Given a non-linear system: y=x^3 - 3x^2 - 1 a) Find the linear approximation of the system at the point (1, -3) b) Plot the system and its linear approximation on a same plot using Matlab. Make sure your plot is clear and having labeling

Answer:   29

Step-by-step explanation:

Let S denotes the total number of people surveyed, A denotes the event of having dog , B denotes the event of having cats and C denotes the event of having fish.

Given :     n(S)=50  ;n(A)=11  ;  n(B) =13  and  n(C)=6

Also, n(A∩B)=5 ;  n(A∩C) = 2 and n(B∩C)=3 and n(A∩B∩C)=1

We know that,

[tex]n(A\cup B\cup C)=n(A)+n(B)+n(C)-n(A \cap B)-n(A \cap C)-n(B \cap C)-n(A \cap B\cap C)\\\\=11+13+6-5-2-3+1=21[/tex]

Now, the number of people owned none of these pets :-

[tex]n(S)-n(A\cup B\cup C)\\\\=50-21=29[/tex]

Hence, the number of people owned none of these pets =29

Answer:

a) -2 from (-2,0) b) 6 from (6,0) c) y-intercept: 12 from (0,12)

Step-by-step explanation:

The X intercepts in a quadratic function are the points of the x-axis crossed by the parabola. One quadratic equation may have up to two points on the X-axis. This or these points in the X-axis, the Zeros of this function,  will be crossed by the parabola.

The Y-intercept is the point of the y-axis crossed by the parabola.

Solving the equation:

[tex]-x^{2} +4x+12=0\\ x'=\frac{-4+\sqrt{64}}{-2} \\ x"=\frac{-4-\sqrt{64}}{-2} \\ x'=-2\\ x"=6\\[/tex]

S={-2, 6} These values, or zeros of this quadratic function are the X, intercepts.

c) The indepent term, or c, in f(x)= ax²+bx+c in this case is 12, also is the Y coordinate for the Parabola Vertex. This point is our intercept for y.

Answer:

If [tex]P_0 (x_0,y_0,z_0)[/tex] is a point on the surface, then the cartesian equation of the  tangent plane at [tex]P_0 (x_0,y_0,z_0)[/tex] is

[tex](\ast)z = z_0 + \frac{\partial z}{ \partial x}(x_0,y_0)\cdot (x -x_0) + \frac{\partial z}{\partial y} (x_0, y_0) (y -y_0)[/tex],

where [tex]z_0 = x_0 f \left ( \frac{y_0}{x_0}\right )[/tex].

Given that

[tex]\frac{\partial z}{\partial x} (x_0 , y_0) = f \left( \frac{y_0}{x_0}\right ) - \frac{y_0}{x_0} \cdot \frac{\partial f}{\partial x}(x_0,y_0) \ , \ \frac{\partial z}{\partial y} (x_0 , y_0)=\frac{\partial f}{\partial y} (x_0,y_0)[/tex], then

[tex](\ast)[/tex] becomes

[tex](\ast \ast) z=x_0 f \left ( \frac{y_0}{x_0}\right ) + f \left( \frac{y_0}{x_0}\right ) - \frac{y_0}{x_0}\cdot \frac{\partial f}{\partial x} (x_0,y_0)\cdot (x -x_0)+\frac{\partial f}{\partial y} (x_0,y_0)\cdot (y -y_0)[/tex].

Finally, replacing [tex] (x,y,z)=(0,0,0)[/tex] in [tex](\ast \ast)[/tex] you have that the equality is true for all [tex]P_0[/tex]. This means that [tex]O(0,0,0)[/tex]

belongs to all tangent planes and therefore, the result follows.      

Answer:

Hello my friend! The answer is 1.75X10^-15

Step-by-step explanation:

If you multiply 2.5 x7 = 17.5

When we do the product of exponential terms with the same base, we can sum de  exponents. In this case (-10) + (-6) = -16.  

