Determine whether the subset of M is a subspace of M with the standard operations of matrix addition and scalar inn nn multiplication The set of all n x n invertible matrices O subspace O not a subspace

Answers

Answer 1

The set of all n×n invertible matrices with the standard operations of matrix addition and scalar multiplication is (b) not a subspace.

A Subspace is defined as a subset of a vector space that is itself a vector space under the same operations of addition and scalar multiplication defined on the original vector space.

To be a subspace of Mₙ,ₙ, a subset of Mₙ,ₙ must satisfy three conditions:

(i) The subset must contain the zero matrix,

(ii) The subset must be closed under matrix addition, meaning that if A and B are in the subset, then (A + B) is also in the subset.

(iii) The subset must be closed under scalar multiplication, meaning that if A is in the subset and c is any scalar, then cA is also in the subset.

The set of all n×n invertible matrices does not contain the zero matrix, as the zero matrix is not invertible.

Therefore, it fails to meet the first condition and cannot be a subspace, the correct option is (b).

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The given question is incomplete, the complete question is

Determine whether the subset of Mₙ,ₙ is a subspace of Mₙ,ₙ with the standard operations of matrix addition and scalar multiplication.

The set of all n×n invertible matrices is

(a) Subspace

(b) Not a subspace.


Related Questions

Correct to 3 significant figures, the of 18.75-(2.11)2

Answers

Answer: 14.5

Step-by-step explanation:

When there is a decimal point, you start counting from the left any number that is not zero. If the zero is at the end, then you count it.

For example, if the answer is 0.000145 then the number of significant figures is still three because you start counting from the first nonzero number from the left.

If the answer is 14.50, then the number of significant figures is four because you start counting from the first nonzero number from the left.

14.53 is the answer to the equation but because you want to correct it to 3 significant figures, you round down because 3 is less than 5 and 14.5 ends up being the final answer.

Which of these act as an unbalanced force to stop objects in motion? Choose more than one answer.

Answers

A catapult, seatbelt, parachute, safety net, or unbalanced force can be used to stop moving objects.

What is unbalanced force to stop objects in motion?

When a seatbelt is fastened during sudden braking or a crash, it exerts pressure on the passenger's body and prevents them from moving forward.

Similar to this, when a parachute is opened, a significant amount of air resistance force is generated, slowing the wearer's descent.

A safety net exerts force on an object that is falling on it as well, absorbing the kinetic energy and stopping it.

A catapult can apply a significant unbalanced force that can halt an object in motion by quickly transferring energy from its springs or elastic materials to the target.

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[tex] \frac{4}{12} \times 3[/tex]
Can you tell me the answer for this .​

Answers

Answer:

1

Step-by-step explanation:

[tex]\frac{4}{12}[/tex]  x 3 = 12/12 = 1

Help find each measure

Answers

The answer of  the given question based on finding each measure of a circle the answer is , (a)  m(MNP) = 12.5° degrees , (b) m(KL) = 102.5° degrees , (c)  m(KJ) = 52.5° degrees , (d)  m(JN) = 102.5° degrees , (e)  m(JLM) =  12.5° degrees.

What is Arc?

In geometry,  arc is a portion of  curved line that can be thought of as  segment of  circle. It is defined by two endpoints on  circle and the arc itself is the part of  circle between those two points. An arc can be measured in degrees, and its measure is equal to  central angle subtended by  arc. The length of  arc can also be calculated using the formula L = rθ, where L is  length of  arc, r is radius of  circle, and θ is  angle (in radians) subtended by  arc at center of circle.

Using the properties of angles and arcs in circles:

a. Angle MNP is inscribed in arc MP, so m(MNP) = 1/2m(MP) = 1/2(25) = 12.5° degrees.

Since angles NPM and MPK are vertical angles, we have m(MPK) = m(NPM) = m(MNP) = 12.5° degrees. Then, m(MN) = m(MPK) + m(KPM) = 12.5 + 40 = 52.5° degrees.

b. Angle LKP is inscribed in arc LP, so m(LKP) = 1/2m(LP) = 1/2(25) = 12.5° degrees.

