fill in the blank. Toward the end of a game of Scrabble, you hold the letters D, O, G, and Q. You can choose 3 of these 4 letters and arrange them in order in ______ different ways. (Give your answer as a whole number.)

As the two triangles are congruent to each other, using that we can get the value of x = 13 and y = 9.

Whether two or more triangles are congruent depends on the size of the sides and angles. As a result, a triangle's three sides and three angles determine its size and shape.

Two triangles are said to be congruent if their respective side and angle pairings are both equal.

Now in the given question,

The triangles are congruent so,

ED = QR

5y -7 = 38

⇒ 5y = 38+7

⇒ y = 45/5

⇒ y = 9

Now as the sum of angles in a triangle are 180°,

∠E +∠D +∠F = 180°

⇒ ∠F = 180 - 123 - 29

⇒ ∠F = 28°

As per congruency,

(2x+2) ° = 28°

⇒ 2x = 28-2

⇒ x = 26/2

⇒ x = 13

To know more about congruent triangles, visit:

https://brainly.com/question/22062407

#SPJ1

The complete question is:

Find the values of x and y. ADEF = AQRS

(5y-7) ft

D

E

123⁰

F

S

S

(2x + 2)° R

= 0, y =

Q

38 ft

P

To find Sn for an arithmetic series, you can use the following formula: Sn = (n/2) * (a1 + an).

In this case, Sn = (14/2)*(3 + 42) = 189.

To explain step-by-step:

1. Find the number of terms in the series, n = 14

2. Find the first term in the series, a1 = 3

3. Find the last term in the series, an = 42

4. Plug the values into the formula, Sn = (n/2)*(a1 + an)

5. Simplify the equation and solve, Sn = (14/2)*(3 + 42) = 189

[tex]80[/tex] billion dollars' worth of net exports were made by China and Germany. The first claim is accurate.

Export is the process of supplying goods and services to some other nation. Contrarily, importing is the act of acquiring goods from outside and transferring them into one's own nation.

The domestic product (GDP) is a measure of all the products and services generated in the United States; thus, changes in exports change significantly in the demand for goods and services made in the United States abroad.

The total of China's and Germany's net exports would be:

[tex]50[/tex] billion + [tex]30[/tex] billion [tex]= 80[/tex] billion

As a result, Germany & China's consolidated net exports amounted to [tex]80[/tex] billion u.s. dollars, reflecting answer option (a).

To know more about exports visit:

https://brainly.com/question/20375046

#SPJ1

Answer:

x=9

Step-by-step explanation:

this is solved by using cover and cancel so we can isolate the x variable

1) first add 72 to both sides.

6x=54

2) second divide both sides by 6

3) x=9

The value of the constant k that makes the function continuous everywhere is [tex]k = \frac{7}{48}$[/tex]

For the function to be continuous everywhere, we need to ensure that the limit of f(x) as x approaches 7 from the left is equal to the limit of f(x) as x approaches 7 from the right.

From the left, we have:

[tex]$$\lim_{x\to7^-}f(x) = \lim_{x\to7^-}kx^2 = k(7)^2 = 49k$$[/tex]

From the right, we have:

[tex]$$\lim_{x\to7^+}f(x) = \lim_{x\to7^+}(x+k) = 7+k$$[/tex]

For the function to be continuous at x=7, we need:

[tex]$$\lim_{x\to7^-}f(x) = \lim_{x\to7^+}f(x)$$[/tex]

Therefore, we need:

[tex]$$49k = 7 + k$$[/tex]

Solving for k:

[tex]$$49k - k = 7$$[/tex]

[tex]$$48k = 7$$[/tex]

[tex]$k = \frac{7}{48}$$[/tex]

Therefore, the value of the constant k that makes the function continuous everywhere is [tex]k = \frac{7}{48}$[/tex]

To know more about Limit visit;

brainly.com/question/29795597

#SPJ1

Answer: the distribution to the x and the 3 makes the solution 4

Step-by-step explanation: i got you broski

The 3-dimensional vector space represented in the form subspace dimensions A and B is given by option B. V = A + B.

