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To find the Taylor polynomial T3(x) for the function f(x) = ex sinx at a = 0, we can use the Taylor series expansion. The Taylor polynomial of degree 3 is given by:

\(T3(x) = f(0) + f'(0)x + (f''(0)/2!)x^2 + (f'''(0)/3!)x^3\)

First, let's find the derivatives of f(x):

f(x) = ex sinx

f'(x) = ex cosx + ex sinx

f''(x) = 2ex cosx

f'''(x) = 2ex cosx - 2ex sinx

Evaluate the derivatives at x = 0:

\(f(0) = e^0 sin(0) = 0\\f'(0) = e^0 cos(0) + e^0 sin(0) = 1\\f''(0) = 2e^0 cos(0) = 2\\f'''(0) = 2e^0 cos(0) - 2e^0 sin(0) = 2\)

Now, substitute these values into the Taylor polynomial formula:

\(T3(x) = 0 + 1x + (2/2!)x^2 + (2/3!)x^3\)

Simplifying:

\(T3(x) = x + x^2 + (1/3)x^3\)

Now, let's evaluate the limit:

\(lim(x- > 0) (e^x sinx / x^3)\)

We can use the Taylor polynomial T3(x) to approximate the function \(e^x sinx\)  as x approaches 0.

The term \(e^x sinx\) can be approximated as x when x is close to 0. Thus, as x approaches 0, the limit becomes:

\(lim(x- > 0) (x / x^3) = lim(x- > 0) (1 / x^2)\)

However, the limit of \(1 / x^2\)as x approaches 0 is infinity.

Therefore, the limit is infinity.

Please note that this is an approximation using the Taylor polynomial, and the actual behavior of the function may differ.

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Answer:

theb correct answer is d VHow does your body’s flexibility naturally change as you age?

Step-by-step explanation:

The amount that must be deposited to get a final amount of
$ 125,000 is $ 26.34 .

What is compound interest and how is it calculated?
The interest that is calculated using both the principal and the interest that has accrued during the previous period is called compound interest. It differs from simple interest in that the principal is not taken into account when determining the interest for the subsequent period with simple interest.
Mathematically, A = P (1 + (R/n))^t

Given, the amount that will be developed after interest is laid
= A = $125,00
Also, number of years for which amount is loaned = n = 16 years
Annual Percentage Rate of the loan availed = R = 7.3
Frequency with which interest is paid out per year = 1
Following the formula established in the literature, we have:
125000 = P(1 + 7.3)⁴ ⇒ P = 125000/(1 + 7.3)⁴ ⇒ P = 125000/8.3⁴ = 26.34
Therefore, the amount that must be deposited to get a final amount of
$ 125,000 is $ 26.34 .

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Using the Newton-Raphson method with an initial guess of x(0) = 1 and a relative error of 0.01, a root of the equation x³ - 2x = 5 is found to be approximately x ≈ 2.094.

To find a root of the equation x³ - 2x = 5 using the Newton-Raphson method, we start with an initial guess of x(0) = 1. The relative error is set to 0.01, which means we continue iterating until the difference between two consecutive approximations is less than 0.01 times the current approximation.

The Newton-Raphson method involves iteratively updating the approximation using the formula:

x(i+1) = x(i) - f(x(i)) / f'(x(i))

Here, f(x) = x³ - 2x - 5 is the given equation. We also need to calculate the derivative f'(x) = 3x² - 2.

Starting with x(0) = 1, we substitute this value into the formula to get x(1). We continue this process until the relative error condition is met.

By applying the Newton-Raphson method, the iteration converges to a root of the equation at approximately x ≈ 2.094.

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Answer:

(-4, -0.5)

Step-by-step explanation:

According to the graph ,

\((-5 , -4) =(x_1,y_1)\\(-3 , 3)=(x_2,y_2)\)

Sorry for the bad handwriting

Workings are in the picture

Answer:

d

Step-by-step explanation:

Answer:

16 lbs

Step-by-step explanation:

total of apples = 1/4 + 3/8 = 2/8 + 3/8 = 5/8

then 10 x 8/5 = 16

Answer: Statistical inference

Step-by-step explanation:

Statistical inference is the process of making conclusions about population  from the sample.

Here,  1,600 registered voters is just a sample which is used to represent the entire population, so the proportion of of all voters approve of the President is representing a statistic .

On absence of complete information of the population the statistic is the the measure to represent entire population and this is a part of statistical inference.

