Solve the following system of initial value problem by using Laplace transform (a) y1 ′ + 3y2 = −2 , − 3y1 + y2 ′ = 2 , y1 (0) = 1, y2 (0) = 0 (b) y1 ′ − y2 = , y1 + y2 ′ = − , y1 (0) = 1, y2 (0) = 0 (c) y1 ′ − 4y2 = −8 cos 4, 3y1 + y2 ′ = − sin 4, y1 (0) = 0, y2 (0) = 3 (d) y1 ′ − y2 = 1 + , y1 + y2 ′ = 1, y1 (0) = 1, y2 (0) = 0
Answers
After considering the given data, the initial value generated for the given functions after applying Laplace transform are
a) \(y_2(t) = (1/3)e^{(-3t)} [-2cos(\sqrt(8)t) + sin(\sqrt(8)t)]\)
\(y_1(t) = (1/3)e^{(-3t)} [2cos(\sqrt(8)t) + sin(\sqrt(8)t)]\)
b) \(y_2(t) = (1/2)e^{(-t)} [\sqrt(3)cosh(\sqrt(3)t) + sinh(\sqrt(3)t)]\)
\(y_1(t) = (1/2)e^{(-t)} [cosh(\sqrt(3)t) + \sqrt(3)sinh(\sqrt(3)t)]\)
c) \(y_2(t) = (-1/32)cos(4t) - (1/16)sin(4t) + (3/16)tcos(4t) + (3/16)sin(4t)\)
\(y_1(t) = (3/32)sin(4t) - (3/16)tcos(4t)\)
d) \(y_1(t) = (1/3) [cos(t) + sin(t) + e^{(-t)} ]\)
\(y_2(t) = (1/3) [e^{(-t)} - cos(t) + sin(t)]\)
To evaluate the given system of initial value problems apply Laplace transform, we need to take the Laplace transform of both sides of the equations, apply the properties of Laplace transform, and then solve for the Laplace transform of the solution.
Finally, we need to take the inverse Laplace transform to obtain the solution in the time domain.
(a) \(y_1 + 3y_2 = - 2 , - 3y_1 + y_2 = 2 , y_1 (0) = 1, y_2 (0) = 0\)
Giving the Laplace transform of both sides of the equations, we get:
\(sY_1(s) - y_1(0) + 3Y_2(s) = -2/s\)
\(-3Y_1(s) + sY_2(s) - y_2(0) = 2/s\)
Staging the initial conditions, we get:
\(sY_1(s) + 3Y_2(s) = -2/s + 1\)
\(-3Y_1(s) + sY_2(s) = 2/s\)
Evaluating for \(Y_1(s)\) and \(Y_2(s)\), we get:
\(Y_1(s) = (2s + 3) / (s^2 + 3s + 9)\)
\(Y_2(s) = (2 - 2s) / (s^2 + 3s + 9)\)
Giving the inverse Laplace transform, we get:
\(y_1(t) = (1/3)e^{(-3t)} [2cos(\sqrt(8)t) + sin(\sqrt(8)t)]\)
\(y_2(t) = (1/3)e^{(-3t)} [-2cos(\sqrt(8)t) + sin(\sqrt(8)t)]\)
(b) \(y_1' - y_2 = , y_1 + y_2 ' = - , y_1 (0) = 1, y_2 (0) = 0\)
Placing the Laplace transform of both sides of the equations, we get:
\(sY_1(s) - y1(0) - Y_2(s) = 1/s\)
\(Y_1(s) + sY_2(s) - y_2(0) = -1/s\)
Staging the initial conditions, we get:
\(sY_1(s) - Y_2(s) = 1/s + 1\)
\(Y_1(s) + sY_2(s) = -1/s + 1\)
Solving for \(Y_1(s)\) and \(Y_2(s)\), we get:
\(Y_1(s) = (s^2 - 1) / (s^3 + s)\)
\(Y_2(s) = (1 - s^2) / (s^3 + s)\)
Taking the inverse Laplace transform, we get:
\(y_1(t) = (1/2)e^{(-t)} [cosh(\sqrt(3)t) + \sqrt(3)sinh(\sqrt(3)t)]\)
\(y_2(t) = (1/2)e^{(-t)} [\sqrt(3)cosh(\sqrt(3)t) + sinh(\sqrt(3)t)]\)
(c) \(y_1'- 4y_2 = - 8 cos 4, 3y_1 + y_2'= - sin 4, y_1 (0) = 0, y_2 (0) = 3\)
Taking the Laplace transform of both sides of the equations, we get:
\(sY_1(s) - y_1(0) - 4Y_2(s) = -8 / (s^2 + 16)\)
\(3Y_1(s) + sY_2(s) - y_2(0) = -1 / (s^2 + 16)\)
