The correct option is option (b). The percent overall error rate for the given confusion matrix is approximately 37.5%.
In the confusion matrix, the diagonal elements represent the correct predictions, while the off-diagonal elements represent the incorrect predictions. The overall error rate is calculated by summing up the incorrect predictions and dividing it by the total number of predictions.
In this case, the total number of predictions is the sum of all the elements in the confusion matrix, which is 80 + 100 + 20 + 120 = 320.
The total number of incorrect predictions is the sum of the off-diagonal elements, which is 100 + 20 = 120.
The percent overall error rate is then calculated by dividing the total number of incorrect predictions by the total number of predictions and multiplying by 100:
(120 / 320) * 100 = 37.5%.
Therefore, the percent overall error rate is approximately 37.5%, which corresponds to option b.
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The range of computer-generated random numbers is
[0, 1)
[–8, 8]
[–8, 0)
[1, 8]
The confusion matrix for a classification method with Class 0 and Class 1 is given below. What is the percent overall error rate?
confusion matrix
actual/predicted 0 1
0 80 100
1 20 120
[tex]a. \( 45.67 \% \)\\b. \( 37.50 \% \)\\ c. \( 55.82 \% \)\\ d. \( 38.70 \% \[/tex]
Which of the following number lines shows the solution to the compound inequality given below?
-2<3r+4<13
Answer:
We get -2 < r < 3
Corresponding to the fourth choice
The fourth number line is the correct option
Step-by-step explanation:
-2 < 3r+4 < 13
We have to isolate r,
subtracting 4 from each term,
-2-4< 3r + 4 - 4 < 13 - 4
-6 < 3r < 9
divding each term by 3,
-6/3 < r < 9/3
-2 < r < 3
so, the interval is (-2,3)
or, -2 < r < 3
this corresponds to
The fourth choice (since there is no equality sign)
Find exact value of the curvature of f(x) = sin^3(x) at
x=pi/2
The exact value of the curvature of f(x) = sin^3(x) at x = π/2 is 3. To find the curvature of the function f(x) = sin^3(x) at x = π/2.Calculate the second derivative of f(x).
2. Substitute x = π/2 into the second derivative.
3. Use the formula for curvature, which is given by the expression |f''(x)| / (1 + [f'(x)]^2)^(3/2).
Let's calculate the curvature of f(x) at x = π/2:
1. Calculating the second derivative of f(x):
f(x) = sin^3(x)
Using the chain rule, we find the first derivative:
f'(x) = 3sin^2(x) * cos(x)
Differentiating again, we find the second derivative:
f''(x) = (6sin(x) * cos^2(x)) - (3sin^3(x))
2. Substituting x = π/2 into the second derivative:
f''(π/2) = (6sin(π/2) * cos^2(π/2)) - (3sin^3(π/2))
Since sin(π/2) = 1 and cos(π/2) = 0, the expression simplifies to:
f''(π/2) = 6 * 0^2 - 3 * 1^3
f''(π/2) = -3
3. Calculating the curvature using the formula:
curvature = |f''(π/2)| / [1 + (f'(π/2))^2]^(3/2)
Since f'(π/2) = 3sin^2(π/2) * cos(π/2) = 0, the denominator becomes 1.
curvature = |-3| / (1 + 0^2)^(3/2)
curvature = 3 / 1^3/2
curvature = 3 / 1
curvature = 3
Therefore, the exact value of the curvature of f(x) = sin^3(x) at x = π/2 is 3.
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Find the absolute maximum and absolute minimum of the function
f(x, y) = y^2 — x^2 + 4xy
over the region R that is the square with vertices (−1,0); (0, 1); (1,0) and (0, –1).
The absolute maximum and absolute minimum of the given function over the region R that is the square with vertices (−1,0); (0, 1); (1,0) and (0, –1) are 1 and -1, respectively.