Hence, the  statement is an example of "statistical inference."

Answer:

10 (p + m) or 10 p + 10 m

Step-by-step explanation:

The value of X = 25.

Solution:

The two equations are

(i) the first condition is difference of x and y is 14

x - y = 14 ---------equation 1

(ii) the second condition of the given data is value of x is 3 times more than two times of y value.

x - 2y = 3 --------equation 2

From equation 2, we have to separate two variables x and y,

x = 3 + 2y  --------equation 3

We have to Substitute equation 3 in equation 1

3 + 2y - y = 14

3 + y = 14

y = 14 - 3

y = 11

Substitute y = 11 in equation 1........

x - 11 = 14

x = 14 + 11

x = 25

So, the value of x is 25 and y is 11.

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Answer:

Step-by-step explanation:

Given the difference of x and y is 14, and the difference of x and 2y is 3, you want two equations, their graph, and the value of x.

The difference of 'a' and 'b' is (a -b). Here, the two differences are expressed as the equations ...

The attachment shows a graph of these equations. Their point of intersection is (25, 11), meaning the value of x is 25.

The graph of the first equation is easily drawn by recognizing the x- and y-intercepts are 14 and -14, respectively.

The graph of the second equation will go through the x-intercept point of (3, 0) and the y-intercept point of (0, -3/2). It is probably easier to graph this by hand by considering the x-intercept point and the slope of 1/2.

Since we're only interested in the value of x, it is convenient to eliminate the variable y. We can to that by subtracting the second equation from twice the first:

  2(x -y) -(x -2y) = 2(14) -(3)

  x = 25 . . . . . . . . . simplify

Answer:

I already answered this for someone, but what you want to do is combine/cancel out like terms. So cancel out x on 2+7x/x to get 2+7/1. The first step in solving most rational equations is combining/canceling like terms, so follow through with that here.

 $1.09 is Total return in investment   .

What is investment?

Investments are assets that are bought or invested in to increase wealth and set aside cash from hard-earned income or gain. The main purpose of an investment is to provide an additional source of income or to generate a profit over a certain amount of time.

                                Any strategy for producing future revenue might be referred to as an investment. This involves, among other things, purchasing securities such as bonds, stocks, or real estate.

Total return in dollars = Profit / Initial investment

Highest = $67

Lowest = $32

                         no of shares     price per share      value

                                    a                      b                       c = a * b

lowest price              1000                32                         32000

highest price             1000               67                          67000

profit = sale value - purchase value                                35000

Total return in investment                                                   $1.09

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Given that the original sample had a variance of 4, variance of the transformed sample is 36.

Given the data in the question;

Now, Lets represent the original random variable by X  

So, Variance of X will be 4

Var(X) = 4

The Variance of the transformation is;

\(Y = 3X + 10\)

By using property of variance [ derived from its definition]

\(Var(Y) = a^2Var(X)\)

Here, X is the variable and "a" is the constant

We substitute

\(Var(Y) = 3^2\ * \ 4\)

\(Var(Y) = 9\ *\ 4\\\)

\(Var(Y) = 36\)

Therefore, given that the original sample had a variance of 4, variance of the transformed sample is 36.

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If the Rockwell hardness of pins of a certain type has a mean value of 50 and a standard deviation of 1.4, then we can use these values to make predictions about the hardness of individual pins.

For example, we might expect that approximately 68% of pins will have a hardness value between 48.6 and 51.4 (i.e., within one standard deviation of the mean), and approximately 95% of pins will have a hardness value between 47.2 and 52.8 (i.e., within two standard deviations of the mean). Additionally, we can use the mean and standard deviation values to calculate the probability of obtaining a particular hardness value or range of values, using tools like the normal distribution or z-scores. Overall, knowing the mean and standard deviation of the Rockwell hardness of pins of a certain type provides valuable information for understanding the variability and distribution of this important characteristic.

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An equation with the steepest graph has the largest absolute value of slope.

The equation with the steepest graph is the equation with the largest absolute value of slope.

A slope is a measure of how steep a line is.

If a line has a positive slope, it is rising to the right.

If a line has a negative slope, it is falling to the right.

If the slope of a line is zero, the line is horizontal.

To multiply the square root of 2 + i and its conjugate, you can use the complex multiplication formula.