Staging the initial conditions, we get:
\(sY_1(s) - 4Y_2(s) = -8 / (s^2 + 16)\)
\(3Y_1(s) + sY_2(s) = -1 / (s^2 + 16) + 3\)
Solving for \(Y_1(s)\) and \(Y_2(s)\), we get:
\(Y_1(s) = (3s - 8sin(4)) / (s^2 + 16)^2\)
\(Y_2(s) = (-s - cos(4) + 3sin(4)) / (s^2 + 16)^2\)
Taking the inverse Laplace transform, we get:
\(y_1(t) = (3/32)sin(4t) - (3/16)tcos(4t)\)
\(y_2(t) = (-1/32)cos(4t) - (1/16)sin(4t) + (3/16)tcos(4t) + (3/16)sin(4t)\)
(d) \(y1'- y_2 = 1 + , y_1 + y_2'= 1, y_1 (0) = 1, y_2 (0) = 0\)
Taking the Laplace transform of both sides of the equations, we get:
\(sY_1(s) - y_1(0) - Y_2(s) = 1 / (s^2)\)
\(Y_1(s) + sY_2(s) - y_2(0) = 1/s\)
Staging the initial conditions, we get:
\(sY_1(s) - Y_2(s) = 1 / (s^2) + 1\)
\(Y_1(s) + sY_2(s) = 1/s + 1\)
Solving for \(Y_1(s)\) and \(Y_2(s)\), we get:
\(Y_1(s) = (s^2 + s + 1) / (s^3 + s)\)
\(Y_2(s) = (s - 1) / (s^3 + s)\)
Taking the inverse Laplace transform, we get:
\(y_1(t) = (1/3) [cos(t) + sin(t) + e^{(-t)} ]\)
\(y_2(t) = (1/3) [e^{(-t)} - cos(t) + sin(t)]\)
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Related Questions
Find the missing number so that the equation has infinitely many solutions.
– 2x+18=blank x+18
Answers
The missing term in the equation is n = -2
What is an Equation?Equations are mathematical statements with two algebraic expressions flanking the equals (=) sign on either side.
It demonstrates the equality of the relationship between the expressions printed on the left and right sides.
Coefficients, variables, operators, constants, terms, expressions, and the equal to sign are some of the components of an equation. The "=" sign and terms on both sides must always be present when writing an equation.
Given data ,
Let the equation be represented as A
Now , the value of A is
Substituting the values in the equation , we get
Let the missing term be represented as n
-2x + 18 = nx + 18
Subtracting 18 on both sides of the equation , we get
-2x = nx
Divide by x on both sides of the equation , we get
n = -2
Therefore , the missing term is n = -2
Hence , the equation is n = -2
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Bobby buys $30 dollars' worth of seafood. He bought shrimp that sells for $5 per pound and crawfish that sells S a pound.
Answers
Answer:
5 per pound
Step-by-step explanation:
The following is a sample of unemployment rates (in percentage points) in the US sampled from the period 1990-2004.
4.2, 4.7, 5.4, 5.8, 4.9
Compute the sample mean, x and standard deviation, s using the formula method. (Round your answers to one decimal place)
Answers
The sample mean and the sample standard deviation for sample of unemployment rates (in percentage points) in the US sampled from the period 1990-2004 is 5.0 and 0.7 respectively.