To find the function's absolute maximum and absolute minimum, f(x, y) = y^2 — x^2 + 4xy, we need to determine the critical points in the given square region R and then use the Second Derivative Test to classify them.
Then we must evaluate the function at each vertex of R and select the greatest and smallest values as the absolute maximum and minimum values of f(x, y), respectively. So let's calculate the critical points of the given function:
∂f/∂x = -2x + 4y = 0 ...............(1)
∂f/∂y = 2y + 4x = 0 ................(2)
From (1) and (2),
we have x = 2y and y = -2x/4
⇒ y = -x/2.
Substituting this value of y in equation (1), we get x = -y.t
Now, we can write the point (x, y) = (-y, -x/2) as the critical point.
To classify these critical points as maximum, minimum or saddle point,
we can write the Second Derivative Test.
D(f(x, y)) = ∂²f/∂x² ∂²f/∂x∂y∂²f/∂y∂x ∂²f/∂y²
= (-2) (4) (4) (-2) - (4)²
= -16 < 0
Thus, we have a saddle point at (-y, -x/2). The greatest and smallest values are the absolute maximum and minimum values of f(x, y), respectively. Thus, we concluded that the absolute maximum and absolute minimum of the given function over the region R that is, the square with vertices (−1,0); (0, 1); (1,0) and (0, –1) are 1 and -1, respectively.
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Find the value or values of c that satisfy the equation f(b)−f(a)/b−a=f′(c) in the conclusion of the Mean Value Theorem for the following function and interval. f(x)=3x2+5x−2,[−2,1].
The value of `c` that satisfies the equation `f(b)−f(a)/b−a=f′(c)` in the conclusion of the Mean Value Theorem for the given function and interval `[a,b]` is `-1/2`.
Given function, `f(x) = 3x² + 5x - 2` in the interval `[-2,1]`.
The Mean Value Theorem(MVT) states that the slope of the tangent line at some point in an interval is equal to the slope of the secant line between the two endpoints.
It means there exists a point `c` in `[a,b]`
such that
`f'(c) = (f(b) - f(a)) / (b - a)`.
We have to find the value of `c` that satisfies the MVT for the given function and interval.
So,
`a = -2,
b = 1` and
`f(x) = 3x² + 5x - 2`.
Now, we need to find `f'(x)`.
`f(x) = 3x² + 5x - 2`
`f'(x) = d/dx(3x² + 5x - 2)``
= 6x + 5`
By MVT,
`f(b) - f(a) / b - a = f'(c)`
Substituting values of `f(a)`, `f(b)`, `a` and `b`, we get;
`[f(1) - f(-2)] / [1 - (-2)] = f'(c)`
Now,
`f(1) = 3(1)² + 5(1) - 2
= 6`
`f(-2) = 3(-2)² + 5(-2) - 2
= 4
`Thus,
`[6 - 4] / [1 - (-2)] = f'(c)`
Simplifying,
`2 / 3 = 6c + 5`
Solving this equation we get, `c = -1/2`.
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Which ONE of the following statements is correct? Select one: Select one: a. As analogue to digital conversion is a dynamic process, each conversion takes a finite amount of time called the quantisati
The correct statement is:
a. As analogue to digital conversion is a dynamic process, each conversion takes a finite amount of time called the quantization time.
Analog-to-digital conversion is the process of converting continuous analog signals into discrete digital representations. This conversion involves several steps, including sampling, quantization, and encoding.
During the quantization step, the continuous analog signal is divided into discrete levels or steps. Each step represents a specific digital value. The quantization process introduces a finite amount of error, known as quantization error, due to the approximation of the analog signal.
Since the quantization process is dynamic and involves the discretization of the continuous signal, it takes a finite amount of time to perform the conversion for each sample. This time is known as the quantization time.
During this time, the analog signal is sampled, and the corresponding digital value is determined based on the quantization levels. The quantization time can vary depending on the specific system and the required accuracy.