(a + bi)(a - bi) = \(a^2 - abi + abi - b^2i^2\)

where the number is √2 + i. Let's do a multiplication with this:

(√2 + i)(√2 - i)

Using the above formula we get:

\((\sqrt{2})^2 - (\sqrt{2})(i ) + (\sqrt{2} )(i) - (i)^2\)

Further simplification:

2 - (√2)(i) + (√2)(i) - (- 1)

Combining similar terms:

2 + 1

results in 3. So (√2 + i)(√2 - i) is 3.

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The point P(12,−2) is not on ⨀C which has a centre at (9, -7) and radius √13

The equation for a circle has the generic form: x² + y² + 2gx + 2fy + c = 0. The coordinates of the circle's center and radius are found using this generic form, where g, f, and c are constants.

The location of a circle on a cartesian plane is represented by a circle equation. If the center and radius of a circle are known, it may be sketched on paper. Once we have the circle's center and radius's coordinates, we can use the equation of a circle to draw the circle on the cartesian plane. The equation of a circle can be represented in several ways, including

General form

Standard form

Parametric form

Polar form

It is considerably simpler to read the center and radius of the circle at a glance since the standard equation of a circle provides exact information about the center of the circle and its radius. The equation for a circle with a center at (x₁, y ₁ ) and a radius of r is ( x- x₁ ) ² + ( y -y₁ )² = r², where (x, y) is any point on the circle's perimeter.

The centre of the cirle is (9, -7) and one point on circumference is (6, -5)

The radius of the circle √{(9 - 6)² + (-7 + 5)²} = √{3² + (-2)²} = √{9+4} = √13

The equation of the circle will be

(x - 9)² + (y + 7)² = 13

Now for P(12, -2)

(12-9)² + (-2 + 7)² = 3² + 5² = 34 ≠ 13

So, the point does not lie on the circle.

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Answer:

k=20

Step-by-step explanation:

when x approaches 4, the denominator x-4 approaches 0

if the denominator is 0, it means that this is invalid

if the function is a number over 0 when x=4, it represents a vertical asymptote, which means no limit

so the only way possible to let there be a limit is to let the function be 0/0 when we plug in x=4

so x^2 + x - k = 0 when x = 4

4^2 + 4 - k = 0 ==> 20 - k = 0 ==> k = 20

Answer:

35

Step-by-step explanation:

Add them together and then divide

Answer:

b + b + h + h

Step-by-step explanation:

You are finding the perimeter of the given rectangle. Note that by definition of a rectangle, the opposite parallel sides will have the same measurement. In this case, Terrance gives the expression of 2b + 2h, and so simply just expand the expression:

2b + 2h = b + b + h + h

~

Answer:

Step-by-step explanation:

a) To determine if the polynomials 1+2t, 3+6t^2, 1+3t+4t^2 are linearly dependent in P2, we need to check if there exist constants c1, c2, and c3 such that c1(1+2t) + c2(3+6t^2) + c3(1+3t+4t^2) = 0, where 0 is the zero polynomial in P2.

Rewriting this equation in terms of the standard basis B = {1, t, t^2}, we have:

(c1 + c3) + (2c1 + 3c3)t + (4c2 + 3c3)t^2 = 0

This gives us the system of equations:

c1 + c3 = 0

2c1 + 3c3 = 0

4c2 + 3c3 = 0

Solving this system of equations, we get c1 = -3c3/2, c2 = -3c3/4. Therefore, any choice of c3 that is not equal to zero would give us a nontrivial solution, which implies that the polynomials are linearly dependent in P2.

b) To determine if the polynomials 1+2t+t^2, 3-9t^2, 1+4t+5t^2 are linearly dependent in P2, we need to check if there exist constants c1, c2, and c3 such that c1(1+2t+t^2) + c2(3-9t^2) + c3(1+4t+5t^2) = 0, where 0 is the zero polynomial in P2.

Rewriting this equation in terms of the standard basis B = {1, t, t^2}, we have:

(c1 + c3) + (2c1 + 4c3)t + (c1 + 5c3)t^2 - 9c2t^2 = 0

This gives us the system of equations:

c1 + c3 = 0

2c1 + 4c3 = 0

c1 + 5c3 - 9c2 = 0

Solving this system of equations, we get c1 = -2c3, c2 = (1/9)(c1 + 5c3). Therefore, any choice of c3 that is not equal to zero would give us a nontrivial solution, which implies that the polynomials are linearly dependent in P2.