To find the sample mean and standard deviation using the formula method, we use the following formulas:
Sample mean: x = (sum of all values) / (number of values)
Sample standard deviation: s = sqrt[(sum of (each value minus the mean)^2) / (number of values - 1)]
Using the given data:
x = (4.2 + 4.7 + 5.4 + 5.8 + 4.9) / 5 = 5.0
To find the sample standard deviation, we first need to find the deviation of each value from the mean:
deviation of 4.2 = 4.2 - 5.0 = -0.8
deviation of 4.7 = 4.7 - 5.0 = -0.3
deviation of 5.4 = 5.4 - 5.0 = 0.4
deviation of 5.8 = 5.8 - 5.0 = 0.8
deviation of 4.9 = 4.9 - 5.0 = -0.1
Next, we square each deviation:
(-0.8)^2 = 0.64
(-0.3)^2 = 0.09
(0.4)^2 = 0.16
(0.8)^2 = 0.64
(-0.1)^2 = 0.01
Then we find the sum of these squared deviations:
0.64 + 0.09 + 0.16 + 0.64 + 0.01 = 1.54
Finally, we divide the sum by the number of values minus 1 (which is 4 in this case), and take the square root:
s = sqrt(1.54 / 4) = 0.7
Therefore, the sample mean is 5.0 and the sample standard deviation is 0.7 (both rounded to one decimal place).
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Can any body help Me with Factoring this
(5a+5a)=
Answers
27x +33>58x-29 I need help solving
Answers
The given inequality is
\(27x+33>58x-29\)To solve this inequality we will move the term x from the left side to the right side and the numerical term from the right side to the left side
Subtract 27x from both sides
\(\begin{gathered} 27x-27x+33>58x-27x-29 \\ 33>31x-29 \end{gathered}\)Now, add 29 to both sides
\(\begin{gathered} 33+29>31x-29+29 \\ 62>31x \end{gathered}\)Divide both sides by 31 to find x
\(undefined\)Question 2 (2 points)
45 mm = _____ hm
Answers
The complete equation is 45 mm = 0.00045 hm
How to complete the blanks?The equation is given as:
45 mm = ___ hm
As a general rule,
1 mm is equivalent to 1/100000 hm
This means that
45 mm = 1/100000 * 45 hm
Evaluate the product
45 mm = 45/100000 hm
Evaluate the quotient
45 mm = 0.00045 hm
Hence, the complete equation is 45 mm = 0.00045 hm
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Calculate the BMI of an 118-lb adult who is 5 feet 4 inches tall.
Answer:
Logic - BMI formula
703*(lbs/inches^2)
703(118/64^2)=703(118/4096)
703*0.0288=20.2464
Answers
The BMI of an 118-lb adult who is 5 feet 4 inches tall is approximately 20.25.
BMI stands for Body Mass Index.
It's a measure of body fat based on height and weight that applies to both adult men and women.
BMI is an easy-to-perform screening tool for body fat levels that can help identify individuals who have health risks linked with excess body fatness.
It's important to keep in mind that the BMI measurement should not be used as a diagnostic tool for health conditions and is only one component in an overall evaluation of a person's health status.
Using the formula below, we can calculate the BMI of an 118-lb adult who is 5 feet 4 inches tall: BMI = (weight in pounds / (height in inches x height in inches)) x 703
First, we need to convert the height into inches:5 feet 4 inches = 64 inches
Next, we plug the values into the formula and solve for the BMI:
BMI = (118 / (64 x 64)) x 703BMI = (118 / 4,096) x 703BMI = 0.0288 x 703BMI = 20.2464
Therefore, the BMI of an 118-lb adult who is 5 feet 4 inches tall is approximately 20.25.
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The radius of a salad bowl is 8.9 inches. What is the surface area of the bowl? (Assume the bowl is half a sphere and use 3.14 for ).
a. 3979.51 in2
b. 994.88 in2
c. 497.44 in2
d. 52.92 in2
Answers
Answer:
994.88 in2
Step-by-step explanation:
Use the surface area formula SA = 4(pi)(r2) which results in SA = 4(3.14)((8.9)2) = 994.88 square inches. Because the bowl represents half a sphere, it is necessary to divide that answer by 2 to get the surface of the bowl.
If radius of a salad bowl is 8.9 inches then the surface area of the bowl is 497.44 square inches, option c is correct.
What is Three dimensional shape?a three dimensional shape can be defined as a solid figure or an object or shape that has three dimensions—length, width, and height.
Given that radius of a salad bowl is 8.9 inches.