Since angles PKL and LKN are vertical angles, we have m(PKL) = m(LKN) = m(LKP) = 12.5° degrees. Then, m(KL) = m(PKL) + m(PKC) = 12.5 + 90 = 102.5° degrees.

c. Angle PKJ is inscribed in arc PJ, so m(PKJ) = 1/2m(PJ) = 1/2(25) = 12.5° degrees.

Since angles LPK and LPJ are vertical angles, we have m(LPK) = m(LPJ) = m(PKJ) = 12.5° degrees. Then, m(KJ) = m(LPK) + m(LPJ) = 12.5 + 40 = 52.5° degrees.

d. Angle JNM is inscribed in arc JM, so m(JNM) = 1/2m(JM) = 1/2(25) = 12.5° degrees.

Since angles KJM and KJN are vertical angles, we have m(KJM) = m(KJN) = m(JNM) = 12.5° degrees. Then, m(JN) = m(KJM) + m(MJN) = 12.5 + 90 = 102.5° degrees.

e. Angle JLM is an inscribed angle that intercepts arc JM, so m(JLM) = 1/2m(JM) = 1/2(25) = 12.5° degrees.

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A group of ten office workers recorded the distance, in kilometres, they walked on Monday. Their results are listed below. (a) Find the mean distance, in kilometres, walked by this group of office workers on Monday. The following box-and-whisker plot represents the results listed above. (b) (i) Find the value of
p
. (ii) Find the interquartile range. Distances less than
r
kilometres walked on Monday are considered outliers. (c) Find the value of
r
. A group of twenty freelancers also recorded the distance, in kilometres, they walked during the same Monday. The mean distance walked by this group of freelancers on the Monday is
1.7 km
with a standard deviation of
1 km
. (d) (i) Find the total distance the group of freelancers walked on the Monday. (ii) Find the combined mean distance that all thirty office workers and freelancers walked on the Monday. (iii) On Tuesday, all twenty freelancers walked double the distance they walked on Monday. Calculate the variance in the distance walked by the freelancers on the Tuesday.

Answers

A)  The mean distance walked by the group of office workers on Monday was approximately 6.89 km.

b) The interquartile range was approximately 2.2 km.

c)   The value of r would be: r ≈ 4.1 km

d)  (i) Total distance walked = 1.7 km * 20 = 34 km

(ii),  the combined mean distance walked by all thirty workers was approximately 3.28 km.

(iii) the standard deviation is not given directly, we can use the fact that the standard deviation is equal to the square

(a) The mean distance walked by the group of ten office workers on Monday can be found by adding up all of their distances and dividing by the number of workers:

Mean = (5 + 2 + 3 + 6 + 5 + 8 + 5 + 7 + 6 + 4) / 10

= 51 / 10

= 5.1 km

(b) (i) The value of p is the median of the data set. Since there are 10 data points, the median is the average of the 5th and 6th smallest values:

p = (5 + 5) / 2

= 5 km

(c) The value of r represents the maximum distance that is not considered an outlier. From the box-and-whisker plot, we can see that the maximum value is 8 km, which is not an outlier. Therefore,

r = 8 km

(d) (i) The total distance walked by the group of 20 freelancers can be found by multiplying their mean distance by the number of freelancers:

Total distance = Mean distance * Number of freelancers

= 1.7 km * 20

= 34 km

(d)

(i) The total distance walked by the group of 20 freelancers can be found by multiplying their mean distance by the number of freelancers:

Total distance = Mean distance * Number of freelancers

= 1.7 km * 20

= 34 km

(ii) The combined mean distance that all 30 workers walked on Monday can be found by adding up the total distance walked by all workers and dividing by the total number of workers:

Combined mean distance = (Total distance by office workers + Total distance by freelancers) / Total number of workers

= (51 km + 34 km) / 30

= 85 / 30

= 2.83 km

(iii) On Tuesday, all 20 freelancers walked double the distance they walked on Monday. The variance in the distance walked by the freelancers on Tuesday can be found using the formula for the variance of a constant times a random variable:

Var(aX) = a^2 Var(X)

Since the freelancers walked double the distance, the variance of the distance they walked on Tuesday is:

Var(Tuesday distance) = 4 * Var(Monday distance) = 4 * (1 km)^2 = 4 km^2

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For the functions f(x)=−7x+3 and g(x)=3x2−4x−1, find (f⋅g)(x) and (f⋅g)(1).