Answer:

80

Step-by-step explanation:

add 60 plus 4o then subtracted from 180

Answer:

x = 100°

Step-by-step explanation:

40 + 60 =

100

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Answer:

Step-by-step explanation:

92 -- 2*2*23

146 ---2*73

The ratio of used books to all books is 7 : 13 and  the ratio of new books to used books is 6 : 7

If there are 13 books on a shelf. 6 of these books are new.

(a) The ratio of used books to all books

Number of all books = 13

Number of used books = total no. of books - no. of new books

                                       =  13 - 6

                                       = 7

So, ratio of used books to all books

= no. of all books : no. of total books

= 7 : 13

Thus, the ratio of used books to all books = 7 : 13

(b) The ratio of new books to used books

Number of new books = 6

Number of used books = 7

So, the ratio of new books to used books

= no. of new books : no. of used books

= 6 : 7

Thus, the ratio of new books to used books = 6 : 7

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Answer:

\(y = \frac{x}{3} \)

Step-by-step explanation:

divide both sides by 3 to get the answer

The  value of the line integral along C is 178/3.

To evaluate the line integral, we need to compute the integral of the given function along each segment of the curve separately and then sum them up.

First, let's consider the line segment from (0, 0) to (5, 1). Parameterizing this segment as x = t and y = t/5 (where t ranges from 0 to 5), we can rewrite the line integral as ∫₀⁵(t + 5(t/5)) dt + ∫₀⁵(t²)(1/5) dt. Simplifying, we get the value of the integral over this segment as (25/2) + (25/3) = 175/6.

Next, for the line segment from (5, 1) to (6, 0), we parameterize it as x = 5 + t and y = 1 - t (where t ranges from 0 to 1). Substituting these values into the line integral expression, we get ∫₀¹((5 + t) + 5(1 - t)) dt + ∫₀¹((5 + t)²)(-dt). Evaluating this integral gives us the value (69/2) - (32/3) = 181/6.

Finally, we add the values obtained from each segment: 175/6 + 181/6 = 356/6 = 178/3.

Therefore, the value of the line integral along C is 178/3.

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To find the probability of the event "the first die is 6 or the sum is 8," we need to count the number of outcomes that satisfy this event and divide it by the total number of possible outcomes.

To find the probability of the event "the first die is 6 or the sum is 8," we need to consider the total possible outcomes when rolling two dice and the favorable outcomes for this event.

Total possible outcomes: 6 sides on each die, so there are 6 x 6 = 36 possible outcomes.

Favorable outcomes:
1. First die is 6: There are 6 possible outcomes (6,1), (6,2), (6,3), (6,4), (6,5), and (6,6).
2. Sum is 8: There are 5 possible outcomes (2,6), (3,5), (4,4), (5,3), and (6,2).

However, (6,2) is counted twice, so we should subtract 1 from the total favorable outcomes: 6 + 5 - 1 = 10.

Probability = Favorable outcomes / Total possible outcomes = 10/36. Simplifying the fraction gives 5/18.

So, the probability of the event "the first die is 6 or the sum is 8" is 5/18.

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\(\\ \sf\longmapsto m=\dfrac{y_2-y_1}{x_2-x_1}\)

\(\\ \sf\longmapsto m=\dfrac{1-3}{-4-3}\)

\(\\ \sf\longmapsto m=\dfrac{-2}{-7}\)

\(\\ \sf\longmapsto m=\dfrac{2}{7}\)

Points are (-7,6),(11,-4)

\(\boxed{\sf slope(m)=\dfrac{y_2-y_1}{x_2-x_1}}\)

\(\\ \sf\longmapsto m=\dfrac{-4-6}{11+7}\)

\(\\ \sf\longmapsto m=\dfrac{-10}{18}\)

\(\\ \sf\longmapsto m=-\dfrac{5}{9}\)

Answer:

Step-by-step explanation:

Slope = \(\frac{y_{2}-y_{1}}{x_{2}-x_{1}}\)

9) Mark any two point on the line

(x₁ , y₁) = (3 , 3)   ;   (x₂, y₂) = (-4 ,1)

\(Slope =\frac{1-3}{-4-3}\\\\=\frac{-2}{-7}\\\\=\frac{2}{7}\)

10) (x₁ , y₁) = ( -7 , 6)   ;   (x₂, y₂) = (11 ,-4)

\(Slope =\frac{-4-6}{11-[-7]}\\\\ =\frac{-4-6}{11+7}\\\\=\frac{-10}{18}\\\\=\frac{-5}{9}\)

Answer:

The following statements are true:

• The square root of a positive number less than 1 is between 0 and 1.

