The root of the equation f(x) = 0 is found by using the Secant method. The initial guesses are x-1 = 3.6 and x0 = 1.5. Given that f(3.6) = 7.1 and f(1.5) = 3.9, the angle the secant line makes with the x axis is ___ (Report your answer in in degrees ; keep 4 decimal places.)

Answer:

8.82 years.

Step-by-step explanation:

Since, the monthly payment formula is,

[tex]P=\frac{PV(r)}{1-(1+r)^{-n}}[/tex]

Where, PV is the present value of the loan,

r is the rate per month,

n is number of months,

Here,

PV =  $ 25,000,

Annual rate = 5.2 % = 0.052 ⇒ Monthly rate, r = [tex]\frac{0.052}{12}[/tex]

( 1 year = 12 months )

P = $ 295,

By substituting the values,

[tex]295=\frac{25000(\frac{0.052}{12})}{1-(1+\frac{0.052}{12})^{-n}}[/tex]

By the graphing calculator,

We get,

[tex]n = 105.84[/tex]

Hence, the time ( in years ) = [tex]\frac{105.84}{12}=8.82[/tex]

Answer:

Odd numbers that are multiple of 5 and are in between 22 and 66 are-

25, 35, 45, 55, 65

Let this set be represented by A

A= {25, 35, 45, 55, 65}

the above form represents the set in its roster form

The set notation for the odd numbers between 22 and 66 that are multiples of 5 is { x ∈ N | x is odd, 22 < x < 66, x ≡ 0 (mod 5) }.

The set notation for the odd numbers between 22 and 66 that are multiples of 5 can be written as:

{ x ∈ N | x is odd, 22 < x < 66, x ≡ 0 (mod 5) }

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Answer:

B. 200 doses

Step-by-step explanation:

Given,

1 dose is required for 100 mg,

Since, 1 mg = 0.001 g,

⇒ 100 mg = 0.1 g

⇒ 1 dost is required for 0.1 g,

Thus, the ratio of doses and quantity ( in gram ) is [tex]\frac{1}{0.1}=10[/tex]

Let x be the doses required for 20 grams,

So, the ratio of doses and quantity is [tex]\frac{x}{20}[/tex]

[tex]\implies \frac{x}{20}=10[/tex]

[tex]\implies x=200[/tex]

Hence, 200 doses can be obtained from 20 grams of the drug.

Option 'B' is correct.

To solve this question, we will follow these steps:
1. We need to ensure that we use the same units for both the drug amount and the dose. Since the drug amount is given in grams and the dose in milligrams, we will convert the drug amount from grams to milligrams.
2. We know that 1 gram is equivalent to 1000 milligrams.
3. Now, let's convert the drug amount from grams to milligrams:
  We have 20 grams of the drug, so the conversion to milligrams is:
  \(20 \text{ grams} \times \dfrac{1000 \text{ milligrams}}{1 \text{ gram}} = 20,000 \text{ milligrams}\)
4. Next, we will divide the total milligrams of the drug by the milligram dosage that is recommended per dose to find out how many doses we can get from the drug amount.
5. Given that each dose is 100 mg, we now divide the total drug amount in milligrams by the dose in milligrams:
  \(20,000 \text{ milligrams} \div 100 \text{ milligrams per dose} = 200 \text{ doses}\)
Therefore, from 20 grams of the drug, we can obtain 200 doses.
The correct answer is:
  B. 200 doses

[tex]f(x)=\displaystyle\int_{-6}^x\sqrt{36-t^2}\,\mathrm dt[/tex]

The integrand is defined for [tex]36-t^2\ge0[/tex], or [tex]-6\le t\le6[/tex], so the domain should be the same, [tex]-6\le x\le6[/tex].

When [tex]x=-6[/tex], the integral is 0.

The integrand is non-negative for all [tex]x[/tex] in the domain, which means the value of [tex]f(x)[/tex] increases monotonically over this domain. When [tex]x=6[/tex], the integral gives the area of the semicircle centered at the origin with radius 6, which is [tex]\dfrac\pi26^2=18\pi[/tex], so the range is [tex]\boxed{0\le f(x)\le 18\pi}[/tex].

The range of the function f(x) is the integral from -6 to x of the square root of the quantity 36 minus t squared is [0, 6*π] because the total area of the semicircle is the maximum value.