Answers

Answer:

Find f(g(x)) f(x)=7x-8 , g(x)=3x-2. f(x)=7x−8 f ( x ) = 7 x - 8 , g(x)=3x−2 g ( x ) = 3 x - 2. Step 1. Set up the composite result function. f(g(x)) f ( g ...

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A card is pulled from a deck of cards and noted. The card is then replaced, the deck is shuffled, and a second card is pulled and noted. What is the probability that both cards are face cards?

Answers

Answer: There is a 5.32544378% Chance of a face card being pulled twice

Step-by-step explanation: If there is 52 cards in a deck, and 12 of them are face cards, there is roughly a 23% (23.0769%) chance of pulling one in the first draw. Multiply .230769 x .230769 and you get .0532544378 which equals 5.32544378%

he buys a 5kg lwisa samp and repacks the samp into 125g packets. determine how many packets will be able to get from one pack of 5kg samp?​

Answers

Answer:5

Step-by-step explanation:5555555

A researcher tests whether smoking by parents influences children’s attitudes toward smoking behavior.
Independent Variable: ________
Quasi-Independent Variable: ________
Dependent Variable: ________

Answers

Independent Variable: Smoking by parents
Quasi-Independent Variable: None (assuming that smoking by parents is a naturally occurring variable and not manipulated by the researcher)
Dependent Variable: Children's attitudes toward smoking behavior

Answer:

Independent Variable: Smoking

Quasi-Independent Variable: Parents to Children (Assuming that the researcher purposely manipulated it, otherwise there is none.)

Dependent Variable: Children's attitudes toward smoking behavior

Find the standard normal area for each of the following(round your answers to 4 decimal places

Answers

With four decimal places added, we have P(2.04 Z 3.04) 0.0189.

Two decimal places are what?

To round a decimal value to two decimal places, use the hundredths place, which is the second place to the right of the decimal point.

Subtracting the area to the left of 1.25 from the area to the left of 2.15 will give us the standard normal area between 1.25 and 2.15.

The area to the left of 1.25 is 0.8944, and the area to the left of 2.15 is 0.9842, according to a conventional normal distribution table or calculator.

So, the standard normal area between 1.25 and 2.15 is:

P(1.25 < Z < 2.15) = 0.9842 - 0.8944 = 0.0898

Rounding to four decimal places, we get:

P(1.25 < Z < 2.15) ≈ 0.0898

We follow the same procedure as before to determine the standard normal region between 2.04 and 3.04:

P(2.04 < Z < 3.04) = P(Z < 3.04) - P(Z < 2.04)\

The area to the left of 2.04 is 0.9798, and the area to the left of 3.04 is 0.9987, according to a conventional normal distribution table or calculator.

So, the standard normal area between 2.04 and 3.04 is:

P(2.04 < Z < 3.04) = 0.9987 - 0.9798 = 0.0189

Rounding to four decimal places, we get:

P(2.04 < Z < 3.04) ≈ 0.0189

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Cari owns a horse farm and a horse trailer that can
transport up to 8, 000 pounds of livestock and tack. She
travels with 5 horses whose combined weight is 6, 240
pounds. Let t represent the average weight of tack per
horse. Which of the following inequalities could be used to
determine the weight of take Cari can allow for each
horse?
O 6240 +t≤ 8000
O 8000 6240t
O 6240 + 5t ≤ 8000
O 8000 +6240 < 5t
O 8000 6240 > 5t
what would the answer be, it’s multiply chose

Answers

Answer:

87

Step-by-step explanation:

87

Brandi has a deck of 12 cards labeled A through L. Brandi draws a card from the deck and returns it, then draws a card again. What is the theoretical probability that Brandi draws a card with a vowel both times?
A) 6.25%

B) 2.08%

C) 25%

D) 75%

Answers

Answer: The probability of drawing a vowel from a deck of 12 cards is 5/12, since there are 5 vowels (A, E, I, O, U) and 12 cards in total.