• The square root of a perfect square is a whole number.

The binary representations are a) 0.11000110, b) 0.10000010 and c) 0.10100000.

Let's convert the given real numbers to binary with 8 binary places after the radix point.

A. 0.11:

To convert 0.11 to binary, we can use the following steps:

Multiply 0.11 by 2:

0.11 × 2 = 0.22

Take the integer part of the result, which is 0, and write it down.

Multiply the decimal part of the result by 2:

0.22 × 2 = 0.44

Again, take the integer part (0) and write it down.

Repeat steps 3 and 4 until you reach the desired precision (8 binary places after the radix point).

0.44 × 2 = 0.88 (integer part: 0)

0.88 × 2 = 1.76 (integer part: 1)

0.76 × 2 = 1.52 (integer part: 1)

0.52 × 2 = 1.04 (integer part: 1)

0.04 × 2 = 0.08 (integer part: 0)

0.08 × 2 = 0.16 (integer part: 0)

0.16 × 2 = 0.32 (integer part: 0)

0.32 × 2 = 0.64 (integer part: 0)

Write down the integer parts obtained in step 4 and 5, in order:

0.11000110

Therefore, the binary representation of 0.11 with 8 binary places after the radix point is 0.11000110.

B. 0.51:

To convert 0.51 to binary, we can use the same steps:

Multiply 0.51 by 2:

0.51 × 2 = 1.02

Take the integer part of the result, which is 1, and write it down.

Multiply the decimal part of the result by 2:

0.02 × 2 = 0.04

Again, take the integer part (0) and write it down.

Repeat steps 3 and 4 until you reach the desired precision (8 binary places after the radix point).

0.04 × 2 = 0.08 (integer part: 0)

0.08 × 2 = 0.16 (integer part: 0)

0.16 × 2 = 0.32 (integer part: 0)

0.32 × 2 = 0.64 (integer part: 0)

0.64 × 2 = 1.28 (integer part: 1)

0.28 × 2 = 0.56 (integer part: 0)

0.56 × 2 = 1.12 (integer part: 1)

0.12 × 2 = 0.24 (integer part: 0)

Write down the integer parts obtained in step 4 and 5, in order:

0.10000010

Therefore, the binary representation of 0.51 with 8 binary places after the radix point is 0.10000010.

C. 0.625:

To convert 0.625 to binary, we can use the same steps:

Multiply 0.625 by 2:

0.625 × 2 = 1.25

Take the integer part of the result, which is 1, and write it down.

Multiply the decimal part of the result by 2:

0.25 × 2 = 0.50

Again, take the integer part (0) and write it down.

Repeat steps 3 and 4 until you reach the desired precision (8 binary places after the radix point).

0.50 × 2 = 1.00 (integer part: 1)

0.00 × 2 = 0.00 (integer part: 0)

0.00 × 2 = 0.00 (integer part: 0)

0.00 × 2 = 0.00 (integer part: 0)

0.00 × 2 = 0.00 (integer part: 0)

0.00 × 2 = 0.00 (integer part: 0)

0.00 × 2 = 0.00 (integer part: 0)

0.00 × 2 = 0.00 (integer part: 0)

Write down the integer parts obtained in step 4 and 5, in order:

0.10100000

Therefore, the binary representation of 0.625 with 8 binary places after the radix point is 0.10100000.

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Answer:

y - 4 = 1/3(x + 5)

Step-by-step explanation:

The general equation for the point-slope form of a line is given by:

y - y1 = m(x - x1), where

Step 1:  To find the equation of the line through (-5, 4) and (1, 6), we first need to find the slope of the line using the slope formula, which is:

m = (y2 - y1) / (x2 - x1), where

We can allow (-5, 4) to be our (x1, y1) point and (1, 6) to be our (x2, y2) point:

m = (6 - 4) / (1 - (-5))

m = 2 / (1 + 5)

m = 2/6

m = 1/3

Now we can use the point-slope form of the equation with either of the two points.  

y - 4 = 1/3(x - (-5))

y - 4 = 1/3(x + 5)

Thus, the point-slope equation of the line through (-5, 4) and (1, 6) is:

y - 4 = 1/3(x + 5)