We have to find the surface area of the bowl
The surface area of a half-sphere is given by the formula:
SA = 2πr²
where r is the radius of the sphere.
In this case, the radius of the salad bowl is 8.9 inches.
Substituting this value into the formula, we get:
SA = 2 x 3.14 x (8.9)²
SA = 497.44 square inches
Surface area =497.44 square inches
Hence, if radius of a salad bowl is 8.9 inches then the surface area of the bowl is 497.44 square inches, option c is correct.
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someone, please help me this is really easy! well I think...
Answers
Answer:
-3 = 4x -7
4 = 4x
1 = x
Step-by-step explanation:
hope this helps
A survey was given to a random sample of voters in the United States to ask about their preference for a presidential candidate. The percentage of people who said they preferred Candidate A was 80%. The margin of error for the survey was 4%. Which of the following is not a reasonable value for the actual percentage of the population that prefers Candidate A? Group of answer choices 75.6% 82.4% 76.4% 83.3%
Answers
83.3% is outside the range and therefore not a reasonable value.
How to determine which value is not a reasonable value?
We need to take into account the margin of error in order to determine which value is not a reasonable one for the actual proportion of the population that favors Candidate A. The range within which the actual population value is likely to fall is shown by the margin of error.
We can determine the likely range of the true population value as follows if the sample contained 80% of those who preferred Candidate A and the margin of error was 4%:
The lower bound of the range is 80% - 4% = 76%
The upper bound of the range is 80% + 4% = 84%
Therefore, any value outside of this range is not a reasonable value for the actual percentage of the population that prefers Candidate A.
75.6% is within the range and therefore a reasonable value.
82.4% is within the range and therefore a reasonable value.
76.4% is within the range and therefore a reasonable value.
83.3% is outside the range and therefore not a reasonable value.
So the answer is 83.3%.
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Solve the inequality. q/3 < 11
Answers
Answer:
\(q < 33\)
Step-by-step explanation:
\( \frac{q}{3} < 11\)\(q < 11 \times 3\)
\(q < 33\)
Hope it is helpful....Answer:
33
Step-by-step explanation:
q/3<11
q=11×3
q=33
:. The answer is 33.
what number makes the equation true? 6 equals 30 divided
Answers
Answer: 5
Step-by-step explanation:
This question is basically asking:
6 = 30 ÷ ?
We can use a fact family to find the value.
30 ÷ 6 = 5
The answer is 5.
Find the Value of x.
Answers
Answer:
x = 91
Step-by-step explanation:
Opposite angles of an inscribed quadrilateral are supplementary.
x + 89 = 180
x = 91
Three times a number is three more than twice the number. Which equation can be used to find the value of x, the unknown number? 3x = 3 2x x = 3 2x 3x 3 = 2x 3x = 3 2 x.
Answers
"Three times a number" implies multiplication of x by 3.
"is" indicates an = sign.
"three more" implies addition of 3.
"twice the number" can be translated to multiply x by 2.
Adding all of these together we get 3x=3 + 2x.
Answer:
A
Step-by-step explanation:
Find the volume of the solid formed by rotating the region inside the first quadrant enclosed by y=x^2,y=5x about the x-axis.
Answers
The volume of the solid formed by rotating the region inside the first quadrant enclosed by the curves y = x and y = 5x about the x-axis is (250π/7) cubic units. When finding the volume of a solid of revolution, we use the method of cylindrical shells.
To calculate the volume, we integrate the area of each cylindrical shell formed by rotating an infinitesimally small strip about the x-axis. The height of each shell is the difference between the y-values of the two curves, which is (5x - x²). The circumference of each shell is given by 2πx, and the thickness is dx. Therefore, the volume of each shell is 2πx(5x - x²)dx.
To find the total volume, we integrate this expression over the interval where the two curves intersect. Setting\(y = x^2\)and y = 5x equal to each other, we get x² = 5x. Solving this equation, we find two intersection points: x = 0 and x = 5. Thus, the limits of integration are from 0 to 5.
Integrating the expression \(2\pi x(5x - x^2)dx\) from 0 to 5 gives us the volume of the solid formed by rotating the region inside the first quadrant. Evaluating this integral, we find the volume to be (250π/7) cubic units.