The function f(x) is the integral from -6 to x of the square root of the quantity 36 minus t squared. This is a known geometrical shape, which is a semicircle with radius 6. To find the range of this function, we need to know the possible outcomes of this function. In general, for a semicircle of radius r, the values of the square root of the quantity r squared minus t squared will vary from 0 to r, both inclusive. So, if you consider the function from -6 to 6, the range would be [0, 6*π] because the total area of the semicircle is the maximum value.

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Answer:

The mean is 5.309.

Step-by-step explanation:

Given group of data,

4.1, 8.9, 3.2, 1.9, 7.3, 6.3, 6.7, 8.6, 3.2, 2.3, 5.9,

Sum = 4.1+ 8.9 + 3.2 + 1.9 + 7.3 + 6.3 + 6.7 + 8.6 + 3.2 + 2.3 + 5.9 = 58.4,

Also, number of observations in the data = 11,

We know that,

[tex]Mean=\frac{\text{Sum of all observation}}{\text{Total observations}}[/tex]

Hence, the mean of given data = [tex]\frac{58.4}{11}=5.30909\approx 5.309[/tex]

The magnetic field H(z,t) of an electromagnetic wave is related to the electric field E(z,t) by a factor of the speed of light. Therefore, if E(z,t) = 4cos(6 x 10^8 t - 2z) +3 sin(6 x 10^8 t -2z), the associated magnetic field would be H(z,t) = (4/c) cos(6 x 10^8 t - 2z) +(3/c) sin(6 x 10^8 t -2z), where c is speed of light, approximately 3 x 10^8 m/s.

The question is asking for the associated magnetic field H(z,t) of an Electromagnetic wave given the electric field E(z,t). A crucial fact to know for this question is that the electric and magnetic fields in an electromagnetic wave are perpendicular to each other and the direction of propagation. They also have a constant ratio of magnitudes in free space or air, which is the speed of light given by c = 1/√εOMO. Because of these relations, we know that we can find the magnetic field by simply dividing the given electric field by the speed of light in units that match the given Electric field.

So, if E(z,t) = 4cos(6 x 10^8 t - 2z) +3 sin(6 x 10^8 t -2z), then the associated magnetic field would be H(z,t) = (4/c) cos(6 x 10^8 t - 2z) +(3/c) sin(6 x 10^8 t -2z), where c is the speed of light, approximately 3 x 10^8 m/s.

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To find the associated magnetic field H(z, t), you can use Faraday's law of electromagnetic induction. This law states that the rate of change of magnetic flux through a surface is equal to the induced electromotive force (EMF) along the boundary of the surface. By following a step-by-step process, you can find the magnetic field B(z, t) using the given electric field E(z, t).

The associated magnetic field H(z, t) can be found by using Faraday's law of electromagnetic induction. Faraday's law states that the rate of change of magnetic flux through a surface is equal to the electromotive force (EMF) along the boundary of the surface. In this case, the magnetic field is changing due to the time-dependent electric field, so we can use Faraday's law to find the magnetic field.

By following these steps, you can find the associated magnetic field H(z, t) using Faraday's law of electromagnetic induction.

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Answer:

72 months approx.

Step-by-step explanation:

Monthly deposit = m = $140

r = 3.8% or 0.038

Amount needed in the account = A = $11300

The formula will be :

[tex]11300=140(\frac{(1+0.038/12)-1}{0.038/12} )[/tex]

[tex]11300=140(\frac{(1+0.038/12)-1}{0.003166})[/tex]

[tex]11300=44219.83((1.003166)^{m}-1)[/tex]

[tex]0.2555=(1.003166)^{m}-1[/tex]

[tex]1.2555=(1.003166)^{m}[/tex]

m=log1.2555/log1.003166

m =71.98 ≈ 72

Hence, it will take 72 months approx.

Answer:

Option b Tyree’s solution is inaccurate. In step 1, he substituted incorrectly.

Step-by-step explanation:

we know that

the formula to calculate the distance between two points is equal to

[tex]d=\sqrt{(y2-y1)^{2}+(x2-x1)^{2}}[/tex]

we have

[tex]A(-4,-2)\\B(-7,-7)[/tex]

step 1

substitute the values in the formula

[tex]d=\sqrt{(-7-(-2))^{2}+(-7-(-4))^{2}}[/tex]

step 2

Simplify

[tex]d=\sqrt{(-7+2)^{2}+(-7+4)^{2}}[/tex]

step 3

[tex]d=\sqrt{(-5)^{2}+(-3)^{2}}[/tex]

step 4

[tex]d=\sqrt{25+9}[/tex]

step 5

[tex]d=\sqrt{34}[/tex]

therefore

Tyree’s solution is inaccurate. In step 1, he substituted incorrectly.