Since Brandi returns the first card drawn to the deck before drawing the second card, the outcome of the first draw does not affect the outcome of the second draw. Therefore, we can treat the two draws as independent events.

The probability of drawing a vowel on the first draw is 5/12, and the probability of drawing a vowel on the second draw is also 5/12. Since we want to find the probability of both events happening (drawing a vowel both times), we can multiply the probabilities:

P(vowel on both draws) = P(vowel on first draw) * P(vowel on second draw)

= (5/12) * (5/12)

= 25/144

Therefore, the theoretical probability that Brandi draws a card with a vowel both times is 25/144, which is approximately 0.1736 or 17.36%.

So the closest answer choice is B) 2.08%.

Step-by-step explanation:

IF A SOUP RECIPE YIELDS 20 GALLONS, HOW MANY 5-FLUID OUNCE PORTIONS WILL THE RECIPE YIELD?

Answers

The number of 5 fluid ounce portions needed in 20 gallons is a total of 512 portions

Calculating the number of ounce portions needed

To solve this problem, we first need to convert the volume of the soup recipe from gallons to fluid ounces, and then divide by the size of each portion to find the total number of portions.

Given that

1 gallon = 128 fluid ounces

Therefore, the recipe yields:

20 gallons x 128 fluid ounces per gallon = 2560 fluid ounces

Now, we can find the number of 5-fluid ounce portions by dividing the total volume by the portion size:

2560 fluid ounces / 5 fluid ounces per portion = 512 portions

Therefore, the soup recipe yields 512 portions of 5 fluid ounces each.

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Equation for the line up passes through the points (-6,-3) and (-8,-4)

Answers

[tex](\stackrel{x_1}{-6}~,~\stackrel{y_1}{-3})\qquad (\stackrel{x_2}{-8}~,~\stackrel{y_2}{-4}) \\\\\\ \stackrel{slope}{m}\implies \cfrac{\stackrel{\textit{\large rise}} {\stackrel{y_2}{-4}-\stackrel{y1}{(-3)}}}{\underset{\textit{\large run}} {\underset{x_2}{-8}-\underset{x_1}{(-6)}}} \implies \cfrac{-4 +3}{-8 +6} \implies \cfrac{ -1 }{ -2 } \implies \cfrac{ 1 }{ 2 }[/tex]

[tex]\begin{array}{|c|ll} \cline{1-1} \textit{point-slope form}\\ \cline{1-1} \\ y-y_1=m(x-x_1) \\\\ \cline{1-1} \end{array}\implies y-\stackrel{y_1}{(-3)}=\stackrel{m}{ \cfrac{ 1 }{ 2 }}(x-\stackrel{x_1}{(-6)}) \implies y +3 = \cfrac{ 1 }{ 2 } ( x +6) \\\\\\ y+3=\cfrac{ 1 }{ 2 }x+3\implies {\Large \begin{array}{llll} y=\cfrac{ 1 }{ 2 }x \end{array}}[/tex]

ivnqa1o
just now
Mathematics
College
Martin has a spinner that is divided into four sections labeled A, B, C, and D. He spins the spinner twice. PLEASE ANSWER THE BOXES ORDER AND RIGHT, HELP EASY THANK UU
Question 1

Drag the letter pairs into ALL the boxes to correctly complete the table and show the sample space of Martin's experiment. PLEASE LIST ALL ORDERSS

Answers

Answer: Below.

Step-by-step explanation:

To find each we must use the left side to put the first letter. The top for the right letter.

First row - AA

Second row - CB

Third row - CD

Fourth row - BD

Find the distance from Link to the Octorok so Link can attack

Answers

The distance  from Link to the Octorok  is 10.63 units.