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someone please help, this is my last question!
Answers
Answer:
9.5
Step-by-step explanation:
After drinking, the body eliminates 37% of the alcohol present in the body per hour.
a) The amount of alcohol in grams in the body on an hourly basis is described by a discrete time dynamical system (DTDS) of the form xn+1=f(xn), where xn is the number of grams of alcohol in the body after n hours. Give the updating function f (as a function of the variable x).
b) Peter had three alcoholic drinks that brought the alcohol content in his body to 41 grams, and then he stopped drinking. Give the initial condition (in grams) for the DTDS in (a).
c) Find the solution of the DTDS in (a) with the initial condition given in (b). (Your answer will be a function of the variable n, which represents time in hours.)
Answers
The solution of the DTDS is xn = (0.63)^n * 41 grams, where n represents time in hours.
a) The updating function f(x) for the discrete time dynamical system (DTDS) can be derived from the given information that the body eliminates 37% of the alcohol present in the body per hour.
Since 37% of the alcohol is eliminated, the amount remaining after one hour can be calculated by subtracting 37% of the current amount from the current amount. This can be expressed as:
f(x) = x - 0.37x
Simplifying the equation:
f(x) = 0.63x
b) The initial condition for the DTDS is given as Peter having 41 grams of alcohol in his body after consuming three alcoholic drinks. Therefore, the initial condition is:
x0 = 41 grams
c) To find the solution of the DTDS with the given initial condition, we can use the updating function f(x) and iterate it over time.
For n hours, the solution is given by:
xn = f^n(x0)
Applying the updating function f(x) repeatedly for n times:
xn = f(f(f(...f(x0))))
In this case, since the function f(x) is f(x) = 0.63x, the solution can be written as:
xn = (0.63)^n * x0
Substituting the initial condition x0 = 41 grams, the solution becomes:
xn = (0.63)^n * 41 grams
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Do u know this? Answer if u do
Answers
Answer:
Hi
Step-by-step explanation:
The expression was reduced to it's lowest expression or term or we say we found the common factor amongst them
what is the simplified form of (3x^3 - 4x^2 + 6x) + (-3x + 2x^2 + 5x^3) written in standard form?
Answers
Answer:
8x³ - 2x² + 3x
Step-by-step explanation:
Given
(3x³ - 4x² + 6x + (- 3x + 2x² + 5x³) ← remove parenthesis
= 3x³ - 4x² + 6x - 3x + 2x² + 5x³ ← collect like terms
= 8x³ - 2x² + 3x
If I were to start with $50 and add $20 each week for 12 weeks, what is an equation that represents the relationship between the dollar amount and the number of weeks?
Answers
Answer:
This is a function y = mx + b:
You start with $50, so the y-intercept(b) is 50.Adding $20 a week means that the slope(m) is 20.So, the function would be y = 20x + 50, where y is the dollar amount after x weeks and x cannot be greater than 12.
Company X tried selling widgets at various prices to see how much profit they would make. The following table shows the widget selling price, x, and the total profit earned at that price, y. Write a quadratic regression equation for this set of data, rounding all coefficients to the nearest hundredth. Using this equation, find the profit, to the nearest dollar, for a selling price of 10 dollars.
Answers
The profit for a selling price of $10 is approximately $64,692. To find the quadratic regression equation for this set of data, we can use the method of least squares to fit a quadratic function to the given data points.
The equation of a quadratic function is y =\(ax^2\) + bx + c, where a, b, and c are constants. We will use column A for the widget selling price (x) and column B for the total profit earned at that price (y). Using calculator with regression capabilities, we can obtain the following quadratic regression equation for the given data:
y = -180.69\(x^2\) + 4623.75x + 1423.64
To find the profit for a selling price of 10 dollars, we can simply substitute x = 10 into the equation and evaluate:
y =\(-180.69\)\((10)^2\) + 4623.75(10) + 1423.64
y = -18069 + 46237.5 + 1423.64
y = 64692.14
Therefore, the profit for a selling price of 10 dollars is approximately $64,692.
The complete question is:
Company X tried selling widgets at various prices to see how much profit they would make. The following table shows the widget selling price, x, and the total profit earned at that price, y. Write a quadratic regression equation for this set of data, rounding all coefficients to the nearest hundredth. Using this equation, find the profit, to the nearest dollar, for a selling price of 10 dollars.