Answer:

(a) a=6 and b≠[tex]\frac{11}{4}[/tex]

(b)a≠6

(c) a=6 and b=[tex]\frac{11}{4}[/tex]

Step-by-step explanation:

writing equation in agumented matrix form

[tex]\begin{bmatrix}1 &1 & 0 &1 &b\\ 2 &3 & 1 &5 &6\\ 0& 0 & 1 &1 &4\\ 0& 2 & 2&a &1\end{bmatrix}[/tex]

now [tex]R_{2} =R_{2}-2\times R_{1}[/tex]

[tex]\begin{bmatrix}1 &1& 0 &1 &b\\ 0 &1& 1 &3 &6-2b\\ 0& 0 & 1 &1 &4\\ 0& 2 & 2&a &1\end{bmatrix}[/tex]

now [tex]R_{4} =R_{4}-2\times R_{2}[/tex]

[tex]\begin{bmatrix}1 &1& 0 &1 &b\\ 0 &1& 1 &3 &6-2b\\ 0& 0 & 1 &1 &4\\ 0& 0 & 0 &a-6 &4b-11\end{bmatrix}[/tex]

a) now for inconsistent

rank of augamented matrix ≠ rank of matrix

for that  a=6 and b≠[tex]\frac{11}{4}[/tex]

b) for consistent w/ a unique solution

rank of augamented matrix = rank of matrix

  a≠6

c) consistent w/ infinitely-many sol'ns

  rank of augamented matrix = rank of matrix < no. of variable

for that condition

 a=6 and b=[tex]\frac{11}{4}

then rank become 3 which is less than variable which is 4.

Final answer:

The probability that the second character will die given that the first character dies is 53.33%. This is known as conditional probability.

Explanation:

To find the probability that the second character will die given that the first character dies, we use the concept of conditional probability.

The formula for conditional probability is P(B|A) = P(A and B) / P(A), where P(B|A) is the probability of event B occurring given that event A has occurred, P(A and B) will be the probability of both events A and B occurring, and P(A) is the probability of event A occurring.

In this scenario, event A is the first character dying, and event B is the second character dying. The student has already stated there is a 70% chance that the first character will survive, which means there is a 30% (100% - 70%) chance that the first character will die.

They've also stated a 16% chance that both characters will die. Applying the formula gives us P(B|A) = P(A and B) / P(A) = 0.16 / 0.30 = 0.5333, or 53.33%.

Therefore, the probability that the second character will die given that the first character dies is 53.33%. This kind of probability is called conditional probability.

Answer:

The answer is - budget constraint

Step-by-step explanation:

The slope of the budget constraint is determined by the relative price of the two goods, which is calculated by taking the price of one good and dividing it by the price of the other good.  

A budget constraint happens when a consumer demonstrates limited consumption patterns by a certain income.

The function log311y can be expressed as the sum of two logarithms, log3(11) + log3(y), according to the product rule of logarithms.

The function log311y represents the logarithm base 3 of the product of the numbers 11 and y. Using the rules of logarithms, we can rewrite this expression as a sum of two logarithms.

According to the product rule of logarithms, the logarithm of a product is the sum of the logarithms of the component numbers. Applying this rule to our expression, log311y becomes:

log3(11) + log3(y)

This is the sum of the logarithm base 3 of 11 and the logarithm base 3 of y. In conclusion, the function log311y can be expressed as the sum of the separate logarithms: log3(11) + log3(y).

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The logarithm log3(11y) can be broken down into two simpler logarithms, log3(11) and log3(y), by using properties of logarithms. This is the sum of the two simpler logarithms.

To express the logarithm log3(11y) as the sum or difference of logarithms, we will utilize the properties of logs:

Applying these properties to the given logarithm, we find:

log3(11y) = log3(11) + log3(y)

Thus, the original logarithm has been expressed as a sum of two simpler logarithms.

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The net force is what remains of the pull when we subtract the friction force:

[tex]F = 64-36 = 28N[/tex]

Now, use the law

[tex]F=ma[/tex]

and solve it for the acceleration

[tex]a = \dfrac{F}{m}[/tex]

to get the result:

[tex]a = \dfrac{28}{14}=2[/tex]

Answer:

2 m/s²

Step-by-step explanation:

F = applied force in the horizontal direction = 64 N

f = frictional force acting between the box and the floor = 36 N

m = mass of the box = 14 kg

a = acceleration of the box = ?