However, to scientific notation, we have to use 1.75

So, the final result wich were 17.5x10-16, will be "1.75x10^-15"

Answer: 0.03

Step-by-step explanation:

Total number of napkins: 5 x 525 = 2,625

75/2626 = 0.02857, which rounds to 0.03

The cost per individual napkin, when rounded to the nearest hundredth, is $0.03.

To begin finding the cost per napkin, we first need to find out how many napkins are purchased every 3 months. Since each box contains 525 napkins and you purchase 5 boxes every 3 months, that would be 525 * 5 = 2625 napkins. The cost of these napkins is $75.00.

So, to find the cost per individual napkin, you would divide the total cost by the total number of napkins. That would be 75 / 2625 = $0.028571... When rounded to the nearest hundredth, this becomes $0.03. So, each individual napkin costs $0.03. Therefore, the correct answer is (C) 0.03.

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Answer:  a.    320 books

Step-by-step explanation:

Given : The total number of books  were placed on the discount shelf for 70% off the regular price = 960

The fraction of books sold = [tex]\dfrac{2}{3}[/tex]

Then, the number of books sold = [tex]\dfrac{2}{3}\times960=640[/tex]

Now, the number of books remain on the discount shelf = [tex]960-640=320[/tex]

Hence, the number of books remain on the discount shelf =320

Answer:

p = 5/7

Step-by-step explanation:

The given function is:

[tex]g(x) = -3x^{2} - 2x + 8[/tex] for -4 ≦ x < 1

[tex]g(x) = -2x + 7p[/tex] for 1 ≦ x ≦ 5

Part a)

A continuous function has no breaks, jumps or holes in it. So, in order for g(x) to be continuous, the point where g(x) stops during the first interval -4 ≦ x < 1 must be equal to the point where g(x) starts in the second interval 1 ≦ x ≦ 5

The point where, g(x) stops during the first interval is at x = 1, which will be:

[tex]-3(1)^{2}-2(1)+8=3[/tex]

The point where g(x) starts during the second interval is:

[tex]-2(1)+7(p) = 7p - 2[/tex]

For the function to be continuous, these two points must be equal. Setting them equal, we get:

3 = 7p - 2

3 + 2 = 7p

p = [tex]\frac{5}{7}[/tex]

Thus the value of p for which g(x) will be continuous is [tex]\frac{5}{7}[/tex].

Part b)

We have to find p by setting the two pieces equal to each other. So, we get the equation as:

[tex]-3x^{2}-2x+8=-2x+7p\\\\ -3x^{2}+8=7p[/tex]

Substituting the point identified in part (a) i.e. x=1, we get:

[tex]-3(1)^{2}+8=7p\\\\ 5=7p\\\\ p=\frac{5}{7}[/tex]

This value agrees with the answer found in previous part.

Answer: Yes , the points (-4,-1), (2,1) and (11,4) are collinear.

Step-by-step explanation:

We know that if three points [tex](x_1,y_1),(x_2,y_2)[/tex] and [tex](x_3,y_3)[/tex] are collinear, then their area must be zero.

The area of triangle passes through points[tex](x_1,y_1),(x_2,y_2)[/tex] and [tex](x_3,y_3)[/tex] is given by :-

[tex]\text{Area}=\dfrac{1}{2}|x_1(y_2-y_3)+x_2(y_3-y_1)+x_3(y_1-y_2)|[/tex]

Given points : (-4,-1), (2,1) and (11,4)

Then, the area of ΔABC will be :-

[tex]\text{Area}=\dfrac{1}{2}|-4(1-4)+(2)(4-(-1))+(11)(-1-1)|\\\\\Rightarrow\text{Area}=\dfrac{1}{2}|-4(-3)+(2)(5)+(11)(-2)||\\\\\Rightarrow\text{Area}=\dfrac{1}{2}|12+10-22|\\\\\Rightarrow\text{Area}=\dfrac{1}{2}|0|=0 [/tex]

Hence, the points (-4,-1), (2,1) and (11,4) are collinear.