Therefore, statement a. accurately states that analog-to-digital conversion is a dynamic process that takes a finite amount of time called the quantization time for each conversion.
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Let F(x)=f(x5) and G(x)=(f(x))5. You also know that a4=10,f(a)=2,f′(a)=4,f′(a5)=4 Then F′(a)= and G′(a)=__
the required values are:F'(a) = 200000G'(a) = 640 Hence, the required answer is F′(a) = 200000 and G′(a) = 640.
Let's use the chain rule of differentiation to calculate F'(a).F(x) = f(x⁵)
Using the chain rule, we get:F'(x) = f'(x⁵) × 5x⁴
Applying this to F(x), we get:F'(x) = f'(x⁵) × 5x⁴Also, substituting x = a, we get:F'(a) = f'(a⁵) × 5a⁴We know that f'(a⁵) = 4 and a⁴ = 10.
Substituting these values, we get:F'(a) = 4 × 5 × 10⁴ = 200000
Now, let's use the chain rule of differentiation to calculate G'(a).G(x) = (f(x))⁵Using the chain rule, we get:G'(x) = 5(f(x))⁴ × f'(x)
Applying this to G(x), we get:G'(x) = 5(f(x))⁴ × f'(x)
Also, substituting x = a, we get:G'(a) = 5(f(a))⁴ × f'(a)
We know that f(a) = 2 and f'(a) = 4.
Substituting these values, we get:G'(a) = 5(2)⁴ × 4 = 640
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In our 6/30 class, we tried to work out the infamous ∫sec^3 xdx, and I made a mistake (anyone who found my error and email me will have extra credit) and got stuck. Now you will do it by following the Integration by Parts:
a. Rewrite it as ∫ (sec x) sec^2 xdx = ∫ udv, what is u and dv?
b. What is du and v?
c. For working on ∫ vdu, transform all expressions to sec x and work out.
Rewrite it as ∫ (sec x) sec^2 xdx = ∫ udv, Let's apply integration by parts. Here, the aim is to determine the integrals of the product of two functions, like f(x)g(x) when the integral of either f(x) or g(x) is unknown. Choose a "u" part of f(x) and the rest as "dv" part. Then apply the formula [uv - ∫vdu] for integration by parts.
Let's do that with the given question. ∫ sec^3 xdxLet's take the u as sec x and dv as sec^2 xdx.The expression is
∫ sec x * sec^2 xdx = ∫ sec x * sec x *
tan x dx = ∫ sec^2 x * tan x dxb. We need to differentiate the u term and integrate the dv term. Let's do that in detail.
u = sec x ⇒ du/dx = sec x * tan x ⇒ du = sec x * tan x dx On integrating dv, we get the following:
v = ∫ sec^2 xdx = tan x Therefore,
dv = sec^2 xdxc.
For working on ∫ vdu, transform all expressions to sec x and work out.Now we need to calculate the value of ∫ vdu. We can now substitute u and v values to this expression and get the answer as shown below:∫ sec^3 x dx = sec x tan x - ∫ tan^2 x dx = sec x tan x - ∫ (sec^2 x - 1) dx = sec x tan x - ln|sec x + tan x| + C.
By applying integration by parts, ∫ sec^3 xdx = sec x tan x - ln|sec x + tan x| + C. We used integration by parts to solve the given expression.
Here, we took the u as sec x and dv as sec^2 xdx. We then differentiated the u term and integrated the dv term. On substituting the values of u and v, we obtained the answer to be sec x tan x - ln|sec x + tan x| + C in the end.
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{-3 + x, if x < 3
Let f(x) = {3 –x, if x ≥
Evaluate the following expressions.
limx→3−f(x)=
limx→3+f(x)=
f(3)=
Is the function f continuous at 3 ?
The function f(x) is defined piecewise as -3 + x for x < 3 and 3 - x for x ≥ 3. We need to evaluate the limits as x approaches 3 from the left and right, find the value of f(3), and determine whether the function is continuous at x = 3.