Force equation along the horizontal direction is given as

F - f = ma

Inserting the values

64 - 36 = 14 a

28 = 14 a

a =  [tex]\frac{28}{14}[/tex]

a = 2 m/s²

Answer:

  (0, -1), (1, 0), (2, 1)

Step-by-step explanation:

I find this easier to do after multiplying the equation by -1:

  y = x - 1

Pick any value for x, then subtract 1 from it to find the corresponding value of y.

Answer: 0.3679

Step-by-step explanation:

The formula for Poisson distribution  :-

[tex]P(x)=\dfrac{e^{-\lambda}\lambda^{x}}{x!}[/tex]

Let x be the number of breakdowns.

Given : The rate of breakdown per week :  0.5

Then , for 2 weeks period the number of breakdowns = [tex]\lambda=0.5\times2=1[/tex]

Then , the probability that there will be no breakdown on his car in the trip is given by :-

[tex]P(x)=\dfrac{e^{-1}1^{0}}{0!}=0.367879441171\approx0.3679[/tex]

Hence, the required probability : 0.3679

Answer:

The future value of the annuity due to the nearest cent is $2956.

Step-by-step explanation:

Consider the provided information:

It is provided that monthly payment is $175, interest is 7% and time is 11 years.

The formula for the future value of the annuity due is:

[tex]FV of Annuity Due = (1+r)\times P[\frac{(1+r)^{n}-1}{r}][/tex]

Now, substitute P = 175, r = 0.07 and t = 11 in above formula.

[tex]FV of Annuity Due = (1+0.07)\times 175[\frac{(1+0.07)^{11}-1}{0.07}][/tex]

[tex]FV of Annuity Due = (1.07)\times 175[\frac{1.10485}{0.07}][/tex]

[tex]FV of Annuity Due = 187.5(15.7835)[/tex]

[tex]FV of Annuity Due = 2955.4789[/tex]

Hence, the future value of the annuity due to the nearest cent is $2956.

Answer:

A. number of decayed atoms = 73.197

Step-by-step explanation:

In order to find the answer we need to use the radioactive decay equation:

[tex]N(t)=N0*e^{kt}[/tex] where:

N0=initial radioactive atoms

t=time

k=radioactive decay constant

In our case, when t=0 we have 1,000,000 atoms, so:

[tex]1,000,000=N0*e^{k*0}[/tex]

[tex]1,000,000=N0[/tex]

Now we need to find 'k'. Using the provied information that after 365 days we have 973,635 radioactive atoms, we have:

[tex]973,635=1,000,000*e^{k*365}[/tex]

[tex]ln(973,635/1,000,000)/365=k[/tex]

[tex] -0.0000732=k[/tex]

A. atoms decayed in a day:

[tex]N(t)=1,000,000*e^{-0.0000732t}[/tex]

[tex]N(1)=1,000,000*e^{-0.0000732*1}[/tex]

[tex]N(1)= 999,926.803[/tex]

Number of atoms decayed in a day = 1,000,000 - 999,926.803 = 73.197

B. Because 'k' represents the probability of decay, then the probability that on a given day 51 radioactive atoms decayed is k=0.0000732.

Answer: [tex]H_0:\mu\geq33[/tex]

[tex]H_a:\mu<33[/tex]

Step-by-step explanation:

Let [tex]\mu[/tex] be the average number of hours a person drive without adding the product.

Given claim : An infomercial claims that a woman drove 33 hours without​ oil.

i.e. [tex]\mu\geq33[/tex]

It is known that the null hypothesis always contains equal sign and alternative hypothesis is just opposite of the null hypothesis.

Thus the null and alternative hypothesis for the given situation will be :-

[tex]H_0:\mu\geq33[/tex]

[tex]H_a:\mu<33[/tex]

Answer:

2/13

Step-by-step explanation:

there are 4 jacks and 4 threes in a standard poker deck.

4+4 is 8

8/52=2/13

The probability of drawing either a Jack or a three from a standard deck of 52 cards is 2/13, because there are 8 such cards in a deck and the total number of cards in the deck is 52.

The question asks for the probability of drawing either a Jack or a three from a standard deck of 52 cards. To solve this, we need to count how many Jacks and threes there are in a deck. Since each suit (hearts, diamonds, clubs, and spades) includes one Jack and one three, there are 4 Jacks and 4 threes in a standard deck. Therefore, there are 8 cards that satisfy the condition (either a Jack or a three).

