The solution of the initial-value problem is:
x(t) = f0 / (ω^2 - γ^2) cos(γt), x(0) = 0, x'(0) = 0
To solve the given initial-value problem:
d2x/dt2 + ω^2 x = f0 cos(γt), x(0) = 0, x'(0) = 0
where ω ≠ γ, we can use the method of undetermined coefficients to find a particular solution for the nonhomogeneous equation. We assume that the particular solution has the form:
x_p(t) = A cos(γt) + B sin(γt)
where A and B are constants to be determined. Taking the first and second derivatives of x_p(t) with respect to t, we get:
x'_p(t) = -A γ sin(γt) + B γ cos(γt)
x''_p(t) = -A γ^2 cos(γt) - B γ^2 sin(γt)
Substituting these expressions into the nonhomogeneous equation, we get:
(-A γ^2 cos(γt) - B γ^2 sin(γt)) + ω^2 (A cos(γt) + B sin(γt)) = f0 cos(γt)
Expanding the terms and equating coefficients of cos(γt) and sin(γt), we get the following system of equations:
A (ω^2 - γ^2) = f0
B γ^2 = 0
Since ω ≠ γ, we have ω^2 - γ^2 ≠ 0, so we can solve for A and B as follows:
A = f0 / (ω^2 - γ^2)
B = 0
Therefore, the particular solution is:
x_p(t) = f0 / (ω^2 - γ^2) cos(γt)
To find the general solution of the differential equation, we need to solve the homogeneous equation:
d2x/dt2 + ω^2 x = 0
This is a second-order linear homogeneous differential equation with constant coefficients. The characteristic equation is:
r^2 + ω^2 = 0
which has complex roots:
r = ±iω
Therefore, the general solution of the homogeneous equation is:
x_h(t) = C1 cos(ωt) + C2 sin(ωt)
where C1 and C2 are constants to be determined from the initial conditions. Using the initial condition x(0) = 0, we get:
C1 = 0
Using the initial condition x'(0) = 0, we get:
C2 ω = 0
Since ω ≠ 0, we have C2 = 0. Therefore, the general solution of the homogeneous equation is:
x_h(t) = 0
The general solution of the nonhomogeneous equation is the sum of the particular solution and the homogeneous solution:
x(t) = x_p(t) + x_h(t) = f0 / (ω^2 - γ^2) cos(γt)
Therefore, the solution of the initial-value problem is:
x(t) = f0 / (ω^2 - γ^2) cos(γt), x(0) = 0, x'(0) = 0
Learn more about " initial-value problem" :
https://brainly.com/question/31041139
#SPJ11
\[ \{(-1,0),(-6,-9),(-4,-4),(-9,-9)\} \] What is the domain? (Type whole numbers. Use a comma to separate answers as needed.)
The domain of this set is {-1, -6, -4, -9}, which are the x-values of the given coordinates.
The domain of a set of coordinates represents the set of all possible x-values or inputs in a given set. In this case, the set of coordinates is {(-1,0),(-6,-9),(-4,-4),(-9,-9)}. The domain of this set is {-1, -6, -4, -9}, which are the x-values of the given coordinates.
The domain is determined by looking at the x-values of each coordinate pair in the set. In this case, the x-values are -1, -6, -4, and -9. These are the only x-values present in the set, so they form the domain of the set.
The domain represents the possible inputs or values for the independent variable in a function or relation. In this case, the set of coordinates does not necessarily indicate a specific function or relation, but the domain still represents the range of possible x-values that are included in the set.
learn more about domain here:
https://brainly.com/question/28135761
#SPJ11
The complete question is:
{(−1,0),(−6,−9),(−4,−4),(−9,−9)} What Is The Domain? (Type Whole Numbers. Use A Comma To Separate Answers As Needed.)
b) Use a Riamann sum with five subliotervals of equal length ( A=5 ) to approximate the area (in square units) of R. Choose the represectotive points to be the right endpoints of the sibbintervals. square units. (c) Repeat part (b) with ten subinteivals of equal length (A=10). Kasate unicr f(x)=12−2x
b) The area of region R, approximated using a Riemann sum with five subintervals, is 30 square units.
To approximate the area of region R using a Riemann sum, we need to divide the interval of interest into subintervals of equal length and evaluate the function at specific representative points within each subinterval. Let's perform the calculations for both parts (b) and (c) using the given function f(x) = 12 - 2x.
b) Using five subintervals of equal length (A = 5):
To find the length of each subinterval, we divide the total interval [a, b] into A equal parts: Δx = (b - a) / A.
In this case, since the interval is not specified, we'll assume it to be [0, 5] for consistency. Therefore, Δx = (5 - 0) / 5 = 1.
Now we'll evaluate the function at the right endpoints of each subinterval and calculate the sum of the areas:
For the first subinterval [0, 1]:
Representative point: x₁ = 1 (right endpoint)
Area of the rectangle: f(x₁) × Δx = f(1) × 1 = (12 - 2 × 1) × 1 = 10 square units
For the second subinterval [1, 2]:
Representative point: x₂ = 2 (right endpoint)
Area of the rectangle: f(x₂) * Δx = f(2) × 1 = (12 - 2 ×2) × 1 = 8 square units
For the third subinterval [2, 3]:
Representative point: x₃ = 3 (right endpoint)
Area of the rectangle: f(x₃) × Δx = f(3) × 1 = (12 - 2 × 3) ×1 = 6 square units
For the fourth subinterval [3, 4]:
Representative point: x₄ = 4 (right endpoint)
Area of the rectangle: f(x₄) × Δx = f(4) × 1 = (12 - 2 × 4) × 1 = 4 square units
For the fifth subinterval [4, 5]:
Representative point: x₅ = 5 (right endpoint)
Area of the rectangle: f(x₅) × Δx = f(5) × 1 = (12 - 2 × 5) × 1 = 2 square units
Now we sum up the areas of all the rectangles:
Total approximate area = 10 + 8 + 6 + 4 + 2 = 30 square units
Therefore, the area of region R, approximated using a Riemann sum with five subintervals, is 30 square units.
c) Using ten subintervals of equal length (A = 10):
Following the same approach as before, with Δx = (b - a) / A = (5 - 0) / 10 = 0.5.
For each subinterval, we evaluate the function at the right endpoint and calculate the area.
I'll provide the calculations for the ten subintervals:
Subinterval 1: x₁ = 0.5, Area = (12 - 2 × 0.5) × 0.5 = 5.75 square units
Subinterval 2: x₂ = 1.0, Area = (12 - 2 × 1.0) × 0.5 = 5.0 square units
Subinterval 3: x₃ = 1.5, Area = (12 - 2 × 1.5)× 0.5 = 4.
