [tex]\frac{seca - coseca}{seca+coseca } = \frac{ \frac{1}{cos a}-\frac{1}{sina} }{\frac{1}{cos a }+\frac{1}{sin a} }[/tex]      

=> [tex]\frac{ \frac{sin a - cosa}{cos a sn a} }{\frac{sin a + cos a}{cos a sin a} }[/tex]  

=> [tex]\frac{sin a - cosa}{cos a+ sin a}[/tex]  

=> [tex]\frac{ \frac{21}{35.8} - \frac{29}{35.6} }{\frac{29}{35.8} + \frac{21}{35.8} }[/tex]  

=> [tex]\frac{21- 29}{29+21}[/tex]  

=>  [tex]\frac{8}{50}[/tex]  

=>  [tex]\frac{4}{25}[/tex]  

Therefore,

The values of sec a - cosec a/sec a+cosec a is 4/25.

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If the temperature dropped at a constant rate, it would have dropped 4°C per hour. This can be calculated by taking the difference in temperature (24°C) and dividing it by the number of hours that passed (4 hours).

The set builder notation is A = { 9x | x ∈ N } which is a set of all x’s containing multiples of 9

The union of two sets A and B is the set of all those elements which are either in A or in B, i.e. A ∪ B, whereas the intersection of two sets A and B is the set of all elements which are common. The intersection of these two sets is denoted by A ∩ B

The union of two sets is a new set that contains all of the elements that are in at least one of the two sets

The intersection of two sets is a new set that contains all of the elements that are in both sets

Given data ,

Let the set be represented as A

Now , the value of A is

A = { 9 , 18 , 27 , 36 , ... }

The set A contains all the positive multiples of 9

So , let x be the set of natural numbers and the value of A is

A =  { 9x | x ∈ N } , and

A = set of all x’s containing multiples of 9

Hence , the set builder notation is { 9x | x ∈ N }

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The equation to represent the total cost, y, for x months is y = 12x + 30.

From the given table, we can observe that the total cost increases by $12 for each additional month of usage.

This means that the rate of change, or the slope of the equation, is $12 per month.

We can represent this relationship using the slope-intercept form of a linear equation: y = mx + b, where m is the slope and b is the y-intercept.

In this case, the slope (rate of change) is $12, and the y-intercept can be determined by examining the table. When x = 0 months, the total cost is $30. Therefore, the y-intercept is $30.

Putting it all together, the equation to represent the total cost, y, for x months is:

y = 12x + 30

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The perimeter of the triangle is equal to 28.7 using the trigonometric ratio

The trigonometric ratios is concerned with the relationship of an angle of a right-angled triangle to ratios of two side lengths. The basic trigonometric ratios includes;

sine, cosine and tangent.

tan60 = q/4 {opposite/adjacent}

q = 4 × tan60 {cross multiplication}

q = 6.9282

sin60 = q/r {opposite/hypotenuse}

sin60 = 6.9282/r

r = 6.9282/sin60 {cross multiplication}

r = 8

sin45 = q/s {opposite/hypotenuse}

sin45 = 6.9282/s

s = 6.9282/sin45 {cross multiplication}

s = 9.8

cos45 = p/s {adjacent/hypotenuse}

cos45 = p/9.8

p = 6.9282 {cross multiplication}

perimeter of the triangle = 4 + p + s + r

perimeter of the triangle = 4 + 6.9282 + 9.8 + 8

perimeter of the triangle = 28.7282

Therefore, the perimeter of the triangle is equal to 28.7 using the trigonometric ratio.

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Answer:

Step-by-step explanation:

The terms "4/10, 2/30, n/60" indicate that the buyer can take a discount of 4% if they pay within 10 days from the invoice date, or pay the full amount within 30 days from the invoice date. If the payment is made after 30 days, the buyer must pay the full amount plus any late fees that may apply.

Since the invoice was dated October 15 and the payment was sent on October 28, the payment was made within 30 days from the invoice date, so the buyer cannot take the 4% discount.

The total cost of 50 pairs of shoes at $85 per pair is 50 * 85 = $4250.

Therefore, the total amount of the check would be $4250, which is the full amount of the purchase price without the 4% discount.