Answer:

The integer representing the change of altitude to avoid the storm is 8,000

The integer representing the altitude after passing the storm is 30,000

Explanation:

The diagram of this question is shown in the attached image

We are given that the initial altitude of the plane was 30,000 ft

1- During the storm:

The plane flew at at altitude of 38,000 feet

To get the change in the altitude, we will subtract the final altitude from the initial one

change of altitude = final altitude - initial altitude

change of altitude = 38,000 - 30,000 = 8,000 ft

Therefore, the integer representing the change of altitude to avoid the storm is 8,000

2- After the storm:

We know that, after the storm, the plane returned to its initial altitude

Given that the initial altitude is 30,000 ft, this would mean that the integer representing the altitude after passing the storm is 30,000

Hope this helps :)

Final answer:

The integer representing the change in altitude when the airplane avoided the storm and then returned to its original altitude is zero. The altitude after passing the storm is 30,000 feet, the same as its original altitude before the ascent.

Explanation:

The integer representing the airplane's change in altitude to avoid the storm is zero because it returned to its original altitude after passing the storm. During the avoidance maneuver, the airplane would have increased in altitude (a positive change) and then decreased the same amount to return to its original altitude (a negative change). The sum of this positive and negative change is zero.

The integer representing the altitude after passing the storm is 30,000 feet, which is the same as the original altitude since the airplane returned to this altitude after avoiding the storm.

When considering aircraft performance, it's vital to take into account the rate of climb and descent, potential energy swaps from kinetic energy, and altitude effects on aircraft performance. However, in this instance, the specific figure for altitude change during the storm avoidance is not given, but the concept of returning to starting altitude implies no net change.

Answer:

(B) 18.

Step-by-step explanation:

We are asked to find the perimeter of a quadrilateral with vertices at (-1, 4), (7,4), (7,5), and (-1. 5).

First of all, we will draw vertices of quadrilateral on coordinate plane and connect the vertices as shown in the attached photo.

We can see that our quadrilateral is a parallelogram, whose parallel sides are equal.

[tex]\text{Perimeter of quadrilateral}=8+1+8+1[/tex]

[tex]\text{Perimeter of quadrilateral}=16+2[/tex]

[tex]\text{Perimeter of quadrilateral}=18[/tex]

Therefore, the perimeter of the given quadrilateral is 18 units.

Answer: 0.339

Step-by-step explanation:

Given : Tour players Harry, Ron, Harmione and Ginny are playing a card game.

. A deck of 52 cards are dealt out equally.

Then, the number of card each person has = [tex]\dfrac{52}{4}=13[/tex]

If Harmione and Ginny have a total of 8 spades among them, then the total cards the total spades left = 13-8=5

Now, the number of ways to get 3 of 5 spades : [tex]^5C_3=\dfrac{5!}{3!2!}=10[/tex]

Number of ways to draw remaining 10 cards :  [tex]^{21}C_{10}=\dfrac{21!}{10!11!}=352716[/tex]

Also, the total cards Harmione and Ginny have = 13+13=26

Then the total cards left = 26

The number of ways to get 13 cards for Harry :

[tex]^{26}C_{13}=\dfrac{26!}{13!(26-13)!}\\\\=\dfrac{26!}{13!13!}=10400600[/tex]

Now, the probability that Harry has 3 of the remaining 5 spades :_

[tex]\dfrac{^5C_3\times ^{21}C_{10}}{^{26}C_{13}}\\\\=\dfrac{10\times352716}{10400600}\\\\=0.339130434783\approx0.339[/tex]

Hence, the probability that Harry has 3 of the remaining 5 spades= 0.339 (approx)

Answer:

$2191.12

Step-by-step explanation:

We are asked to find the value of a bond after 10 years, if you invest $1000 in a savings bond that pays 4% interest, compounded semi-annually.