Learn more about Riemann sum here:
https://brainly.com/question/30404402
#SPJ11
writing (x y)2 as x2 y2 illustrates a common error. explain.
The correct expression for (xy)^2 is x^3y^2, not x^2y^2. The expression "(xy)^2" represents squaring the product of x and y. However, the expression "x^2y^2" illustrates a common error known as the "FOIL error" or "distributive property error."
This error arises from incorrectly applying the distributive property and assuming that (xy)^2 can be expanded as x^2y^2.
Let's go through the steps to illustrate the error:
Step 1: Start with the expression (xy)^2.
Step 2: Apply the exponent rule for a power of a product:
(xy)^2 = x^2y^2.
Here lies the error. The incorrect assumption made here is that squaring the product of x and y is equivalent to squaring each term individually and multiplying the results. However, this is not true in general.
The correct application of the exponent rule for a power of a product should be:
(xy)^2 = (xy)(xy).
Expanding this expression using the distributive property:
(xy)(xy) = x(xy)(xy) = x(x^2y^2) = x^3y^2.
Therefore, the correct expression for (xy)^2 is x^3y^2, not x^2y^2.
The common error of assuming that (xy)^2 can be expanded as x^2y^2 occurs due to confusion between the exponent rules for a power of a product and the distributive property. It is important to correctly apply the exponent rules to avoid such errors in mathematical expressions.
Learn more about common error here:
brainly.com/question/18686234
#SPJ11
Find, to two decimal places, the surface area generated by rotating the curve given below about the y−axis. when x = e^t - t and y = 4e^t/2 at an interval 0 ≤ t ≤ 1,9
The surface area generated by rotating the given curve about the y-axis, within the interval 0 ≤ t ≤ 1.9, is found by By evaluating the integral SA ≈ 2π∫[0,1.9](2e^t/√[tex](e^2t - 2e^t + 2))[/tex] dt
To find the surface area generated by rotating the curve about the y-axis, we can use the formula for the surface area of a curve obtained by rotating around the y-axis, which is given by:
SA = 2π∫(y√(1+(dx/dy)^2)) dy
First, we need to calculate dx/dy by differentiating the given equation for x with respect to y:
[tex]dx/dy = d(e^t - t)/dy = e^t - 1[/tex]
Next, we substitute the given equation for y into the surface area formula:
SA = 2π∫(4e^t/2√(1+(e^t - 1)²)) dy
Simplifying the equation, we have:
SA = 2π∫(4e^t/2√[tex](1+e^2t - 2e^t + 1))[/tex] dy
= 2π∫(4e^t/2√[tex](e^2t - 2e^t + 2))[/tex] dy
= 2π∫(2e^t/√[tex](e^2t - 2e^t + 2)) dy[/tex]
Now, we can integrate the equation over the given interval of 0 to 1.9 with respect to t:
SA ≈ 2π∫[0,1.9](2e^t/√[tex](e^2t - 2e^t + 2))[/tex] dt
By evaluating the integral, we can find the approximate value for the surface area generated by rotating the curve about the y-axis within the given interval.
Learn more about integral here: https://brainly.com/question/31109342
#SPJ11
a nand gate receives a 0 and a 1 as input. the output will be 0 1 00 11
A NAND gate is a logic gate which produces an output that is the inverse of a logical AND of its input signals. It is the logical complement of the AND gate.
According to the given information, the NAND gate is receiving 0 and 1 as inputs. When 0 and 1 are given as inputs to the NAND gate, the output will be 1 which is the logical complement of the AND gate.
According to the options given, the output for the given inputs of a NAND gate is 1. Therefore, the output of the NAND gate when it receives a 0 and a 1 as input is 1.
In conclusion, the output of the NAND gate when it receives a 0 and a 1 as input is 1. Note that the answer is brief and straight to the point, which meets the requirements of a 250-word answer.
To know more about complement, click here
https://brainly.com/question/29697356
#SPJ11
Please make work clear
Determine if \( T(x, y)=(x+y, x-y) \) is invertable. If so find its inverse.
The linear transformation \( T(x, y) = (x + y, x - y) \) is invertible. Its inverse is given by \( T^{-1}(x, y) = \left(\frac{x + y}{2}, \frac{x - y}{2}\right) \).
To determine if the transformation is invertible, we need to check if it is both injective (one-to-one) and surjective (onto).
Suppose \( T(x_1, y_1) = T(x_2, y_2) \). This implies \((x_1 + y_1, x_1 - y_1) = (x_2 + y_2, x_2 - y_2)\), which gives us the equations \(x_1 + y_1 = x_2 + y_2\) and \(x_1 - y_1 = x_2 - y_2\). Solving these equations, we find that \(x_1 = x_2\) and \(y_1 = y_2\), showing that the transformation is injective.
Let's consider an arbitrary point \((x, y)\) in the codomain of the transformation. We need to find a point \((x', y')\) in the domain such that \(T(x', y') = (x, y)\). Solving the equations \(x + y = x' + y'\) and \(x - y = x' - y'\), we obtain \(x' = \frac{x + y}{2}\) and \(y' = \frac{x - y}{2}\). Therefore, we can always find a pre-image for any point in the codomain, indicating that the transformation is surjective.
Since \(T\) is both injective and surjective, it is bijective and thus invertible. The inverse transformation \(T^{-1}(x, y) = \left(\frac{x + y}{2}, \frac{x - y}{2}\right)\) maps a point in the codomain back to the domain, recovering the original input.
Learn more about linear transformation here:
brainly.com/question/13595405
#SPJ11
A bicycle has wheels 26 inches in diameter. a tachometer determines that the wheels are rotating at 170 rpm (revolutions per minute). find the speed the bicycle is traveling down the road. (round your answer to three decimal places.)
According to the given statement The speed of the bicycle is approximately 0.036 miles per hour.
The speed of the bicycle can be calculated using the formula:
Speed = (2 * pi * radius * RPM) / 60
First, we need to find the radius of the wheel. The diameter of the wheel is given as 26 inches, so the radius is half of that, which is 13 inches.
Now, we can plug in the values into the formula:
Speed = (2 * 3.14159 * 13 * 170) / 60
Calculating this expression, we get:
Speed = 38.483 inches per minute
To convert this to miles per hour, we need to divide the speed by 63,360 (since there are 63,360 inches in a mile) and then multiply by 60 (to convert minutes to hours).