To evaluate limx→3⁻ f(x), we substitute x = 3 into the piece of the function that corresponds to x < 3. In this case, f(x) = -3 + x, so limx→3⁻ f(x) = -3 + 3 = 0.
To evaluate limx→3⁺ f(x), we substitute x = 3 into the piece of the function that corresponds to x ≥ 3. In this case, f(x) = 3 - x, so limx→3⁺ f(x) = 3 - 3 = 0.
To find f(3), we substitute x = 3 into the piece of the function that corresponds to x ≥ 3. In this case, f(x) = 3 - x, so f(3) = 3 - 3 = 0.
Since the limits from the left and right, as well as the function value at x = 3, are all equal to 0, we can conclude that the function f(x) is continuous at x = 3. This is because the left-hand and right-hand limits exist and are equal to each other, and they both match the value of the function at x = 3.
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Use implicit differentiation to find the slope of the tangent line to the curve defined by 9xy + xy = 10 at
the point (1, 1). The slope of the tangent line to the curve at the given point is Preview
The slope of the tangent line to the curve defined by 9xy + xy = 10 at the point (1, 1) is -1.
To find the slope of the tangent line to the curve defined by the equation 9xy + xy = 10 at the point (1, 1), we can use implicit differentiation.
Let's start by differentiating both sides of the equation with respect to x.
Differentiating the left side of the equation:
d/dx(9xy + xy) = d/dx(10)
Using the product rule for differentiation, we differentiate each term separately:
d/dx(9xy) + d/dx(xy) = 0
Now, let's calculate the derivatives of each term:
For the first term, 9xy:
Using the product rule, we have:
d/dx(9xy) = 9y * dx/dx + 9x * dy/dx
= 9y + 9x * dy/dx
For the second term, xy:
Using the product rule again, we have:
d/dx(xy) = y * dx/dx + x * dy/dx
= y + x * dy/dx
Substituting these results back into our equation, we get:
9y + 9x * dy/dx + y + x * dy/dx = 0
Combining like terms, we have:
10y + 10x * dy/dx = 0
Now, let's find the value of dy/dx at the point (1, 1). We substitute x = 1 and y = 1 into the equation:
10(1) + 10(1) * dy/dx = 0
Simplifying further:
10 + 10 * dy/dx = 0
Dividing both sides by 10:
1 + dy/dx = 0
Finally, subtracting 1 from both sides:
dy/dx = -1
Therefore, the slope of the tangent line to the curve defined by 9xy + xy = 10 at the point (1, 1) is -1.
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Find the slope of the curve at the indicated point.
y = x^2 + 5x +4, x = -1
o m = 3
o m=7
o m = -4
o m = -2
`m = 3` is the slope of the curve at the indicated point. Hence, the correct option is `o m = 3`.
To find the slope of the curve at the indicated point, given
`y = x^2 + 5x +4, x = -1`,
we will use the first principle of differentiation.
The slope of the curve can be obtained by finding the derivative of the given equation.
First, we differentiate the function with respect to `x` using the first principle of differentiation.
This is given as:
`(dy)/(dx) = [f(x+h) - f(x)]/h`
Let
`f(x) = x^2 + 5x + 4`.
Then
`f(x + h) = (x + h)^2 + 5(x + h) + 4
= x^2 + 2hx + h^2 + 5x + 5h + 4`
Substituting the values in the formula:
`(dy)/(dx) = lim (h→0) [f(x+h) - f(x)]/h
= lim (h→0) [(x^2 + 2hx + h^2 + 5x + 5h + 4) - (x^2 + 5x + 4)]/h` `
= lim (h→0) [2hx + h^2 + 5h]/h
= lim (h→0) [2x + h + 5]`
Thus, the slope of the curve at the given point is:
`m = (dy)/(dx)
= 2x + 5
= 2(-1) + 5
= 3`.