V be 3-dimensional vector space.

Subspace of dimensions of A and B are 2 and 1 respectively.

And A ∩ B = 0.

It follows that every vector in A is linearly independent of every vector in B.

This implies,

Any vector v in V can be expressed uniquely as a sum of a vector in A and a vector in B.

Let v be an arbitrary vector in V.

A has dimension 2, it has a basis of two linearly independent vectors.

Let {a1, a2} be such a basis.

B has dimension 1, it has a basis consisting of a single nonzero vector b.

Then, any vector v in V can be expressed uniquely as

v = c1a1 + c2a2 + cb,

where c1, c2, and c are scalars.

Thus,

V = A + B.

Therefore, the correct answer to represents 3 dimensional vector space V  as option(B).  V = A + B.

Learn more about vector space here

brainly.com/question/17135326

#SPJ4

Step-by-step explanation:

a probability is always the ratio

desired cases / totally possible cases

the probably for the second roll to roll the same number as in the first roll again, is actually pretty high.

it means that we "accept" any result in the first roll. that makes this a 6/6 = 1 probability.

and then for the second roll we have the usual 1/6 probability (to roll the same number again).

so, we have

6/6 × 1/6 = 1/6 = 0.166666666...

P(even, odd) is the probability to roll first an even number (3 out of 6 = 3/6 = 1/2) and then an odd number (again 3 out of 6 = 1/2) :

1/2 × 1/2 = 1/4 = 0.25

so, this is more likely.

P(2, 5) is the probability to roll first a 2 (1 out of 6 is 1/6) and then a 5 (again 1 out of 6 is 1/6) :

1/6 × 1/6 = 1/36 = 0.027777777...

so, this is less likely.

P(odd, 1) is the probability to roll first an odd number (3 out of 6 = 1/2) and then a 1 (1 out of 6 = 1/6) :

1/2 × 1/6 = 1/12 = 0.083333333...

so, it is less likely.

1/42,1/22,1/10,1/6 are domain of function .

What are a function's domain and range?

The set of all possible inputs and outputs is known as a function's domain, and the same is true for its range. Important features of a function are the domain and range.

                               The range contains all of the function's output values, while the domain contains all of the real numbers that can be used as input values.

g(x) = 1/(x²+6)

Domain: {-6, -4, -2, 0, 2, 4, 6}​

G(-6) = 1/(-6² + 6) = 1/42

G(-4) = 1/(-4² + 6) = 1/22

G(-2) = 1/(-2² + 6) = 1/10

G(0) = 1/(0+6)    = 1/6

 G(2) = 1/(2² + 6) = 1/10

 G(4) = 1/(4² + 6) = 1/22

 G(6) = 1/(6² + 6) = 1/42

Learn more about domain and range

brainly.com/question/29452843

#SPJ1

According to the question,the equation of the parabola is y = (x + 4)² - 6.

An equation is a statement that equates two expressions using mathematical symbols. It is a mathematical statement that two expressions are equal in value. Equations can involve numbers, variables, and constants. Equations are used to solve real-world problems such as determining the speed of a car from the distance traveled and time elapsed.

The equation of a parabola with a focus at (-4, 7) and a directrix of y = 1 is given by:

y = (x + 4)² + 4.

This equation is derived from the standard equation of a parabola:

y = (x - h)² + k,

where (h, k) is the coordinates of the focus.

In this case, the coordinates of the focus are (-4, 7), so the equation becomes:

y = (x + 4)² + 7.

The directrix of the parabola is a line, so its equation is given by:

y = 1.

Substituting this equation into the equation of the parabola, we get:

(x + 4)² + 7 = 1

(x + 4)² = -6

y = (x + 4)² - 6.

To learn more about equation

https://brainly.com/question/17145398

#SPJ1

The perimeter of the figure is 24 feet.