Answer:

0.0075

Step-by-step explanation:

Answer:

Below

Step-by-step explanation:

Put '9' where  'n' is and compute

(9)^2 + 6(9) ÷ 3 =

81  +    54 ÷3  =

81  + 18 =   99

Using the a pair of adjacent angles, we will see that:

Z° = 137°

When we have an intersection between two lines, we know that any pair of adjacent angles (they share a side) are congruent angles, thus, their measures add up to 180°.

Then here we can write a linear equation for z.

z + 43 = 180

Solving that for z, we will get:

z = 180 - 43

z = 137

That is the measure of the angle.

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Using a system of equations, the number of each type of coffee that Julia and her husband should buy is as follows:

A system of equations is two or more equations solved concurrently or at the same time.

A system of equations can also be described as simultaneous equations.

The cost per pound of City Roast Columbian coffee = $6.50

The cost per pound of French Roast Columbian coffee = $8.20

The total quantity of the blend = 34 pounds

The cost per pound of the blend = $6.90

The total cost of the 34 pounds = $234.60 ($6.90 x 34)

Let the number of pounds of City Roast coffee = x

Let the number of pounds of French Roast coffee = y

x + y = 34 ... Equation 1

6.5x + 8.2y = 234.6 ... Equation 2

Multiply Equation 1 by 6.5:

6.5x + 6.5y = 221 ... Equation 3

Subtraction Equation 3 from Equation 2:

6.5x + 8.2y = 234.6

-

6.5x + 6.5y = 221

1.7y = 13.6

y = 8

Substitute, y = 8 in Equation 1:

x + y = 34

x = 34 - 8

x = 26

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The two lengths that Bottom 7%: L= 6.04 cm

Top 7%: L= 6.22 cm

What does "z-score" mean?

The Z-score provides information on how far a given value deviates from the standard deviation. The amount of standard deviations a given data point is above or below the mean is represented by the Z-score, also known as the standard score.

                                       The level of variability within a particular data collection is effectively reflected in the standard deviation.

Mean length of the population (μ) = 6.13 cm

Standard deviation (σ) = 0.06 cm

The z-score for any given length 'X' is

        Z = X - μ/σ

     What we want to know is the length at the 7-th percentile and at the 93-rd percentile.

According to a z-score table, the 7-th percentile has a correspondent z-score of -1.476 and the 93-rd percentile has a z-score of 1.476. Therefore, the bottom 7% and top 7% are separated by the following lengths

      Z(Xb) = Xb - μ/σ

  - 1.476 = Xb - 6.13/0.06

      Xb = 6.04

  z(Xt ) = Xt - μ/σ

   1.476 = Xt - 6.13/0.06

     Xt = 6.22

Bottom 7%: L= 6.04 cm.

Top 7%: L= 6.22 cm.

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Answer: In each question you have two options, T and F. Then in 6 questions you have 2 options for the first, two for the second, two for the third... At the end you get 2·2·2·2·2·2 = 26 options and 26 = 64.

Step-by-step explanation:

The perimeter of the parallelogram is; 184

In geometry, the perimeter of a shape is defined as the total length of its boundary.

Now, from the given image, we don't know the vertical sides XW and YZ and we can find it from Pythagoras theorem;

XW = √(68² - 60²)

XW = 32

Thus, the perimeter of the parallelogram is;

Perimeter = 2(32 + 60)

= 2 * 92

= 184

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Answer:

Step-by-step explanation:

inequality form

d≤ 1

Interval Notation:

(−∞,1]

Answer: w=9cm l=21cm

Step-by-step explanation

l=3w-6

l+l+w+w=60

3w-6+3w-6+w+w=60

8w-12=60

8w=60+12=72

w=72/8

w=9

l=3w-6

l=27-6

l=21

16 sample points (possible outcomes as to the classifications) exist.

What is the combination?

Combinations are ways to choose elements from a collection in mathematics where the order of the selection is irrelevant. Let's say we have a trio of numbers: P, Q, and R. Then, combination determines how many ways we can choose two numbers from each group.

Here, we have

Given: An experiment consists of selecting a student body president and vice president. All undergraduate students (freshmen through seniors) are eligible for these offices.

There are 4 possible classifications for the president (freshman, sophomore, junior, senior) and 4 for the vice president.

A total of (4)(4) = 16 sample points.

Hence, 16 sample points (possible outcomes as to the classifications) exist.

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