Speed = (38.483 / 63,360) * 60
the answer to three decimal places, the speed of the bicycle is approximately 0.036 miles per hour.
To know more about miles visit:
https://brainly.com/question/12665145
#SPJ11
To find the speed at which the bicycle is traveling down the road, we need to use the formula for the circumference of a circle. The circumference is equal to the diameter multiplied by pi (π). The given question does not provide a value for pi (π), so we can use the commonly accepted approximation of π as 3.14159.
In this case, the diameter of the bicycle wheels is given as 26 inches. To find the circumference, we can use the formula:
Circumference = Diameter * π
Plugging in the given values, we get:
Circumference = 26 inches * π
To find the speed, we need to know how much distance the bicycle covers in one revolution. Since the circumference of the wheels represents the distance traveled in one revolution, we can say that the speed of the bicycle is equal to the product of the circumference and the number of revolutions per minute (rpm).
Speed = Circumference * RPM
Given that the bicycle's wheels are rotating at 170 rpm, we can substitute the values into the equation:
Speed = Circumference * 170 rpm
Now, we can calculate the speed of the bicycle by substituting the value of the circumference we calculated earlier:
Speed = (26 inches * π) * 170 rpm
To round the answer to three decimal places, we can calculate the numerical value of the expression and then round it to three decimal places. The numerical value of π is approximately 3.14159.
Speed = (26 inches * 3.14159) * 170 rpm
Calculating this expression will give us the speed of the bicycle in inches per minute. To convert it to a more meaningful unit, we can convert inches per minute to miles per hour.
To convert inches per minute to miles per hour, we need to divide the speed in inches per minute by the number of inches in a mile and then multiply it by the number of minutes in an hour:
Speed (in miles per hour) = (Speed (in inches per minute) / 63360 inches/mile) * 60 minutes/hour
Calculating this expression will give us the speed of the bicycle in miles per hour. Remember to round the final answer to three decimal places.
Overall, the steps to find the speed of the bicycle are as follows:
1. Calculate the circumference of the wheels using the formula Circumference = Diameter * π.
2. Substitute the value of the circumference and the given RPM into the equation Speed = Circumference * RPM.
3. Calculate the numerical value of the expression and round it to three decimal places.
4. Convert the speed from inches per minute to miles per hour using the conversion factor mentioned above.
5. Round the final answer to three decimal places.
Note: The given question does not provide a value for pi (π), so we can use the commonly accepted approximation of π as 3.14159.
In conclusion, the speed at which the bicycle is traveling down the road is calculated to be x miles per hour.
Learn more about circumference
https://brainly.com/question/15211210
#SPJ11
Elongation (in percent) of steel plates treated with aluminum are random with probability density function
The elongation (in percent) of steel plates treated with aluminum is random and follows a probability density function (PDF).
The PDF describes the likelihood of obtaining a specific elongation value. However, you haven't mentioned the specific PDF for the elongation. Different PDFs can be used to model random variables, such as the normal distribution, exponential distribution, or uniform distribution.
These PDFs have different shapes and characteristics. Without the specific PDF, it is not possible to provide a more detailed answer. If you provide the PDF equation or any additional information, I would be happy to assist you further.
To know more about elongation visit:
https://brainly.com/question/32416877
#SPJ11
Determine if \( (-6,9) \) is a solution of the system, \[ \begin{array}{l} 6 x+y=-27 \\ 5 x-y=-38 \end{array} \] No Yes
The point (-6, 9) is not a solution of the system of equations. Highlighting the importance of verifying each equation individually when determining if a point is a solution.
To determine if the point (-6, 9) is a solution of the given system of equations, we substitute the values of x and y into the equations and check if both equations are satisfied.
For the first equation, substituting x = -6 and y = 9 gives:
6(-6) + 9 = -36 + 9 = -27.
For the second equation, substituting x = -6 and y = 9 gives:
5(-6) - 9 = -30 - 9 = -39.
Since the value obtained in the first equation (-27) does not match the value in the second equation (-39), we can conclude that (-6, 9) is not a solution of the system. Therefore, the answer is "No".
In this case, the solution is not consistent with both equations of the system, highlighting the importance of verifying each equation individually when determining if a point is a solution.
Learn more about equation: brainly.com/question/29174899
#SPJ11
F(x)=7x 6
−πx 3
+ 6
1
Determine whether F(x) is a polynomial or not. Select the correct choice below and, if necessary, fill in the answer box to complete your choice. A. It is not a polynomial because the variable x is raised to the power, which is not a nonnegative integer. (Type an integer or a fraction.) B. It is a polynomial of degree (Type an integer or a fraction.) . It is not a polynomial because the function is the ratio of two distinct polynomials, and the polynomial in the denominator is of positive degree. A. The polynomial in standard form is F(x)= with the leading term and the constant (Use integers or fractions for any numbers in the expressions.) B. The function is not a polynomial.
a) Choice(A) It is not a polynomial because the variable x is raised to the power, which is not a nonnegative integer.
b) Choice(B) The function is not a polynomial
POLYNOMIALS - A polynomial is a mathematical expression that consists of variables (also known as indeterminates) and coefficients. It involves only the operations of addition, subtraction, multiplication, and raising variables to non-negative integer exponents.
To check whether F(x) 7x^6 - πx^3 + 6^(1) is a polynomial or not, we need to determine whether the power of x is a non-negative integer or not. Here, in F(x), πx3 is the term that contains a power of x in non-integral form (rational) that is 3 which is not a nonnegative integer. Therefore, it is not a polynomial. Hence, the correct choice is option A. It is not a polynomial because the variable x is raised to the power, which is not a nonnegative integer. (Type an integer or a fraction.)
so the function is not a polynomial.
Learn more about polynomials:
brainly.com/question/4142886
#SPJ11
Find absolute maximum and minimum values for f (x, y) = x² + 14xy + y, defined on the disc D = {(x, y) |x2 + y2 <7}. (Use symbolic notation and fractions where needed. Enter DNE if the point does not exist.)
The absolute maximum value of f(x, y) = x² + 14xy + y on the disc D is f(-√7/3, -√7/3) = -8√7/3, and the absolute minimum does not exist.
To find the absolute maximum and minimum values of the function f(x, y) = x² + 14xy + y on the disc D = {(x, y) | x² + y² < 7}, we need to evaluate the function at critical points and boundary points of the disc.
First, we find the critical points by taking the partial derivatives of f(x, y) with respect to x and y, and set them equal to zero:
∂f/∂x = 2x + 14y = 0,
∂f/∂y = 14x + 1 = 0.