Since the total number of cards in the deck is 52, the probability of drawing either a Jack or a three is calculated as the number of favorable outcomes (drawing a Jack or a three) divided by the total number of outcomes (drawing any card from the 52-card deck). Thus, the probability is:

Probability = (Number of Jacks + Number of threes) / Total number of cards = (4 + 4) / 52 = 8 / 52 = 2 / 13

Therefore, the probability of drawing either a Jack or a three from a standard deck of 52 cards is 2/13.

Answer:

.188 pounds per day

Step-by-step explanation:

Given

Divide the amount of pounds gained by the total number of days

35/186 = .188

Answer

Approximately .188 pounds per day.

Answer:

Yes

Step-by-step explanation:

We are given that a function [tex]\phi(x)=x^2-x^{-1}[/tex]

We have to find that given function is an explicit solution to the linear equation

[tex]\frac{d^2y}{dx^2}-\frac{2}{x^2}y=0[/tex]

If given function is an explicit solution of given linear equation then it satisfied the given differential equation

Differentiate w.r.t x

Then we get [tex]\phi'(x)=2x+x^{-2}[/tex]

Again differentiate w.r.t x

Then we get

[tex]\phi''(x)=2-\frac{2}{x^3}[/tex]

Substitute all values in the given differential equation

[tex]2-\frac{2}{x^3}-\frac{2}{x^2}(x^2-x^{-1})[/tex]

=[tex]2-\frac{2}{x^3}-2+\frac{2}{x^3}=0[/tex]

Hence, given function is an explicit solution of given differential equation.

Therefore, answer is yes.

Answer: 0.33

Step-by-step explanation:

Let,

In this question we are using the Bayes' theorem,

where,

P(E1) = P(E2) = P(E3) = [tex]\frac{1}{3}[/tex]

As there is an equal probability assign for choosing a coin.

Given that,

it comes up heads

so, let A be the event that heads occurs

then,

P(A/E1) = 1

P(A/E2) = 0

P(A/E3) =  [tex]\frac{1}{2}[/tex]

Now, we have to calculate the probability that the opposite side of coin is tails.

that is,

P(E3/A) = ?

∴ P(E3/A) = [tex]\frac{P(E3)P(A/E3)}{P(E1)P(A/E1) + P(E2)P(A/E2) + P(E3)P(A/E3) }[/tex]

= [tex]\frac{(1/3)(1/2)}{(1/3)(1) + 0 + (1/2)(1/3)}[/tex]

= [tex]\frac{1}{6}[/tex] × [tex]\frac{6}{3}[/tex]

= [tex]\frac{1}{3}[/tex]

= 0.3333 ⇒ probability that the opposite face is tails.

Given a double-headed coin, a double-tailed coin, and a regular coin, the probability that the opposite face is tails after tossing a head is 33.33%, assuming we picked one coin randomly and tossed it to see a head.

The student is asking about a problem involving conditional probability, with the specific condition that one of the sides that came up is a head. We are given three coins: a double-headed coin, a double-tailed coin, and a regular coin. The aim is to calculate the probability that the opposite face is tails given that the tossed coin shows heads.

First, we need to consider the total number of heads that can come up when choosing any coin. This yields two heads from the double-headed coin, and one head from the regular coin, resulting in three possible heads. However, only the regular coin has a tail on the opposite side.

Consequently, the probability that the opposite face is tails given that a head has been tossed is 1 out of 3, or 33.33%.

Answer:

CF+PI=[tex]c_1e^{2x}+c_2e^{x}[/tex]+[tex]2e^{3t}[/tex]

Step-by-step explanation:

we have given y"-3y'=2y=[tex]4e^{3t}[/tex]

this differential equation solution have two part that CF and PI

CALCULATION OF CF :

[tex]m^2-3m+2=0[/tex]

[tex]m^2-2m-m+2=0[/tex]

[tex](m-1)(m-2)=0[/tex]

m=1 and m=2

so CF=[tex]c_1e^{2x}+c_2e^{x}[/tex]

CALCULATION OF PI :

PI =   [tex]\frac{4e^{3t}}{(m-1)(m-2)}[/tex]

at m= 3 in PI

[tex]PI=\frac{4e^{3t}}{2}=2e^{3t}[/tex]

so the complete solution is

CF+PI=[tex]c_1e^{2x}+c_2e^{x}[/tex]+[tex]2e^{3t}[/tex]

Answer:

[tex]y = 700 + 50x[/tex]

Step-by-step explanation:

Hello, great question. These types are questions are the beginning steps for learning more advanced Algebraic Equations.