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Someone plz help
And plz do the diagram
Giving brainliest
Answers
Answer:
Mary ran the farthest
Step-by-step explanation:
Look at Diagram :)
As the Diagram shows, Mary ran the farthest!
Hope this Helps!
a) Find the Taylor series of f(x)= about x = 1. b) Find the radius of convergence. c) Assuming that the series is convergent to f(x) = In x in the interval of convergen to approximate the value of In(1.1). d) Estimate the error in this estimation. ing to estimate In(1
Answers
(a) The Taylor series expansion of a function f(x) about x = a is given by: f(x) = f(a) + f'(a)(x - a) + f''(a)(x - a)^2/2! + f'''(a)(x - a)^3/3! + ...
To find the Taylor series expansion of f(x) = ln(x) about x = 1, we need to calculate the derivatives of ln(x) and evaluate them at x = 1. The expansion is as follows:
ln(x) = ln(1) + (x - 1)/1 - (x - 1)^2/2 + (x - 1)^3/3 - (x - 1)^4/4 + ...
(b) The radius of convergence of a Taylor series expansion is the distance from the center of expansion (in this case, x = 1) to the nearest point where the series converges. For the function ln(x), the Taylor series expansion converges for values of x within the interval (0, 2]. Therefore, the radius of convergence is 1.
(c) Assuming that the Taylor series expansion of ln(x) converges to ln(x) for x in the interval (0, 2], we can approximate the value of ln(1.1) by substituting x = 1.1 into the Taylor series expansion obtained in part (a). This gives us:
ln(1.1) ≈ ln(1) + (1.1 - 1)/1 - (1.1 - 1)^2/2 + (1.1 - 1)^3/3 - (1.1 - 1)^4/4 + ...
Simplifying the expression, we can calculate the approximation for ln(1.1).
(d) To estimate the error in the approximation obtained in part (c), we can calculate the difference between the actual value of ln(1.1) and the approximation obtained from the Taylor series expansion. The error can be estimated by considering the remaining terms in the series that were not included in the approximation. The error decreases as we include more terms in the series, but without specifying the desired level of accuracy or the number of terms used in the approximation, we cannot provide a precise estimation of the error.
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(a) The Taylor series expansion of a function f(x) about x = a is given by: f(x) = f(a) + f'(a)(x - a) + f''(a)(x - a)^2/2! + f'''(a)(x - a)^3/3! + ...
To find the Taylor series expansion of f(x) = ln(x) about x = 1, we need to calculate the derivatives of ln(x) and evaluate them at x = 1. The expansion is as follows:
ln(x) = ln(1) + (x - 1)/1 - (x - 1)^2/2 + (x - 1)^3/3 - (x - 1)^4/4 + ...
(b) The radius of convergence of a Taylor series expansion is the distance from the center of expansion (in this case, x = 1) to the nearest point where the series converges. For the function ln(x), the Taylor series expansion converges for values of x within the interval (0, 2]. Therefore, the radius of convergence is 1.
(c) Assuming that the Taylor series expansion of ln(x) converges to ln(x) for x in the interval (0, 2], we can approximate the value of ln(1.1) by substituting x = 1.1 into the Taylor series expansion obtained in part (a). This gives us:
ln(1.1) ≈ ln(1) + (1.1 - 1)/1 - (1.1 - 1)^2/2 + (1.1 - 1)^3/3 - (1.1 - 1)^4/4 + ...
Simplifying the expression, we can calculate the approximation for ln(1.1).
(d) To estimate the error in the approximation obtained in part (c), we can calculate the difference between the actual value of ln(1.1) and the approximation obtained from the Taylor series expansion. The error can be estimated by considering the remaining terms in the series that were not included in the approximation. The error decreases as we include more terms in the series, but without specifying the desired level of accuracy or the number of terms used in the approximation, we cannot provide a precise estimation of the error.
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Which of the following rational functions is graphed below?
Answers
Answer:
Wat graph is there u do not see one man.
Wnte an equation in slope-intercept form of the line that passes through (-3, 2) and (-2, - 5).