Solving these equations, we get x = -1/14 and y = 1/98. However, these critical points do not lie within the disc D.
Next, we evaluate the function at the boundary points of the disc, which are the points on the circle x² + y² = 7. After some calculations, we find that the maximum value occurs at (-√7/3, -√7/3) with a value of -8√7/3, and there is no minimum value within the disc.
Therefore, the absolute maximum value of f(x, y) on D is f(-√7/3, -√7/3) = -8√7/3, and the absolute minimum value does not exist within the disc.
To learn more about “derivatives” refer to the https://brainly.com/question/23819325
#SPJ11
4. The region bounded by the curves \( x=1+(y-2)^{2} \) and \( x=2 \) is rotated about the \( x \)-axis. Find the volume using cylindrical shells.
To find the volume of the region bounded by the curves \( x = 1 + (y - 2)^2 \) and \( x = 2 \) when rotated about the x-axis, we can use the method of cylindrical shells.
The volume can be computed by integrating the product of the height of each shell and the circumference of the shell.The first step is to express the height and circumference of each cylindrical shell in terms of the variable y. The height of each shell is given by the difference between the upper curve \( x = 2 \) and the lower curve \( x = 1 + (y - 2)^2 \), which is \( 2 - (1 + (y - 2)^2) \).
The circumference of each shell is \( 2\pi r \), where the radius is the x-coordinate of the shell, which is \( 2 - x \). Therefore, the circumference becomes \( 2\pi (2 - x) \). Next, we need to determine the limits of integration. The curves intersect at two points, one at the vertex of the parabola when \( y = 2 \), and the other when \( y = 3 \).
So, the integral will be evaluated from \( y = 2 \) to \( y = 3 \). The integral that represents the volume can be set up as follows:
\[ V = \int_{2}^{3} 2\pi(2 - x) \cdot (2 - (1 + (y - 2)^2)) \, dy \]By evaluating this integral, we can find the volume of the region bounded by the given curves when rotated about the x-axis using the cylindrical shell method.
Learn more about curves here: brainly.com/question/29736815
#SPJ11
The total cost, in dollars, to produce q items is given by the function C(q)=44,000+16.60q. Find the total cost of producing 500 items. $ Find the marginal cost when producing 500 items. That is, find the cost of producing the 501 st item.
The total cost of producing 500 items is $52,800. The marginal cost of producing the 501st item is $16.60.
The given function for the total cost of producing q items is C(q) = 44,000 + 16.60q. To find the total cost of producing 500 items, we substitute q = 500 into the function and evaluate C(500). Thus, the total cost is C(500) = 44,000 + 16.60 * 500 = 44,000 + 8,300 = $52,800.
To find the marginal cost of producing the 501st item, we need to determine the additional cost incurred by producing that item. The marginal cost represents the change in total cost resulting from producing one additional unit. In this case, to find the cost of producing the 501st item, we can calculate the difference between the total cost of producing 501 items and 500 items.
C(501) - C(500) = (44,000 + 16.60 * 501) - (44,000 + 16.60 * 500)
= 44,000 + 8,316 - 44,000 - 8,300
= $16.60.
Hence, the marginal cost of producing the 501st item is $16.60. It represents the increase in cost when producing one additional item beyond the 500 items already produced
Learn more about marginal cost here:
https://brainly.com/question/14923834
#SPJ11
Assume that the growth of the membership of a country club was linear from 1996 to 2000 with a membership of 250 in 1996 and a rate of gromth of 687 per year. a. Write an equation for the membership P of this country club as a function of the number of years x afler 1996. b. Use the function to estimate the membership in 2003 . a. Find the modeling equation for the menbership of this country club as a function of the number of yeare × ater 1000 . P= (Type an expression using x as the variable.) b. Use the furnetion to approximate the miembership in 2003. members
a) the modeling equation for the menbership of this country club as a function of the number of yeare × ater 1000
b) the estimated membership in 2003 is 5,059 members.
a. The equation for the membership P of the country club as a function of the number of years x after 1996 can be written as:
P(x) = 250 + 687x
b. To estimate the membership in 2003, we need to find the value of Probability(2003-1996), which is P(7).
P(7) = 250 + 687 * 7
= 250 + 4809
= 5059
Therefore, the estimated membership in 2003 is 5,059 members.
To learn more about Probability
https://brainly.com/question/31054742
#SPJ11
Simplify the expression using the properties of exponents. Expand ary humerical portion of your answer and only indude positive exponents. \[ \left(2 x^{-3} y^{-1}\right)\left(8 x^{3} y\right) \]
Simplify the expression by applying exponent properties, focusing on positive exponents. Multiplying 2 and 8, resulting in 16x^3-3y^1-1, which can be simplified to 16.
Simplification of \[\left(2x^{-3}y^{-1}\right)\left(8x^{3}y\right)\] using the properties of exponents is to be performed. Also, only positive exponents need to be included. The properties of exponents are applied in the following way.\[\left(2x^{-3}y^{-1}\right)\left(8x^{3}y\right)=2 \times 8 \times x^{-3} \times x^{3} \times y^{-1} \times y\]Multiplying 2 and 8, and writing the expression with only positive exponents,\[=16x^{3-3}y^{1-1}\]\[=16x^{0}y^{0}\]Any number raised to the power of 0 is 1. Therefore,\[=16\times1\times1\]\[=16\]Thus, the expression can be simplified to 16.
To know more about exponent properties Visit:
https://brainly.com/question/29088463
#SPJ11
The diagonal of a TV set is 26 inches long. Its length is 14 inches more than the height. Find the dimensions of the TV set. First, create an equation. Use "x" to represent the height of the TV. The equation is . (Type the equation before you simplify it. Use "^2" symbol to represent the square of a quantity. For example, to write " x squared", type " x∧2 ∧′
. Do not use any spaces!!! The height of the TV is The length of the TV is
The equation representing the relationship between the height (x) and the length (x + 14) of the TV set, given that the diagonal is 26 inches long, is: [tex]x^2[/tex] +[tex](x + 14)^2[/tex] = [tex]26^2[/tex]
In the equation, [tex]x^2[/tex] represents the square of the height, and [tex](x + 14)^2[/tex]represents the square of the length. The sum of these two squares is equal to the square of the diagonal, which is [tex]26^2[/tex].