The perimeter of a shape is the distance between its edges. Learn how to calculate the perimeter of various shapes by adding the side lengths.

To calculate the perimeter of the figure, add the lengths of all the sides. To calculate the perimeter of the figure, add the lengths of all the sides. Label the diagram's points as follows:

Beginning at the top left and working our way clockwise, we have:

A at (0, 4)

B at (6, 4)

C at (6, 0)

D at (4, 0)

E at (4, 2)

F at (2, 2)

G at (2, 0)

H at (0, 0)

Using the distance formula, we can now calculate the length of each side:

[tex]AB = \sqrt{((6-0)^2 + (4-4)^2)}= 6 feet[/tex]

[tex]BC = \sqrt{((6-6)^2 + (0-4)^2)} = 4 feet[/tex]

[tex]CD = \sqrt{((4-6)^2 + (0-0)^2)} = 2 feet[/tex]

[tex]DE = \sqrt{((4-4)^2 + (2-0)^2)} = 2 feet[/tex]

[tex]EF = \sqrt{((2-4)^2 + (2-2)^2)} = 2 feet[/tex]

[tex]FG = \sqrt{((2-2)^2 + (0-2)^2)} = 2 feet[/tex]

[tex]GH = \sqrt{((0-2)^2 + (0-0)^2) }= 2 feet[/tex]

[tex]HA = \sqrt{((0-0)^2 + (4-0)^2)}= 4 feet[/tex]

The perimeter is equal to the sum of these lengths:

Perimeter = 6 + 4 + 2 + 2 + 2 + 2 + 2 + 4

= 24 feet

As a result, the figure's perimeter is 24 feet.

To know more about perimeter visit:

brainly.com/question/6465134

#SPJ1

a) The maximum value for this data set is 31, the minimum value is 11 and the median value is 15.5.

b) Q1 13.5

Q2 15.5

Q3 24

c) 10.5

d) lower than 11 or higher than 31

e) Yes.

f) This data set is slightly positively skewed.

The median is the middle value of a set of data when the values are arranged in numerical order. It is used to find the middle value of a set of numbers or observations.

A. The values can be plotted on the boxplot as follows:

Maximum: 31

Minimum: 11

Median: 15.5

B. To find Q1, Q2 and Q3, the data must first be sorted from lowest to highest: 11; 13; 14; 15; 15; 16; 21; 26; 30; 31. Q1 is the median of the lower half of the data set (11, 13, 14, 15 and 15), which is 13.5. Q2 is the median of the entire data set (15.5). Q3 is the median of the upper half of the data set (16, 21, 26, 30 and 31), which is 24. The values can be plotted on the boxplot as follows:

Q1: 13.5

Q2: 15.5

Q3: 24

C. The Interquartile Range (IQR) is calculated by subtracting Q1 from Q3: IQR = Q3 - Q1 = 24 - 13.5 = 10.5.

D. Outliers are values that are significantly greater or lower than the majority of the data set. In this data set, values that are lower than 11 or higher than 31 would be considered outliers.

E. Yes, this data set has outliers. The value 11 is significantly lower than the majority of the data set, and the value 31 is significantly higher than the majority of the data set.

F. This data set is slightly positively skewed, as the majority of the data points are clustered around the lower end of the range, and there is a longer tail of higher values. This is indicated by the fact that the median (15.5) is lower than the mean (19.4).

For more questions related to mean

https://brainly.com/question/20118982

#SPJ1

a) The maximum value of the given dataset is 31, while the minimum value is 11 and the median is 15.5.

b) Q1 = 14,

Q2 = 15.5,

Q3 = 21

c) (IQR) = 7

d) Any values below 11 or above 31

e) Yes

f) slightly positively skewed.

Median is a measure of central tendency that is calculated by taking the middle value of a set of numbers when the values are arranged in ascending or descending order.

a) Maximum value: 31,

Minimum value: 11,

Median value: 15.5

b) Q1 = 14,

Q2 = 15.5,

Q3 = 21

c) The Interquartile Range (IQR) is calculated by subtracting Q1 from Q3:

Inter Quartile Range (IQR) = Q3-Q1

=21-14

= 7

d) Any values below 11 or above 31 can be considered as outliers.