Therefore, `m = 3` is the slope of the curve at the indicated point. Hence, the correct option is `o m = 3`.
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price: house price, 1000s
lotsize: size of lot in square feet
sqrft: size of house in square feet
bdrms: number of bedrooms
(a) Write down the definition of homoskedasticity and heteroskedasticity in the context of
the regression equation given in (1).
(b)Do you think that the errors term may be homoskedastic or heteroskedastic? Briefly
explain your reasoning.
a. In mathematical terms, Var(ε) = f(x), where f(x) represents a function of the independent variables. b. the spread or dispersion of the residuals in the regression equation will not be constant across all levels of the predictors, indicating the presence of heteroskedasticity.
(a) In the context of a regression equation, homoskedasticity and heteroskedasticity refer to the characteristics of the error terms or residuals in the model. The error term represents the difference between the observed dependent variable and the predicted value from the regression equation.
Homoskedasticity, also known as homogeneity of variance, implies that the error terms have constant variance across all levels of the independent variables. In other words, the spread or dispersion of the residuals is the same regardless of the values of the predictors. Mathematically, it can be represented as Var(ε) = σ², where Var(ε) denotes the variance of the error term ε, and σ² represents a constant value.
On the other hand, heteroskedasticity means that the error terms have non-constant variance. This implies that the spread or dispersion of the residuals varies across different levels of the independent variables. In mathematical terms, Var(ε) = f(x), where f(x) represents a function of the independent variables.
(b) Based on the given information about house price, lot size, square footage, and number of bedrooms, it is reasonable to suspect that the error term may exhibit heteroskedasticity. This is because various factors can influence the variability of house prices, such as the size of the lot, square footage, and the number of bedrooms.
For instance, larger houses or lots may tend to have higher price fluctuations due to differences in demand, location, or amenities. Similarly, the number of bedrooms may impact the price variability as houses with more bedrooms often cater to different buyer segments, leading to varying preferences and potential price differences.
Therefore, it is likely that the spread or dispersion of the residuals in the regression equation will not be constant across all levels of the predictors, indicating the presence of heteroskedasticity.
In summary, considering the nature of the variables involved in the regression equation (house price, lot size, square footage, and number of bedrooms), it is reasonable to expect that the error term will exhibit heteroskedasticity. The factors influencing house prices are diverse and can lead to variations in price volatility, suggesting that the spread or dispersion of the residuals will likely differ across different levels of the independent variables.
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Consider the given PDA with bottom stack symbol \( X \). It corresponds to a regular language. Create a regular expression for the language accepted by this PDA. Draw a PDA for the palindromes of odd
PDA with bottom stack symbol \(X\) corresponds to a regular language.We can create a regular expression for the language accepted by the PDA with bottom stack symbol \(X\) by constructing a DFA from the given PDA and then converting the DFA to a regular expression.
The PDA accepts palindromes of odd length. Here, we use three states. The symbols \(a,b\) are the input symbols, and \(Y,Z\) are the stack symbols.The transition table for the PDA is given below:For state 0, we have two transitions. The transition with symbol \(a\) pushes \(Y\) onto the stack, and the transition with symbol \(b\) pushes \(X\) onto the stack.For state 1, we have two transitions. The transition with symbol \(a\) pops \(Y\) off the stack, and the transition with symbol \(b\) pushes \(Y\) onto the stack.
For state 2, we have two transitions. The transition with symbol \(a\) pushes \(Y\) onto the stack, and the transition with symbol \(b\) pops \(X\) off the stack.For state 3, we have two transitions. The transition with symbol \(a\) pushes \(Y\) onto the stack, and the transition with symbol \(b\) pushes \(Z\) onto the stack.For state 4, we have two transitions. The transition with symbol \(a\) pushes \(a\) onto the stack, and the transition with symbol \(b\) pushes \(b\) onto the stack.For state 5, we have two transitions. The transition with symbol \(a\) pops \(b\) off the stack, and the transition with symbol \(b\) pops \(a\) off the stack.