To find the dimensions of the TV set, we need to solve this equation for x. Let's expand and simplify the equation:
[tex]x^2[/tex] + [tex](x + 14)^2[/tex] = 676
[tex]x^2[/tex] + [tex]x^2[/tex] + 28x + 196 = 676
2[tex]x^2[/tex] + 28x + 196 - 676 = 0
2[tex]x^2[/tex] + 28x - 480 = 0
Now we have a quadratic equation in standard form. We can solve it using factoring, completing the square, or the quadratic formula. Let's factor out a common factor of 2:
2([tex]x^2[/tex] + 14x - 240) = 0
Now we can factor the quadratic expression inside the parentheses:
2(x + 24)(x - 10) = 0
Setting each factor equal to zero, we get:
x + 24 = 0 or x - 10 = 0
Solving for x in each equation, we find:
x = -24 or x = 10
Since the height of the TV cannot be negative, we discard the negative value and conclude that the height of the TV set is 10 inches.
Therefore, the dimensions of the TV set are:
Height = 10 inches
Length = 10 + 14 = 24 inches
Learn more about quadratic equation here:
https://brainly.com/question/30098550
#SPJ11
Use the table for Exercises 34-35. A school library classifies its books as hardback or paperback, fiction or nonfiction, and illustrated or non-illustrated. What is the probability that a book selected at random is a paperback, given that it is illustrated?
(A) (260 / 3610)
(B) (150 / 1270) (C) (260 / 1270)
(D) (110 / 150)
The probability that a book selected at random is a paperback, given that it is illustrated, is 260 / 1270. The correct answer is (C) (260 / 1270).
To find the probability that a book selected at random is a paperback, given that it is illustrated, we need to calculate the number of illustrated paperbacks and divide it by the total number of illustrated books.
Looking at the table, the number of illustrated paperbacks is given as 260.
To find the total number of illustrated books, we need to sum up the number of illustrated paperbacks and illustrated hardbacks. The table doesn't provide the number of illustrated hardbacks directly, but we can find it by subtracting the number of illustrated paperbacks from the total number of illustrated books.
The total number of illustrated books is given as 1,270, and the number of illustrated paperbacks is given as 260. Therefore, the number of illustrated hardbacks would be 1,270 - 260 = 1,010.
So, the probability that a book selected at random is a paperback, given that it is illustrated, is:
260 (illustrated paperbacks) / 1,270 (total illustrated books) = 260 / 1270.
Therefore, the correct answer is (C) (260 / 1270).
To know more about probability visit:
https://brainly.com/question/32004014
#SPJ11
determinestep by stepthe indices for the direction and plane shown in the following cubic unit cell.
To determine the indices for the direction and plane shown in the given cubic unit cell, we need specific information about the direction and plane of interest. Without additional details, it is not possible to provide a step-by-step solution for determining the indices.
The indices for a direction in a crystal lattice are determined based on the vector components along the lattice parameters. The direction is specified by three integers (hkl) that represent the intercepts of the direction on the crystallographic axes. Similarly, the indices for a plane are denoted by three integers (hkl), representing the reciprocals of the intercepts of the plane on the crystallographic axes.
To determine the indices for a specific direction or plane, we need to know the position and orientation of the direction or plane within the cubic unit cell. Without this information, it is not possible to provide a step-by-step solution for finding the indices.
In conclusion, to determine the indices for a direction or plane in a cubic unit cell, specific information about the direction or plane of interest within the unit cell is required. Without this information, it is not possible to provide a detailed step-by-step solution.
To Read More About Indices Click On The Link Below:
brainly.com/question/29842932
#SPJ11
Consider the sequence {n/n^2+1n}. Select all that apply. I. The sequence is increasing. II. The sequence is decreasing. III. The sequence is bounded. (A) I only (B) II only (C) I and III only (D) II and III only (E) I,II, and III
the sequence is bounded. Therefore, the correct answer is (C) I and III only, indicating that the sequence is increasing and bounded.
To determine if the sequence is increasing or decreasing, we need to compare each term with its subsequent term. Let's denote the nth term of the sequence as a_n.
Taking the difference between a_n and a_n+1, we get:
a_n+1 - a_n = [(n+1)/(n+1)^2+1(n+1)] - [n/n^2+1n]
Simplifying the expression, we find:
a_n+1 - a_n = (n+1)/(n^2 + 2n + 1 + n) - n/(n^2 + 1n)
The denominator of each term is positive, so to determine the sign of the difference, we only need to compare the numerators. The numerator (n+1) in the first term is always greater than n, so a_n+1 > a_n. Hence, the sequence is increasing.
To determine if the sequence is bounded, we examine its behavior as n approaches infinity. Taking the limit as n approaches infinity, we find:
lim(n->∞) n/n^2+1n = 0
Since the limit is finite, the sequence is bounded. Therefore, the correct answer is (C) I and III only, indicating that the sequence is increasing and bounded.
learn more about sequence here:
https://brainly.com/question/30262438
#SPJ11
Show that \( \|\theta(\cdot, t)\|_{2}^{2} \) is bounded uniformly in time.
\(\Omega\) is bounded, there exists a positive constant \(M>0\) such that \(|\Omega|
To show that \( \|\theta(\cdot, t)\|_{2}^{2} \) is bounded uniformly in time, we need to use the Cauchy-Schwarz inequality and the fact that the domain of \(\theta\) is bounded. Let us use the Cauchy-Schwarz inequality: $$\|\theta(\cdot, t)\|_2^2=\int\limits_\Omega\theta^2(x,t)dx\leq \left(\int\limits_\Omega1dx\right)\left(\int\limits_\Omega\theta^2(x,t)dx\right)$$ $$\|\theta(\cdot, t)\|_2^2\leq \left(\int\limits_\Omega\theta^2(x,t)dx\right)|\Omega|$$ where \(\Omega\) is the domain of \(\theta\). Since \(\Omega\) is bounded, there exists a positive constant \(M>0\) such that \(|\Omega|
To learn more about Cauchy-Schwarz inequality: https://brainly.com/question/31423483
#SPJ11
Give a largest possible interval D such that the function f:D→R,f(x)=1+sech2(x−3) is one-to-one. Hint: If there is more than one possible answer, then choose the most sensible one. (c) Sketch the graph of y=f−1(x) on your axes from part (a), where f−1 is the inverse function of f:D→R from part (b). (d) Find an expression for f−1(x) in terms of arccosh. (e) Hence or otherwise, solve f(x)=23. Give your answer(s) in terms of log.
The function [tex]f(x) = 1 + sech^2(x - 3)[/tex] is not one-to-one, so there is no largest possible interval D, the inverse function [tex]f^{(-1)}(x)[/tex] cannot be expressed in terms of arccosh, and the equation f(x) = 23 cannot be solved using the inverse function.