As 11 is the lowest value given and 31 is the highest value.

e) Yes, this data set has outliers. The values 31 and 11 are both outliers since they are beyond the range of the Inter Quartile Range (IQR).

The value 11 is significantly lower than the majority of the data set, and the value 31 is significantly higher than the majority of the data set.

f) This data set is slightly positively skewed.

As the majority of the data points are clustered around the lower end of the range, and there is a longer tail of higher values.

This is indicated by the fact that the median (15.5) is lower than the mean.

For more questions related to dataset

https://brainly.com/question/28168026

#SPJ1

As all the sides of the closed figure are equal to each other, the quadrilateral here is a square.

A square is a closed, two-dimensional (2D), object with four corners. With four sides and four vertices, a quadrilateral is referred to as a square. All four sides of a square are equal and parallel.

In other words, a square is a polygon or quadrilateral with four sides. An equiangular quadrilateral is a shape in which all of the angles are of equal size.

Here in the given figure, we can see a quadrilateral is given.

We can see that all the sides of the quadrilateral are given to be equal to each other.

We can conclude from the observation that the quadrilateral is a square as the sides are all equal to each other.

To know more about squares, visit:

https://brainly.com/question/16865200

#SPJ1

Answer:

Equation of B represents this graph

(a) There are 20 ways to choose 2 colors without replacement when the order is relevant. (b) There are 10 ways to choose 2 colors without replacement when the order is not relevant.

(a) If the order of the choices is relevant, then we need to use the permutation formula. The number of ways to choose 2 colors from 5, without replacement and with order considered, is given by P(5,2) = 5!/(5-2)! = 5x4 = 20.

(b) If the order of the choices is not relevant, then we need to use the combination formula. The number of ways to choose 2 colors from 5, without replacement and with order not considered, is given by C(5,2) = 5!/(2!3!) = 10.

Learn more about permutation formula here: brainly.com/question/1216161

#SPJ4

Answer:

p (even,odd)- more likely

p(2,5) - less likely

p(odd,1)- less likely

Using trigonometric ratio it is proved that sin(B + 2C) + sin(C + 2A) + sin(A + 2B) = 4sin(A-B)/2.cos(B-C)/2.cos(C-A)/2 when A + B + C = 180.

What is trigonometric ratio?

Triangle side length ratios are known as trigonometric ratios. In trigonometry, these ratios show how the ratio of a right triangle's sides to each angle. Sine, cosine, and tangent ratios are the three fundamental trigonometric ratios.

We can start by using the sine addition formula to expand each of the sine terms in the left-hand side of the equation -

sin(B + 2C) = sin(B)cos(2C) + cos(B)sin(2C) = 2sin(B)cos(C)²

sin(C + 2A) = sin(C)cos(2A) + cos(C)sin(2A) = 2sin(C)cos(A)²

sin(A + 2B) = sin(A)cos(2B) + cos(A)sin(2B) = 2sin(A)cos(B)²

Substituting these expressions into the left-hand side of the equation, we get -

2sin(B)cos(C)² + 2sin(C)cos(A)² + 2sin(A)cos(B)²

Factoring out the 2, we can rewrite this as -

2(sin(B)cos(C)² + sin(C)cos(A)² + sin(A)cos(B)²)

Using the trigonometric ratio identity sin(2x) = 2sin(x)cos(x), we can rewrite each of the cosine squared terms as a product of sines and cosines -

cos(C)² = (1/2)(1 + cos(2C)) = (1/2)(1 + 2cos(C)sin(C))

cos(A)² = (1/2)(1 + cos(2A)) = (1/2)(1 + 2cos(A)sin(A))

cos(B)² = (1/2)(1 + cos(2B)) = (1/2)(1 + 2cos(B)sin(B))