How to find the distance?

We know that the distance between two points (x₁, y₁) and (x₂, y₂) is given by the formula below:

distance = √( (x₂ - x₁)² + (y₂ - y₁)²)

Here we want to find the distance from Link to the Octorok so Link can attack, so we need to get the distance between the points (-4, -5) and (3, 3).

The distance will be:

distance = √( (3 + 4)² + (3 + 5)²)

distance = √( (7)² + (8)²)

distance = √113

distance = 10.63

The distance is 10.63 units.

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He has 2 pens. His friend gives him 2 more pens. How many pens he has?

Answers

Answer:

4 pens.

Step-by-step explanation:

Now he has 2 pens.

Then his friend gives him two more.

As a sum, that's:

2 + 2 = 4 pens.

Answer:

He has 4 pens now.

Step-by-step explanation:

2+2=4

(a) Show that if λ is an eigenvalue of A, then λ is an eigenvalue of [tex]A^{T}[/tex]. Show with an example that the eigenvectors of A and [tex]A^{T}[/tex] are not the same.

(b) Show that if λ is an eigenvalue of A, and A is invertible, then λ^-1 is an eigenvalue of A^-1.

Answers

If λ is an eigenvalue of A, then λ is an eigenvalue of [tex]A^T[/tex]. Show with an example that the eigenvectors of A and [tex]A^T[/tex] are not the same.

What are eigenvalues and eigenvectors?

The equation Av = λv, where v is a non-zero vector, is satisfied by an eigenvector v and an eigenvalue given a square matrix A. In other words, the eigenvector v is multiplied by the matrix A to produce a scalar multiple of v. Due to their role in illuminating the behaviour of linear transformations and differential equation systems, eigenvectors play a crucial role in many branches of mathematics and science. When the eigenvector v is multiplied by A, the eigenvalue indicates how much it is scaled.

The eigenvalue and eigenvector states that, let v be a non-zero eigenvector of A corresponding to the eigenvalue λ.

Then, we have:

Av = λv

Taking transpose on both sides we have:

[tex]v^T A^T = \lambda v^T[/tex]

The above equations thus relates transpose of vector and transpose of A to λ.

Now, consider a matrix:

[tex]\left[\begin{array}{cc}1&2\\3&4\\\end{array}\right][/tex]

Now, the eigen values of this matrix are λ1 = -0.37 and λ2 = 5.37.

The eigenvectors are:

[tex]v1 = [-0.8246, 0.5658]^T\\v2 = [-0.4159, -0.9094]^T[/tex]

Now, for transpose of A:

[tex]A^T=\left[\begin{array}{cc}1&3\\2&4\\\end{array}\right][/tex]

The eigen vectors are:

[tex]u1 = [-0.7071, -0.7071]^T\\u2 = [0.8944, -0.4472]^T[/tex]

Hence, we see that, if λ is an eigenvalue of A, then λ is an eigenvalue of [tex]A^T[/tex]. Show with an example that the eigenvectors of A and [tex]A^T[/tex] are not the same.

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I need help with this

Answers

Answer: a

Step-by-step explanation: because a is my name

Given: -4+16-64 +256..., Sn=3,276
Find the number of terms in the series.

Answers

There are 4 terms in the series.

what is an arithmetic sequence?

An arithmetic sequence is a sequence of numbers where each term is equal to the previous term plus a fixed constant difference, called the common difference.

The given series is an arithmetic sequence with a common difference of 16. We can find the first term by plugging in n=1 into the sequence:

a1 = -4 + 16(-1)^1 = -4 + 16 = 12

Using the formula for the sum of an arithmetic sequence, we have:

Sn = n/2(2a1 + (n-1)d)

where n is the number of terms, a1 is the first term, and d is the common difference. Plugging in the given values, we have:

3276 = n/2(2(12) + (n-1)(16))

Simplifying this equation, we get:

3276 = n/2(28 + 16n - 16)

3276 = n/2(12 + 16n)

Multiplying both sides by 2 and rearranging, we get:

6544 = n(6 + 8n)

Dividing both sides by 2, we get:

3272 = n(3 + 4n)

We can see that n must be an even number, since the left side is even. We can also see that n cannot be too large, since the right side increases much faster than the left side. Trying some even values of n, we find that:

n=8 -> 3272 = 8(3 + 4(8)) is false

n=6 -> 3272 = 6(3 + 4(6)) is false

n=4 -> 3272 = 4(3 + 4(4)) is true

Therefore, there are 4 terms in the series.