[tex]FV=C_0\times (1+r)^n[/tex], where,

[tex]C_0=\text{Initial amount}[/tex],

r = Rate of return in decimal form.

n = Number of periods.

Since interest is compounded semi-annually, so 'n' will be 2 times 10 that is 20.

[tex]4\%=\frac{4}{100}=0.04[/tex]

[tex]FV=\$1,000\times (1+0.04)^{20}[/tex]

[tex]FV=\$1,000\times (1.04)^{20}[/tex]

[tex]FV=\$1,000\times 2.1911231430334194[/tex]

[tex]FV=\$2191.1231430334194[/tex]

[tex]FV\approx \$2191.12[/tex]

Therefore, the bond would be $2191.12 worth in 10 years.

Answer:

6233500

Step-by-step explanation:

We are given that CEO of a company that sells car stereos has determined the profit x number of stereos.

The profit of selling x number of stereos is given by

[tex]P(x)=-0.04x^2=100x-16500[/tex]

We have to find the value of profit when the company selling 12500 stereos.

Substitute the value of x=12500

Then, we get

[tex]P(12500)=-.04(12500)^2+1000(12500)-16500[/tex]

[tex]P(12500)=-6250000+12500000-16500=-6266500+12500000[/tex]

[tex]P(12500)=6233500[/tex]

Hence, the company should expect profit 6233500 from selling 12500 stereos.

Answer:

$767.49

Step-by-step explanation:

given,

Amount of money = $16,500

quarterly rate = 3.6/4 = 0.9 %

times = 6 × 4 = 24 quarters.                            

[tex]A =\dfrac{P(r(1+r)^n)}{(1+r)^n-1}\\\\A =\dfrac{16500\times(0.009(1+0.009)^{24})}{(1+0.009)^{24}-1}\\A = \$ 767.49[/tex]          

hence, the payment to amortize the dept  will be equal to $767.49  .

Answer: a) 15:26, and b) 15:11.

Step-by-step explanation:

Since we have given that

Number of graded tests = 15

Number of total tests = 26

Number of ungraded tests is given by

[tex]26-15\\\\=11[/tex]

a) Proportion of all exams has Dr. Fitxgerald graded is given by

15:26.

b) Ratio of graded to ungraded tests is given by 15:11

Hence, a) 15:26, and b) 15:11.

(a) The proportion of all exams graded by Dr. Fitzgerald is [tex]\(\frac{15}{26}\)[/tex].

(b) The ratio of graded to ungraded tests is [tex]\(\frac{15}{26 - 15}\) or \(\frac{15}{11}\)[/tex].

(a) To find the proportion of exams graded by Dr. Fitzgerald, we divide the number of exams graded by the total number of exams. This gives us the fraction:

[tex]\[ \text{Proportion graded} = \frac{\text{Number of exams graded}}{\text{Total number of exams}} = \frac{15}{26} \][/tex]

This fraction represents the part of the whole set of exams that has been graded.

(b) To find the ratio of graded to ungraded tests, we take the number of exams that have been graded and divide it by the number of exams that have not been graded. The number of ungraded exams is the total number of exams minus the number of graded exams:

[tex]\[ \text{Number of ungraded exams} = \text{Total number of exams} - \text{Number of exams graded} = 26 - 15 = 11 \][/tex]

Now, we can find the ratio:

[tex]\[ \text{Ratio of graded to ungraded tests} = \frac{\text{Number of exams graded}}{\text{Number of exams ungraded}} = \frac{15}{11} \][/tex]

This ratio tells us how many times greater the number of graded exams is compared to the number of ungraded exams.

Answer:

1. The college (Arts and Science, Business, etc.) you are enrolled in

Nominal

2. The number of students in a statistics course  Ratio

3. The age of each of your classmates  Ratio

4. Your hometown  Nominal

Step-by-step explanation:

Nominal, ordinal, interval, or ratio data are the four fundamental levels of measurement scales that are used to capture data.