Substituting these expressions into the previous equation, we get -

2(sin(B)(1/2)(1 + 2cos(C)sin(C)) + sin(C)(1/2)(1 + 2cos(A)sin(A)) + sin(A)(1/2)(1 + 2cos(B)sin(B)))

Simplifying and grouping the terms, we get -

sin(B)sin(C)cos(C) + sin(C)sin(A)cos(A) + sin(A)sin(B)cos(B)

Using the sine addition formula again, we can rewrite each of the cosine terms as a product of sines -

cos(C) = sin(A + B)

cos(A) = sin(B + C)

cos(B) = sin(C + A)

Substituting these expressions into the previous equation, we get -

sin(B)sin(C)sin(A + B) + sin(C)sin(A)sin(B + C) + sin(A)sin(B)sin(C + A)

We can rearrange this expression by factoring out a sin(A-B)/2 sin(B-C)/2 sin(C-A)/2 term -

sin(A-B)/2 sin(B-C)/2 sin(C-A)/2 (cos(A) - cos(B) + cos(B) - cos(C) + cos(C) - cos(A))

Simplifying the terms in parentheses, we get -

sin(A-B)/2 sin(B-C)/2 sin(C-A)/2 (0)

Therefore, the left-hand side of the equation simplifies to 0, which is equal to the right-hand side of the equation -

4sin(A-B)/2.cos(B-C)/2.cos(C-A)/2

Therefore, we have proven that sin(B + 2C) + sin(C + 2A) + sin(A + 2B) = 4sin(A-B)/2.cos(B-C)/2.cos(C-A)/2.

To learn more about trigonometric ratio from the given link

https://brainly.com/question/13276558

#SPJ1

Alberto's statement is flawed because all squares can be called rectangles, but not vice versa

Alberto's reasoning is flawed because all squares can be called rectangles, but not all rectangles are squares.

While it is true that squares meet the definition of rectangles, not all rectangles meet the definition of squares.

A square is a special type of rectangle with all sides equal in length.

Therefore, Alberto's argument violates the logical concept of implication, where the truth of one proposition (squares can be called rectangles) does not necessarily imply the truth of the converse (all rectangles must be called squares).

Read more about rectangles at

https://brainly.com/question/25292087

#SPJ1

The rental cost for each movie and each video game is $3.5 and $5.5 respectively.

Let

cost of each movie = x

Cost of each video game = y

5x + 3y = 34

2x + 12y = 73

Multiply (1) by 4

20x + 12y = 136

2x + 12y = 73

subtract the equations to eliminate y

18x = 63

divide both sides by 18

x = 63/18

x = 3.5

Substitute x = 3.5 into (1)

5x + 3y = 34

5(3.5) + 3y = 34

17.5 + 3y = 34

3y = 34 - 17.5

3y = 16.5

y = 16.5/3

y = 5.5

Therefore, $3.5 and $5.5 is the rental cost of each movie and video game respectively.

Read more on rental cost:

https://brainly.com/question/11959610

#SPJ1

Perfect numbers between 1 and n (where n is a positive integer) are 6, 28, 496, 8128.

A positive integer that is the sum of its appropriate divisors is referred to as a perfect number. The sum of the lowest perfect number, 6, is made up of the digits 1, 2, and 3. The digits 28, 496, and 8,128 are also ideal.

Perfect numbers are whole numbers that are equal to the sum of their positive divisors, excluding the number itself. Examples of perfect numbers include 6 (1 + 2 + 3 = 6), 28 (1 + 2 + 4 + 7 + 14 = 28) and 496 (1 + 2 + 4 + 8 + 16 + 31 + 62 + 124 + 248 = 496).

To learn more about Perfect numbers link is here

brainly.com/question/29291022

#SPJ4

The complete question is:

What are all the perfect numbers between 1 and n (where n is a positive integer)?

The company can produce approximately 425 widgets for $2127.

What is cost function ?