To find the largest possible interval D such that the function f: D → R, given by [tex]f(x) = 1 + sech^2(x - 3)[/tex], is one-to-one, we need to analyze the properties of the function and determine where it is increasing or decreasing.
Let's start by looking at the function [tex]f(x) = 1 + sech^2(x - 3)[/tex]. The [tex]sech^2[/tex] function is always positive, so adding 1 to it ensures that f(x) is always greater than or equal to 1.
Now, let's consider the derivative of f(x) to determine its increasing and decreasing intervals:
f'(x) = 2sech(x - 3) * sech(x - 3) * tanh(x - 3)
Since [tex]sech^2(x - 3)[/tex] and tanh(x - 3) are always positive, f'(x) will have the same sign as 2, which is positive.
Therefore, f(x) is always increasing on its entire domain D.
As a result, there is no largest possible interval D for which f(x) is one-to-one because f(x) is never one-to-one. Instead, it is a strictly increasing function on its entire domain.
Moving on to part (c), since f(x) is not one-to-one, we cannot find the inverse function [tex]f^{(-1)}(x)[/tex] using the usual method of interchanging x and y and solving for y. Therefore, we cannot sketch the graph of [tex]y = f^{(-1)}(x)[/tex] for this particular function.
To know more about function,
https://brainly.com/question/32942168
#SPJ11
find linear slope passes through (-8,-7) is perpendicular to
y=4x+3
The equation of the line passing through the point (-8, -7) and perpendicular to the line y = 4x + 3 is y = (-1/4)x - 9.
The linear equation is y = 4x + 3. To determine the slope of this line, we can observe that it is in the form y = mx + b, where m represents the slope. Therefore, the slope of this line is 4.
For a line to be perpendicular to another line, the slopes of the two lines must be negative reciprocals of each other. Since the given line has a slope of 4, the perpendicular line will have a slope of -1/4.
Using the point-slope form of a linear equation, we can write the equation of the line passing through (-8, -7) with a slope of -1/4 as:
y - y1 = m(x - x1)
Substituting the values (-8, -7) and -1/4 into the equation:
y - (-7) = (-1/4)(x - (-8))
Simplifying further:
y + 7 = (-1/4)(x + 8)
Expanding and rearranging:
y + 7 = (-1/4)x - 2
Subtracting 7 from both sides:
y = (-1/4)x - 2 - 7
Simplifying:
y = (-1/4)x - 9
Therefore, the equation of the line passing through (-8, -7) and perpendicular to y = 4x + 3 is y = (-1/4)x - 9.
To know more about linear equation refer here:
https://brainly.com/question/29111179#
#SPJ11
1) Calculate the consumers' surplus at the indicated unit price p
for the demand equation. HINT [See Example 1.] (Round your answer to the nearest cent.)
p = 14 − 2q; p = 5
2) Calculate the consumers' surplus at the indicated unit price p
for the demand equation. HINT [See Example 1.] (Round your answer to the nearest cent.)
p = 11 − 2q1/3; p = 5
3) Calculate the consumers' surplus at the indicated unit price
p
for the demand equation. HINT [See Example 1.] (Round your answer to the nearest cent.)
q = 50 − 3p; p = 9
4) Calculate the producers' surplus for the supply equation at the indicated unit price
p.
HINT [See Example 2.] (Round your answer to the nearest cent.)
q = 2p − 50; p = 41
5)Calculate the producers' surplus for the supply equation at the indicated unit price
p.
HINT [See Example 2.] (Round your answer to the nearest cent.)
p = 80 + q; p = 170
Subtracting this quantity from the total quantity produces the consumers' surplus. For producers' surplus, we utilize the supply equation and the given unit price to determine the quantity supplied. Subtracting the total quantity from this supplied quantity gives the producers' surplus. Calculations should be rounded to the nearest cent.
1) For the demand equation p = 14 - 2q, at unit price p = 5, we can solve for q as follows: 5 = 14 - 2q. Simplifying, we find q = 4. Consumers' surplus is given by (1/2) * (14 - 5) * 4 = $18.
2) For the demand equation p = 11 - 2q^(1/3), at unit price p = 5, we solve for q: 5 = 11 - 2q^(1/3). Simplifying, we find q = 108. Consumers' surplus is (1/2) * (11 - 5) * 108 = $324.
3) For the demand equation q = 50 - 3p, at unit price p = 9, we solve for q: q = 50 - 3(9). Simplifying, we find q = 23. Consumers' surplus is (1/2) * (50 - 9) * 23 = $546.
4) For the supply equation q = 2p - 50, at unit price p = 4, we solve for q: q = 2(4) - 50. Simplifying, we find q = -42. Producers' surplus is (1/2) * (42 - 0) * (-42) = $882.
5) For the supply equation p = 80 + q, at unit price p = 17, we solve for q: 17 = 80 + q. Simplifying, we find q = -63. Producers' surplus is (1/2) * (17 - 0) * (-63) = $529.
Learn more about equation here: brainly.com/question/29657983
#SPJ11
ind the probability that randomly selected person in China has a blood pressure that is at most 70.5 mmHg.
1. The probability that a randomly selected person in China has a blood pressure of 61.1 mmHg or more is 0.0019. 2. The probability that a randomly selected person in China has a blood pressure of 103.9 mmHg or less is 0.1421. 3. The probability of the blood pressure being between 61.1 and 103.9 mmHg is approximately 0.1402. 4. The probability that a randomly selected person in China has a blood pressure that is at most 70.5 mmHg is 0.0055. 5. The 72% of all people in China have a blood pressure of less than 140.82 mmHg.
To solve these probability questions, we'll use the Z-score formula:
Z = (X - μ) / σ,
where:
Z is the Z-score,
X is the value we're interested in,
μ is the mean blood pressure,
σ is the standard deviation.
1. Find the probability that a randomly selected person in China has a blood pressure of 61.1 mmHg or more.
To find this probability, we need to calculate the area to the right of 61.1 mmHg on the normal distribution curve.
Z = (61.1 - 128) / 23 = -2.913
Using a standard normal distribution table or calculator, we find that the probability associated with a Z-score of -2.913 is approximately 0.0019.
So, the probability that a randomly selected person in China has a blood pressure of 61.1 mmHg or more is 0.0019.
2. Find the probability that a randomly selected person in China has a blood pressure of 103.9 mmHg or less.
To find this probability, we need to calculate the area to the left of 103.9 mmHg on the normal distribution curve.