Nominal, are used for labeling variables, without any quantitative value.

Ordinal, the order of the values is what is significant, but the differences between each one is not really known.

Interval, we know both, the order and the exact differences between the values

Ratio, they have the order, the exact value between units, and have an absolute zero

Answer:

1360.78 g

Step-by-step explanation:

1 lb = 453.592 g

3 lbs = 3 * 453.592 g = 1360.78 g

Answer:    

[tex]x-1[/tex] is a factor of [tex]x^n - 1[/tex]

Step-by-step explanation:

[tex]x-1[/tex] is a factor of [tex]x^n - 1[/tex]

We will prove this with the help of principal of mathematical induction.

For n = 1, [tex]x-1[/tex] is a factor [tex]x-1[/tex], which is true.

Let the given statement be true for n = k that is [tex]x-1[/tex] is a factor of [tex]x^k - 1[/tex].

Thus, [tex]x^k - 1[/tex] can be written equal to  [tex]y(x-1)[/tex], where y is an integer.

Now, we will prove that the given statement is true for n = k+1

[tex]x^{k+1} - 1\\=(x-1)x^k + x^k - 1\\=(x-1)x^k + y(x-1)\\(x-1)(x^k + y)[/tex]

Thus, [tex]x^k - 1[/tex] is divisible by [tex]x-1[/tex].

Hence, by principle of mathematical induction, the given statement is true for all natural numbers,n.

The digital scale should display 3.0 when weighing 3 lbs of ham. This is because 3 pounds exactly can be displayed as the decimal 3.0 after converting the number into a fraction, 3/1, and dividing the numerator by the denominator.

When Daniel wants to buy 3 lb of ham, the digital scale at the grocery store should display the decimal 3.0. This is because 3 pounds exactly translates to 3.0 in decimal terms.

The process of converting a number like 3 into a fraction would begin by writing it as 3/1 (as any number can be written over 1).

To convert that into decimal form, you would divide the top number (numerator) by the bottom number (denominator), so 3 ÷ 1 = 3.0.

Thus, the digital scale should read 3.0 when Daniel weighs out his 3 lbs of ham.

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Hi!

You know that if f(p) = 0, then (x-p) is a factor of the polynomial f(x)

Then, f(3)=0 is the case p=0, son the factor is (x-3)

Answer: Since we are told that f(3) = 0, we know that (x-3) is a factor.

Step-by-step explanation:

(12 + -2 )/2

10/2

5 im pretty sure

Answer: 0.3399

Step-by-step explanation:

The binomial probability distribution formula to find the probability of getting success in x trial:-

[tex]P(x)=^nC_xp^x(1-p)^{n-x}[/tex], where n is the number of trials and p is the probability of getting success in each trial.

Given : Jorgensens received a shipment of 12 DVD Players. Shortly after arrival the  manufacturer called to say that that he had accidentally shipped  five defective units with the shipment.

i.e. The proportion of the defective units : [tex]p=\dfrac{5}{12}\approx0.417[/tex]

Also, Mr. Jorgensen pulled ten of the units and tested two of them.

For n=2, the  probability that neither of them was defective:-

[tex]P(x=0)=^{2}C_0(0.417)^0(1-0.417)^{2}\\\\=(1)(1)(0.583)^{2}\ \ \ [\text{ Since}^nC_0=1]\\\\=0.339889\approx0.3399[/tex]

Hence, the  probability that neither of them was defective = 0.3399

Answer:    a) 45    b) 60

Step-by-step explanation:

Given : A small restaurant has a menu with 2 appetizers, 5 main courses, and 3 desserts.

a) Number of meals includes an main course , a dessert and a appetizer, :-

[tex]2\times5\times3=30[/tex]

Number of meals includes an main course and a dessert and but not appetizer , then total possible meals:-

[tex]5\times3=15[/tex]

Then, the number of meals are possible if each includes an main course and a dessert, but may or may not include an appetizer= 30+15=45

b) Number of meals includes an main course and appetizer but not dessert:

[tex]5\times2=10[/tex]

Number of meals includes only main course =5

Now, the number of meals if dessert is also not required= 45+5+10=60

Answer:

D. A, B, and C

Step-by-step explanation:

Option (A) is true because when two straight lines intersect to each other we get two pair of vertically opposite angles and the angles opposite to each other is always equal.