The key concept used here is the concept of cost functions, which is an important concept in economics and business. A cost function is a mathematical function that expresses the total cost of production as a function of the level of output produced. In this case, the cost function is a linear function of the form C = a + bx, where C is the total cost, a is the fixed cost, b is the variable cost per unit, and x is the level of output.

Finding the number of widgets the company can produce given a fixed cost and a variable cost per widget :

To solve this problem, we can set up an equation that relates the total cost to the number of widgets produced.

Let x be the number of widgets produced.

The total cost C is given by:

C = fixed cost + variable cost

C = 1277 + 1.93x

We want to find the number of widgets produced for a total cost of $2127. So we can set up an equation:

2127 = 1277 + 1.93x

Subtracting 1277 from both sides gives:

850 = 1.93x

Dividing both sides by 1.93 gives:

x ≈ 439.9

Since we can't produce a fractional number of widgets, we need to round down to the nearest integer. Therefore, the company can produce approximately 425 widgets for $2127.

To know more about cost visit :

brainly.com/question/14083670

#SPJ1

We have proved that quadrilateral ABCD is a square below.

The perimeter of ABCD = 4 × AB

The area of ABCD= (AB)² .

A square is a two-dimensional, four-sided shape with all sides of equal length and all angles of 90 degrees. I

Proof that Quadrilateral ABCD is a Square:

We can prove that quadrilateral ABCD is a square by using the properties of a square.

A square is also a quadrilateral with four equal sides and four right angles.

In the image, we can see that side AB = side CD, and side AD = side BC.

Furthermore, we can see that the angles A, B, C, and D are all right angles=90 degree

This means that the quadrilateral ABCD is a square.

Finding the Perimeter of ABCD:

The perimeter of a square is equal to the sum of its four sides.

In the image, we can see that the sides AB, CD, AD, and BC are all equal. Therefore, the perimeter of ABCD=

4 × AB

= 4 × CD

= 4 × AD

= 4 × BC.

Finding the Area of ABCD:

The area of a square is equal to the length of one of its sides squared. In the image, we can see that the sides AB, CD, AD, and BC are all equal. Therefore, the area of ABCD= (AB)²

= (CD)²

= (AD)²

= (BC)².

For more questions related to quadrilateral

https://brainly.com/question/27991573

#SPJ1

We have proved that quadrilateral ABCD is a square below.

Perimeter of ABCD is 4 × AB

The area of ABCD= (AB)².

A square is a two-dimensional, four-sided shape with equal-length edges and 90-degree angles on each side.

The quadrilateral ABCD is a square, therefore:

Using the characteristics of a square, we can demonstrate that the parallelogram ABCD is a square.

A quadrilateral with four equal edges and four right angles is a square.

As seen in the picture, side AB corresponds to side CD and side AD corresponds to side BC.

In addition, we can see that the orientations A, B, C, and D are all 90-degree right angles.

The parallelogram ABCD is a square as a result.

Finding ABCD's Perimeter:

The sum of a square's four edges determines its perimeter.

The sides AB, CD, AD, and BC are equal on the picture, as can be seen. Consequently, the ABCD boundary

4 × AB

= 4 × CD

= 4 × AD

= 4 × BC

Finding the Area of ABCD:

A square's area is equivalent to the square of one of its sides. The sides AB, CD, AD, and BC are equal on the picture, as can be seen.

Consequently, ABCD's area equals (AB)².

= (CD)²

= (AD)²

= (BC)².

For more questions related to quadrilateral, visit:

brainly.com/question/27991573

#SPJ1

The clear image of quadrilateral attached below,

Step 1: x + 0.5 = 5

Step 2: x = 4.5

Step 3: 2*x^2 - 3x + 6 - 3/x +2/x^2 = 2(4.5)^2 - 3(4.5) + 6 - (3/4.5) + (2/(4.5)^2)

Step 4: 2*4.5^2 - 3*4.5 + 6 - 3/4.5 + 2/(4.5)^2 = 44.25 - 13.5 + 6 - 0.666666667 + 0.044444444 = 36.04444444

At equilibrium we expect [Reactants} < [Products)  from the the profile that has been shown.