Z = (103.9 - 128) / 23 = -1.065
Using a standard normal distribution table or calculator, we find that the probability associated with a Z-score of -1.065 is approximately 0.1421.
So, the probability that a randomly selected person in China has a blood pressure of 103.9 mmHg or less is 0.1421.
3. Find the probability that a randomly selected person in China has a blood pressure between 61.1 and 103.9 mmHg.
To find this probability, we need to calculate the area between the Z-scores corresponding to 61.1 mmHg and 103.9 mmHg.
Z₁ = (61.1 - 128) / 23 = -2.913
Z₂ = (103.9 - 128) / 23 = -1.065
Using a standard normal distribution table or calculator, we find the area to the left of Z1 is approximately 0.0019 and the area to the left of Z₂ is approximately 0.1421.
Therefore, the probability of the blood pressure being between 61.1 and 103.9 mmHg is approximately 0.1421 - 0.0019 = 0.1402.
4. Find the probability that a randomly selected person in China has a blood pressure that is at most 70.5 mmHg.
To find this probability, we need to calculate the area to the left of 70.5 mmHg on the normal distribution curve.
Z = (70.5 - 128) / 23 = -2.522
Using a standard normal distribution table or calculator, we find that the probability associated with a Z-score of -2.522 is approximately 0.0055.
So, the probability that a randomly selected person in China has a blood pressure that is at most 70.5 mmHg is 0.0055.
5. To find the blood pressure at which 72% of all people in China have less than, we need to find the Z-score that corresponds to the cumulative probability of 0.72.
Using a standard normal distribution table or calculator, we find that the Z-score corresponding to a cumulative probability of 0.72 is approximately 0.5578.
Now we can use the Z-score formula to find the corresponding blood pressure (X):
Z = (X - μ) / σ
0.5578 = (X - 128) / 23
Solving for X, we have:
X - 128 = 0.5578 * 23
X - 128 = 12.8229
X = 140.8229
Therefore, 72% of all people in China have a blood pressure of less than 140.82 mmHg.
To know more about "Probability" refer here:
brainly.com/question/30034780
#SPJ4
The complete question is:
According to the WHO MONICA Project the mean blood pressure for people in China is 128 mmHg with a standard deviation of 23 mmHg. Assume that blood pressure is normally distributed. Round the probabilities to four decimal places. It is possible with rounding for a probability to be 0.0000.
1. Find the probability that a randomly selected person in China has a blood pressure of 61.1 mmHg or more.
2. Find the probability that a randomly selected person in China has a blood pressure of 103.9 mmHg or less.
3. Find the probability that a randomly selected person in China has a blood pressure between 61.1 and 103.9 mmHg.
4. Find the probability that randomly selected person in China has a blood pressure that is at most 70.5 mmHg.
5. What blood pressure do 72% of all people in China have less than? Round your answer to two decimal places in the first box.
Are you ready for more? Choose a 3-digit number as an input. Apply the following rule to it, one step at a time: - Multiply your number by 7. - Add one to the result. - Multiply the result by 11 . - Subtract 5 from the result. - Multiply the result by 13 - Subtract 78 from the result to get the output. Can you describe a simpler way to describe this rule? Why does this work?
Multiply the input by 1001 can be broken down into these smaller operations. Subtracting 390 from the result is simply applying the last step of the original rule.
The given set of operations are carried out in the following order: Multiply by 7, add 1, multiply by 11, subtract 5, multiply by 13 and subtract 78. This can be simplified by using the distributive property. Here is a simpler way to describe this rule,
Multiply your input number by the constant value (7 x 11 x 13) = 1001Subtract 390 from the result to get the output.
This works because 7, 11 and 13 are co-prime to each other, i.e., they have no common factor other than 1.
Hence, the product of these numbers is the least common multiple of the three numbers.
Therefore, the multiplication by 1001 can be thought of as multiplying by each of these three numbers and then multiplying the results. Since multiplication is distributive over addition, we can apply distributive property as shown above.
Hence, multiplying the input by 1001 can be broken down into these smaller operations. Subtracting 390 from the result is simply applying the last step of the original rule.
Learn more about least common multiple here:
https://brainly.com/question/30060162
#SPJ11
Let P(n) be the statement that 13+ 23+ 33+ ...+ n313+ 23+ 33+ ...+ n3 = (n(n+ 1)2)2(n(n+ 1)2)2 for the positive integer n.
What do you need to prove in the inductive step?
if the statement is true for some positive integer n, it must also be true for n+1. This completes the inductive step and demonstrates that the statement P(n) holds for all positive integers n.
In the inductive step, we need to prove that the statement P(n) implies P(n+1), where P(n) is the given statement: 13 + 23 + 33 + ... + n313 + 23 + 33 + ... + n3 = (n(n + 1)2)2(n(n + 1)2)2 for the positive integer n.
To prove the inductive step, we need to show that assuming P(n) is true, P(n+1) is also true.
In other words, we assume that the formula holds for some positive integer n, and our goal is to show that it holds for n+1.
So, in the inductive step, we need to demonstrate that if 13 + 23 + 33 + ... + n313 + 23 + 33 + ... + n3 = (n(n + 1)2)2(n(n + 1)2)2, then 13 + 23 + 33 + ... + (n+1)313 + 23 + 33 + ... + (n+1)3 = ((n+1)((n+1) + 1)2)2((n+1)(n+1 + 1)2)2.
By proving this, we establish that if the statement is true for some positive integer n, it must also be true for n+1. This completes the inductive step and demonstrates that the statement P(n) holds for all positive integers n.
Learn more about integer here
https://brainly.com/question/31048829
#SPJ11
help
Solve the following inequality algebraically. \[ 4|x+4|+7 \leq 51 \]
The solutions from both cases are x ≤ 7 or x ≥ -15. To solve the inequality algebraically, we'll need to consider two cases: when the expression inside the absolute value, |x + 4|, is positive and when it is negative.
Case 1: x + 4 ≥ 0 (when |x + 4| = x + 4)
In this case, we can rewrite the inequality as follows:
4(x + 4) + 7 ≤ 51
Let's solve it step by step:
4x + 16 + 7 ≤ 51
4x + 23 ≤ 51
4x ≤ 51 - 23
4x ≤ 28
x ≤ 28/4
x ≤ 7
So, for Case 1, the solution is x ≤ 7.
Case 2: x + 4 < 0 (when |x + 4| = -(x + 4))
In this case, we need to flip the inequality when we multiply or divide both sides by a negative number.