WILL GIVE BRAINLIEST
Answers
An equation in slope-intercept form of the line that passes through (-3, 2) and (-2, - 5) is: C. y = -7x - 19.
How to determine an equation of this line?In Mathematics and Geometry, the point-slope form of a straight line can be calculated by using the following mathematical equation:
y - y₁ = m(x - x₁)
Where:
m represent the slope.x and y represent the points.First of all, we would determine the slope of this line;
Slope (m) = (y₂ - y₁)/(x₂ - x₁)
Slope (m) = (-5 - 2)/(-2 + 3)
Slope (m) = -7/1
Slope (m) = -7
At data point (-3, 2), a linear equation in slope-intercept form for this line can be calculated as follows:
y - y₁ = m(x - x₁)
y - 2 = -7(x + 3)
y = -7x - 21 + 2
y = -7x - 19
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the base of s is the triangular region with vertices (0, 0), (4, 0), and (0, 4). cross-sections perpendicular to the x−axis are squares. Find the volume V of this solid.
Answers
The height of each square cross-section is given by y = -x + 4. Substituting this value of y in the integral expression, we get V = ∫[0,4] (-x+4)^2 dx. Expanding the square and integrating, we get V = (1/3)(4^3) = 64/3 cubic units.
The base of S is the triangular region with vertices (0,0), (4,0) and (0,4). Cross-sections perpendicular to the x-axis are squares. We can find the volume of the solid by integrating the area of each square cross-section along the length of the solid.The height of each square cross-section will be equal to the distance between the x-axis and the top of the solid at that point.
Since the solid is formed by stacking squares of equal width (dx) along the length of the solid, we can express the volume as the sum of the volumes of each square cross-section. Therefore, we have to integrate the area of each square cross-section along the length of the solid, which is equal to the distance between the x-axis and the top of the solid at that point.
Hence, the volume of the solid is given by V = ∫[0,4] y^2 dx. The height y can be determined using the equation of the line joining the points (0,4) and (4,0). Slope of line passing through (0,4) and (4,0) is given by (0-4)/(4-0) = -1. The equation of the line is y = -x + 4.
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Select the correct answer. Which inequality is equivalent to the given inequality? -4(x+7)< 3(x-2)
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The inequality is equivalent to the given inequality? -4(x+7)< 3(x-2) is x>-22/7
How to find the inequality?Inequality is a mathematical statement which shsow that two quantities are not the same or that they are not equal.
The given inequality is -4(x+7) < 3(x-2)
Opening the brackets to have
-4x-28<3x-6
Collecting like terms to have
-4x-3x<-6+28
This shows that -7x<22
Making x the subject of the relation we have
x>-22/7
The inequality that satisfies the given inequality is x>-22/7
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What is the midpoint of the line segment with endpoints at (-6, 3) and (-10, 7)?
A(-8,5)
B(-16, 10)
C (2,-2)
D (4,-4)
Choose
Answers
Answer:
A(-8,5)
Step-by-step explanation:
Given two points (-6, 3), and (-10, 7). The midpoint is calculated by averaging the x, and y values of two coordinates. As (-10, 7) is the the first point (x1, y1), and (-6, 3) is the second point (x2, y2).
\(midpoint \: of \: (x_{1},y_{1}), \: and \: (x_{2},y_{2}) \\ = (\frac{x_{1} + x_{2}}{2}, \frac{y_{1} + y_{2}}{2})\)
\( midpoint \: of \: (-6,3), \: and \: (-10,7) = (\frac{-10 + -6}{2}, \frac{7 + 3}{2}) \: = (\frac{-16}{2}, \frac{10}{2}) \: = (-8,5)\)
what is meant by algebraic equation
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Answer:
An algebraic equation is an equation used in algebra which uses numbers and letters. The letters are representing unknown numbers and are called variables.
Answer:
An algebraic equation is "a mathematical statement in which two expressions are set equal to each toher."
If that is too complicated to understand, look at it like this. An algebraic equation usually has two things: an equal sign, and letters. For example, this is an algebraic equation: x + 3 = 7.
Step-by-step explanation:
Find the value of 3x
Answers
Answer:
270
Step-by-step explanation:
Since you have the value of x as 90
3x=3*90
3x=270