An energy profile diagram, also known as an energy diagram or reaction coordinate diagram, is a graphical representation of the energy changes that occur during a chemical reaction or a physical process. It shows the energy levels of the reactants, products, and any intermediate species that may form during the reaction.

The horizontal axis of an energy profile diagram represents the reaction coordinate, which is a measure of the progress of the reaction from the reactants to the products. The vertical axis represents the energy of the system.

Learn more about energy profile:https://brainly.com/question/13189077

#SPJ1

The probability of selecting a blue marble on the first draw and a green marble on the second draw is 0.09.

What is probability?

Probability is a measure of the likelihood of an event occurring. It is a number between 0 and 1, where 0 means the event is impossible and 1 means the event is certain to happen.

There are 12 marbles in total in the bag, so the probability of selecting a blue marble on the first draw is 4/12.

After the first marble is drawn, there are 11 marbles left in the bag, so the probability of selecting a green marble on the second draw, given that the first marble was blue and has already been removed, is 3/11.

To determine the probability of both events occurring together, we multiply the probabilities. Therefore, the probability of selecting a blue marble on the first draw and a green marble on the second draw is:

(4/12) * (3/11) = 0.0909

Rounding to the hundredth place, the probability is approximately 0.09.

Therefore, the probability of selecting a blue marble on the first draw and a green marble on the second draw is 0.09.

To learn more about probability from the given link:

https://brainly.com/question/30034780

#SPJ1

The 66th derivative of f(x) is the same as the second derivative of f(x). Thus, we can calculate the 66th derivative as follows f''(x) = -4sin(x).

The derivative of a function is the rate at which the function changes with respect to its input variable. It is a fundamental concept in calculus and is used in many areas of mathematics, science, and engineering.

The derivative of the function f(x) = 4sin(x) with respect to x is:

f'(x) = 4cos(x)

Taking the derivative again, we get:

f''(x) = -4sin(x)

Taking the derivative 3 times, we get:

f'''(x) = -4cos(x)

Taking the derivative 4 times, we get:

f''''(x) = 4sin(x)

We notice that the derivative of f(x) repeats every 4 times, alternating between sin(x) and cos(x) with a sign change. Therefore, to find the 66th derivative of f(x), we can simplify the calculation by considering the remainder when 66 is divided by 4:

66 mod 4 = 2

This means that the 66th derivative of f(x) is the same as the second derivative of f(x). Thus, we can calculate the 66th derivative as follows:

f''(x) = -4sin(x)

Therefore, the 66th derivative of f(x) is:

f^(66)(x) = f''(x) = -4sin(x)

To know more about derivative visit:

https://brainly.com/question/25324584

#SPJ1

The real solutions of the following equation are 5 and 1

[ Graph is attached below ]

To solve the given equation graphically we need to find the coordinate points that pass through the graph.

This can be done by taking 'x' values and solving for them 'y' values.  

Now draw the graph using the above coordinates and find the solution as shown below.

Here we have

x³ - 6x²+ 5x = 0  

Let y = x³ - 6x²+ 5x  

To draw the graph find the coordinates of points as follows

At x = 0 =>  y = (0) + (0) + (0) = 0

At x = 1 => y = (1)³ - 6(1)² + 5(1) = 0

At x = -1 => y = (-1)³ - 6(-1)² + 5(-1) = - 12

At x = 2 => y = (2)³ - 6(2) + 5(2) = 6

From the above calculation,

The coordinates of the points to draw the graph are (0, 0), (1, 0), (-1, -12), and (2, 6)

Here the solutions of the graph are the x-coordinate of points where the graph cuts the x-axis

From the figure, the graph will cut the x-axis at 1 and 5  

Therefore,

The real solutions of the following equation are 5 and 1

[ Graph is attached below ]

Learn more about Graphs at

https://brainly.com/question/8210152

#SPJ1