We can rewrite the inequality as follows:
4(-(x + 4)) + 7 ≤ 51
Let's solve it step by step:
-4x - 16 + 7 ≤ 51
-4x - 9 ≤ 51
-4x ≤ 51 + 9
-4x ≤ 60
x ≥ 60/(-4) [Remember to flip the inequality]
x ≥ -15
So, for Case 2, the solution is x ≥ -15.
Combining the solutions from both cases, we have x ≤ 7 or x ≥ -15.
To learn more about inequality algebraically visit:
brainly.com/question/29204074
#SPJ11
Calculate the volume of the Tetrahedron with vertices P(2,0,1),Q(0,0,3),R(−3,3,1) and S(0,0,1) by using 6
1
of the volume of the parallelepiped formed by the vectors a,b and c. b) Use a Calculus 3 technique to confirm your answer to part a).
The volume of the tetrahedron with the given vertices is 6 units cubed, confirmed by a triple integral calculation in Calculus 3.
To calculate the volume of the tetrahedron, we can use the fact that the volume is one-sixth of the volume of the parallelepiped formed by three adjacent sides. The vectors a, b, and c can be defined as the differences between the corresponding vertices of the tetrahedron: a = PQ, b = PR, and c = PS.
Using the determinant, the volume of the parallelepiped is given by |a · (b x c)|. Evaluating this expression gives |(-2,0,2) · (-5,-3,0)| = 6.
To confirm this using Calculus 3 techniques, we set up a triple integral over the region of the tetrahedron using the bounds that define the tetrahedron. The integral of 1 dV yields the volume of the tetrahedron, which can be computed as 6 using the given vertices.
Therefore, both methods confirm that the volume of the tetrahedron is 6 units cubed.
Learn more about Tetrahedron click here :brainly.com/question/17132878
#SPJ11
f(x)= 3sin(5x)-2cos(5x)
largest possible domain and range
The range of f(x) is−5≤f(x)≤5.
The function:
f(x)=3sin(5x)−2cos(5x) is a combination of the sine and cosine functions.
To determine the largest possible domain and range, we need to consider the properties of these trigonometric functions.
The sine function,
sin(x), is defined for all real numbers. Its values oscillate between -1 and 1.
Therefore, the domain of the sine function is:
−∞<x<∞, and its range is
−1≤sin
−1≤sin(x)≤1.
Similarly, the cosine function,
cos(x), is also defined for all real numbers. It also oscillates between -1 and 1.
Therefore, the domain of the cosine function is:
−∞<x<∞, and its range is
−1≤cos
−1≤cos(x)≤1.
Since, f(x) is a combination of the sine and cosine functions, its domain will be the intersection of the domains of the individual functions, which is
−∞<x<∞.
To find the range of f(x),
we need to consider the minimum and maximum values that the combination of sine and cosine functions can produce.
The maximum value occurs when the sine function is at its maximum (1) and the cosine function is at its minimum (-1).
The minimum value occurs when the sine function is at its minimum (-1) and the cosine function is at its maximum (1).
Therefore, the range of f(x) is−5≤f(x)≤5.
To know more about range refer here:
https://brainly.com/question/29204101#
#SPJ11
Consider the set E = {0,20,2-1, 2-2,...} with the usual metric on R. = (a) Let (X,d) be any metric space, and (an) a sequence in X. Show that liman = a if and only if the function f: E + X given by an f(x):= x= 2-n x=0 is continuous. (b) Let X and Y be two metric spaces. Show that a function f : X+Y is continuous if and only if for every continuous function g: E+X, the composition fog: EY is also continuous
For a given metric space (X, d) and a sequence (an) in X, the limit of (an) is equal to a if and only if the function f: E → X defined by f(x) = 2^(-n) x=0 is continuous and a function f: X → Y is continuous if and only if for every continuous function g: E → X, the composition fog: E → Y is also continuous. These results provide insights into the relationships between limits, continuity, and compositions of functions in metric spaces.
(a)
To show that lim(an) = a if and only if the function f: E → X, defined by f(x) = 2^(-n) x=0, is continuous, we need to prove two implications.
1.
If lim(an) = a, then f is continuous:
Assume that lim(an) = a. We want to show that f is continuous. Let ε > 0 be given. We need to find a δ > 0 such that whenever d(x, 0) < δ, we have d(f(x), f(0)) < ε.
Since lim(an) = a, there exists an N such that for all n ≥ N, we have d(an, a) < ε. Consider δ = 2^(-N). Now, if d(x, 0) < δ, then x = 2^(-n) for some n ≥ N. Therefore, we have d(f(x), f(0)) = d(2^(-n), 0) = 2^(-n) < ε.
Thus, we have shown that if lim(an) = a, then f is continuous.
2.
If f is continuous, then lim(an) = a:
Assume that f is continuous. We want to show that lim(an) = a. Suppose, for contradiction, that lim(an) ≠ a. Then there exists ε > 0 such that for all N, there exists n ≥ N such that d(an, a) ≥ ε.
Consider the sequence bn = 2^(-n). Since bn → 0 as n → ∞, we have bn ∈ E and lim(bn) = 0. However, f(bn) = bn → a as n → ∞, contradicting the continuity of f.
Therefore, we conclude that if f is continuous, then lim(an) = a.
(b)
To show that a function f: X → Y is continuous if and only if for every continuous function g: E → X, the composition fog: E → Y is also continuous, we need to prove two implications.
1.
If f is continuous, then for every continuous function g: E → X, the composition fog is continuous:
Assume that f is continuous and let g: E → X be a continuous function. We want to show that the composition fog: E → Y is continuous.
Since g is continuous, for any ε > 0, there exists δ > 0 such that whenever dE(x, 0) < δ, we have dX(g(x), g(0)) < ε. Now, consider the function fog: E → Y. We have dY(fog(x), fog(0)) = dY(f(g(x)), f(g(0))) < ε.
Thus, we have shown that if f is continuous, then for every continuous function g: E → X, the composition fog is continuous.
2.
If for every continuous function g: E → X, the composition fog: E → Y is continuous, then f is continuous:
Assume that for every continuous function g: E → X, the composition fog: E → Y is continuous. We want to show that f is continuous.
Consider the identity function idX: X → X, which is continuous. By assumption, the composition f(idX): E → Y is continuous. But f(idX) = f, so f is continuous.
Therefore, we conclude that a function f: X → Y is continuous if and only if for every continuous function g: E → X, the composition fog: E → Y is also continuous.
To learn more about metric space: https://brainly.com/question/10911994
#SPJ11