If the company has a fixed cost (fixed being a keyword) of $700, then that cost will be a steady value before they even start to manufacture the bicycles. Afterwards they have to spend $50 on each bicycle they produce. Since we do not know the amount of bicycles that have been produced we can use the variable x to represent this.

[tex]y = 700 + 50x[/tex]

The equation above states that the company pays $700 plus $50 for every bike produced which comes out to a total of y.

A horizontal line is a line where all of the [tex]y[/tex] values are the same. In this case, [tex]\boxed{y=5}[/tex], so that is the equation.

A vertical line is where all of the [tex]x[/tex] values are the same.  Here, [tex]\boxed{x=7}[/tex], so that's the equation.

Answer:

see below

Step-by-step explanation:

A horizontal line has the same y value  and has a constant y value

y=5

A vertical line has the same x value  and has a constant x value

x=7

The given sequence can be expressed as a sequence of partial sums by finding the differences between terms and adding them to the previous term. The closed formula for the nth partial sum is Sn = n/2(3n - 1), where Sn represents the nth partial sum.

To express the given sequence as a sequence of partial sums, we can find the differences between consecutive terms:

5 - 1 = 4

12 - 5 = 7

22 - 12 = 10

35 - 22 = 13

51 - 35 = 16

From these differences, we can observe that each term in the sequence is obtained by adding the difference to the previous term. Therefore, the sequence can be written as a sequence of partial sums:

1, 1+4, 1+4+7, 1+4+7+10, 1+4+7+10+13, ...

To find a closed formula for the nth partial sum, we can use the formula for the sum of an arithmetic series:

Sn = n/2(a1 + an), where Sn represents the nth partial sum, a1 is the first term, and an is the nth term.

For the given sequence, a1 = 1 and the difference between consecutive terms is 3, so the nth term can be represented as an = 1 + 3(n-1). Substituting these values into the formula, we get:

Sn = n/2(1 + 1 + 3(n-1)) = n/2(2 + 3(n-1)) = n/2(3n - 1).

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Answer:

(A+B)(A+B)=A.A+B.A+A.B+B.B

Step-by-step explanation:

Given that matrices A and B are nxn matrices

We need to find (A+B)(A+B)

For understanding the multiplication of matrices let'take A is mxn and B is pxq matrices,we can multiple only when n=p,so our Ab matrices will be mxq.

We know that that in matrices AB is not equal to BA.

Now find  

(A+B)(A+B)=A.A+B.A+A.B+B.B

So from we can say that (A+B)(A+B) is not equal to A.A+2B.A+B.B because AB is not equal to BA in matrices.

So (A+B)(A+B)=A.A+B.A+A.B+B.B

The derivative of [tex]f(x,y,z)[/tex] at a point [tex]p_0=(x_0,y_0,z_0)[/tex] in the direction of a vector [tex]\vec a=a_x\,\vec\imath+a_y\,\vec\jmath+a_z\,\vec k[/tex] is

[tex]\nabla f(x_0,y_0,z_0)\cdot\dfrac{\vec a}{\|\vec a\|}[/tex]

We have

[tex]f(x,y,z)=3e^x\cos(yz)\implies\nabla f(x,y,z)=3e^x\cos(yz)\,\vec\imath-3ze^x\sin(yz)\,\vec\jmath-3ye^x\sin(yz)\,\vec k[/tex]

and

[tex]\vec a=-\vec\imath+2\,\vec\jmath+3\,\vec k\implies\|\vec a\|=\sqrt{(-1)^2+2^2+3^2}=\sqrt{14}[/tex]

Then the derivative at [tex]p_0[/tex] in the direction of [tex]\vec a[/tex] is

[tex]3\,\vec\imath\cdot\dfrac{-\vec\imath+2\,\vec\jmath+3\,\vec k}{\sqrt{14}}=-\dfrac3{\sqrt{14}}[/tex]

Answer with explanation:

Matrix A= (2 × 2) Matrix

Trace A= -9

Also,Determinant A= |A|=18

⇒Characteristics Polynomial is given by

Δ(A)=A² -A ×trace (A)+Determinant (A)

=A²+9 A+18

=(A+6)(A+3)

So, eigenvalues can be obtained by substituting :

 Δ(A)=0

(A+6)(A+3)=0

A= -6 ∧ A= -3

Two Eigenvalues are = -6, -3

[tex]HTT,HTH,HHH,TTT,THT,THH,TTH[/tex]