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PLEASE HELP ASAP!! 25 POINTS AND BRAINLIEST

Answers

Answer: 62°

Step-by-step explanation:

All angles of a triangle add up to 180°.

So, add up all the other angles.

15 + 25 + 39 = 79

180 - 79 = 101

Then, to find x, subtract 39 from 101, to get 62!

QUICK ANSWER THIS PLEASE What is the constant of proportionality between the corresponding areas of the two pieces of wood?




3



6



9



12

Answers

Answer:

Step-by-step explanation:

D

p(s) = s³ + 10s
f(s) = 6s - 3
Find p(2)-f(2)

Answers

Answer:

19

Step-by-step explanation:

We are given the following two functions of s

[tex]p(s) = s^3 + 10s\\f(s) = 6s - 3\\\\\text{To find p(2) substitute 2 for s in p(s)}\\p(2) = (2)^3 + 10(2) = 8 + 20 = 28\\\\[/tex]

[tex]\text{To find f(2) substitute 2 for s in f(s)}\\f(2) = 6(2) - 3= 12 - 3= 9\\[/tex]

[tex]p(2) - f(2) = 28 - 9 = 19[/tex]

The length of each side of a square is 30 cm. If the length of its sides are now decreased by 20%, find the percentage decrease in the area of the square.

Answers

Answer:

80%

Step-by-step explanation:

each side of a square is 30 cm.

decreased by 20%.

find the percentage decrease in the area of the square.

20%x4side= 80%

A triangle has an area of 144 square feet. The height is 24 feet. What is the length of the base (in feet)?

Answers

We can use the formula for the area of a triangle to solve for the length of the base:

Area = (1/2) x base x height

We know that the area is 144 square feet and the height is 24 feet, so we can substitute those values into the formula:

144 = (1/2) x base x 24

To solve for the length of the base, we can first simplify the right-hand side of the equation:

144 = 12 x base

Dividing both sides by 12, we get:

12 = base

Therefore, the length of the base of the triangle is 12 feet.

NEED HELP ASAP!!!
Which polynomial best represents the scenario below? The total cost of Andre's school supplies depends on the price of each item. Each notebook (n) is $5, each folder (f) is $4, and each backpack (b) is $15.

Answers

Answer:

C = 5n + 4f + 15b

a point is chosen at random on ak. what is the probability that the point will be on bg. dont forget to reduce

Answers

there is a 20% chance that the point chosen at random will lie on bg.

To find the probability that a point chosen at random will be on the line segment bg, we need to consider the length of bg in relation to the length of the entire line segment ak.

Let us assume that ak is a straight line segment, and bg is a smaller segment that lies entirely within it. To find the probability, we need to divide the length of bg by the length of ak.

Let the length of bg be x and the length of ak be y. Then the probability that a point chosen at random will be on bg is:

Probability = Length of bg / Length of ak

Probability = x / y

However, we need to be careful here. If we choose a point anywhere on ak, it may not necessarily lie on bg. There are an infinite number of points on ak, but only one segment bg. Therefore, the probability we are looking for is actually the ratio of the lengths of bg to ak.

So, if we know the lengths of bg and ak, we can find the probability by dividing them. For example, if bg is 2 units long and ak is 10 units long, the probability of choosing a point on bg is:

Probability = 2 / 10

Probability = 0.2 or 20%

In this case, there is a 20% chance that the point chosen at random will lie on bg.

In conclusion, the probability of a point chosen at random on ak being on bg is directly proportional to the length of bg in relation to the length of ak. Therefore, we need to find the ratio of the lengths of the two line segments to determine the probability.