Option (B) is also correct as If the sum of two angles is equal to 180°, then they are supplementary to each other.

Option (C) is also correct as If the sum of the two angles is equal to 90°, then they are Complementary to each other.

Hence, Option (D) is correct.

Answer: A programmer is the person who is responsible for making  computer programs.He/she makes sure that the program is created according to the requirement and  accurate performing operations .The goals of the programmer are as follows:-

Programmer is indulged in these goals because there are always upcoming new technologies in the field of programming so, to keep theirselves updates and maintain their skill they improve theirselves time to time. Also it can affect the job of the programmer if they are not aware about programming skills quite well or might end up losing the job.

Answer: c.  $4668.30

Explanation:

Given:

Sales = $513000

Sales made by an individual = 14% of $513000

Sales made by an individual = [tex]\frac{14}{100}\times 513000[/tex]

Sales made by an individual = $71280

Commission made on this sales = 6.5% of  $71280

Commission made on this sales = [tex]\frac{6.5}{100}\times 71280[/tex]

Commission made on this sales = $4668.30

[tex]2xyy'+y^2-4x^3=0[/tex]

Let [tex]z(x)=y(x)^2[/tex], so that [tex]z'(x)=2y(x)y'(x)[/tex] (which appears in the first term on the left side):

[tex]xz'+z=4x^3[/tex]

This ODE is linear in [tex]z[/tex], and we don't have to find any integrating factor because the left side is already the derivative of a product:

[tex](xz)'=4x^3\implies xz=x^4+C\implies z=\dfrac{x^4+C}x[/tex]

[tex]\implies y(x)=\sqrt{\dfrac{x^4+C}x}[/tex]

With [tex]y(1)=2[/tex], we get

[tex]2=\sqrt{1+C}\implies C=3[/tex]

so the solution is as given in your post.

Answer:

20 cm

Step-by-step explanation:

We are given a trapezoid, where the length of shorter base or on of the parllel line is 16 cm and the length of other parallel side is 24 cm.

Let the two parallel sides be x and y that is x = 16 cm and y = 24 cm.

A median of a trapezoid is a line segment that divides the non parallel sides of a trapezoid equally or a line segment that passes through the mid points of non-parallel sides of a trapezoid.

The length of median of a trapezoid = [tex]\frac{\text{Sum of parallel sides}}{2}[/tex] = [tex]\frac{16+24}{2}[/tex] = 20 cm.

Thus, the length of median of trapezoid is 20 cm.

Answer:

Yolanda had the same number of penalties in both games.

Step-by-step explanation:

Both of these penalties can be modeled by a first order equation.

Game of Meteor-Mania:

In a game of Meteor-Mania, each penalty is -5 seconds. So the expression for the total of penalties is:

Tp(n) = -5*n, where n is the number of penalties.

In the game of Meteor-Mania, Yolanda had penalties totaling -25 seconds. So

-25 = -5*n *(-1)

5n = 25

n = 25/5

n = 5

Yolanda had 5 penalties in the game of Meteor-Mania

Game of Cosmic Calamity

In a game of Meteor-Mania, each penalty is -7 seconds. So the expression for the total of penalties is:

Tp(n) = -7*n, where n is the number of penalties.

In the game of Cosmic Calamity, Yolanda had penalties totaling -35 seconds. So

-35 = -7n *(-1)

7n = 35

n = 35/7

n = 5

Yolanda had 5 penalties in the game of Cosmic Calamity

Yolanda had the same number of penalties in both games.