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what is the probability  a point is chosen at random on ak and then the point will be on bg. dont forget to reduce the products?

Construct triangle PQR in which angle Q = 30 deg , angle R=60^ and PQ + QR + RP = 10cm​

Answers

We can see here that in order to construct a triangle PQR in which angle Q = 30°, angle R=60° and PQ + QR + RP = 10cm​, here is a guide:

Draw a line segment AB = 10 cm.Construct angle 30° at point A and angle 60° at point B.Draw angle bisectors to angles A and B.Make sure these angle bisectors intersect at point P.Draw perpendicular bisector to line segment AP.Let this bisector meet AB at Q.Then draw perpendicular bisector to line segment BP.Let this bisector meet AB at R.Join PQ and PR.PQR is the required triangle.

What is a triangle?

A triangle is a geometric shape that is defined as a three-sided polygon, where each side is a line segment connecting two of the vertices, or corners, of the triangle. The interior angles of a triangle always add up to 180 degrees.

Triangles can be classified into different types based on their side lengths and angles, such as equilateral triangles with three equal sides and three equal angles, isosceles triangles with two equal sides and two equal angles, and scalene triangles with no equal sides or angles.

Triangles are used in many areas of mathematics and science, including geometry, trigonometry, and physics.

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Find the derivative of the function g(t), below. It may be to your advantage to simplify before differentiating. g(t)=tan(ln(t)) g'(t)=

Answers

The derivative of the function g(t) =  -2tsint - sint + 2tcost.

In mathematics, the derivative of a function of a real variable measures the sensitivity of the function's value (the output value) to changes in its independent variable (the input value). Derivatives are a fundamental tool of calculus. The process of finding derivatives is called differentiation. The reverse process is called retro differentiation.

The fundamental theorem of calculus associates inverse differentiation with integration. Differentiation and integration form the two basic operations in univariate calculus.

Given that:

g(t) = tan(ln(t)) g'(t)

Now, Differentiating for the first time:

  (t²+1) d/dt (sin t) + Sin t d/dt (t²+1)

= (t²+1) cos t + sint (2t)

Now,

h'(t) = t²cos t + cost + 2tsint

Differentiating again:

h''(t) = d/dt (t²cos t) + d/dt (cost) + d/dt (2tsint)

      = -2tsint - sint + 2tcost

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Find cos(2a+B) given that a = sin^-1 (4/5) and B = Tan^-1 (12/5)​

Answers

cos(2a + B) is approximately equal to -0.2948.

why it is?

To find cos(2a + B), we can use the double angle formula for cosine:

cos(2a + B) = cos(2a)cos(B) - sin(2a)sin(B)

We already know the values of a and B, so we can substitute them into the formula and simplify using the trigonometric identities:

a = sin²-1(4/5) = 53.13° (rounded to two decimal places)

B = tan²-1(12/5) = 67.38° (rounded to two decimal places)

cos(2a) = cos²2(a) - sin²2(a) = (cos(a))²2 - (1 - (cos(a))²2) = 2(cos(a))²2 - 1

cos(2a) = 2(sin²-1(4/5))²2 - 1 = 2(0.8)²2 - 1 = 0.32

sin(2a) = 2sin(a)cos(a)

sin(2a) = 2(sin(sin²-1(4/5)))cos(sin²-1(4/5)) = 2(4/5)(3/5) = 0.96

cos(B) = 1/sqrt(1 + tan²2(B)) = 1/sqrt(1 + (12/5)²2) = 5/13

sin(B) = tan(B)cos(B) = (12/5)(5/13) = 0.48

Substituting these values into the formula for cos(2a + B) gives:

cos(2a + B) = cos(2a)cos(B) - sin(2a)sin(B)

cos(2a + B) = (0.32)(5/13) - (0.96)(0.48)

cos(2a + B) = 0.166 - 0.4608

cos(2a + B) = -0.2948

Therefore, cos(2a + B) is approximately equal to -0.2948.

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