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Let f(x,y,z)=5x^3−y^3+z^2. Find the maximum value M for the directional derivative at the point (1,−2,1).
(Use symbolic notation and fractions where needed.)
M = ____________
The directional derivative is a measure of the rate at which the function f(x, y, z) changes in the direction of a vector v = under the unit vector u, denoted by Duf.
The formula for the directional derivative is given as:
`D_u(f(x, y, z)) = grad(f) . u`.
Where, grad(f) is the gradient of the function f(x, y, z) and . represents the dot product .
Thus, the maximum value of the directional derivative at point (1, -2, 1) is `-42/sqrt(29)` in the direction of `<3, 4, -2>`.
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"""
Sample code for question 2
We will solve the following equation
2t^2*y''(t)+3/2t*y'(t)-1/2t^2*y(t)=t
"""
import numpy as np
import as plt
from scipy.integrate import odeint
#De
The general solution to the non-homogeneous equation is,
y(t) = c₁[tex]t^{1/2}[/tex] + c2/t + t - 1/(2t³)
where c₁ and c₂ are constants determined by the initial or boundary conditions of the problem.
Now, For this differential equation, we will use the method of undetermined coefficients.
We first need to find the general solution to the homogeneous equation:
2t²*y''(t) + (3/2t)*y'(t) - (1/2t²)*y(t) = 0
We assume a solution of the form y_h(t) = [tex]t^{r}[/tex]. Substituting this into the equation, we get:
2t²r(r-1)*[tex]t^{r - 2}[/tex] + (3/2t)*r * [tex]t^{r - 1}[/tex] - (1/2t²)* [tex]t^{r}[/tex] = 0
Simplifying, we get:
2r*(r-1) + (3/2)*r - (1/2) = 0
Solving for r, we get:
r = 1/2, -1
Therefore, the general solution to the homogeneous equation is:
y_h(t) = c₁[tex]t^{1/2}[/tex] + c₂/t
To find a particular solution to the non-homogeneous equation, we assume a solution of the form y_p(t) = At + B.
Substituting this into the equation, we get:
2t²y''(t) + (3/2t)y'(t) - (1/2t²)*y(t) = t
Differentiating twice, we get:
2t²*y'''(t) + 6ty''(t) - 3y'(t) + (1/t²)*y(t) = 0
Substituting y_p(t) into this equation, we get:
2t²0 + 6tA - 3A + (1/t²)(At + B) = 0
Simplifying, we get:
(A/t)*[(2t³ - 1)B + t⁴] = t
Since this equation must hold for all values of t, we equate the coefficients of t and 1/t:
(2t³ - 1)B + t⁴ = 0
A/t = 1
Solving for A and B, we get:
A = 1
B = -1/(2t³)
Therefore, a particular solution to the non-homogeneous equation is:
y_p(t) = t - 1/(2t³)
So, The general solution to the non-homogeneous equation is the sum of the homogeneous and particular solutions:
y(t) = c₁[tex]t^{1/2}[/tex] + c2/t + t - 1/(2t³)
where c₁ and c₂ are constants determined by the initial or boundary conditions of the problem.
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By using one-sided limits, determine whether each limit exists. Illustrate yOUr results geometrically by sketching the graph of the function.
limx→5 ∣x−5∣ / x−5
The limit as x approaches 5 of |x - 5| / (x - 5) does not exist. There is a discontinuity at x = 5, which prevents the existence of the limit at that point.
To determine the existence of the limit, we evaluate the left-sided and right-sided limits separately.
Left-sided limit:
As x approaches 5 from the left side (x < 5), the expression |x - 5| / (x - 5) simplifies to (-x + 5) / (x - 5). Taking the limit as x approaches 5 from the left side, we substitute x = 5 into the expression and get (-5 + 5) / (5 - 5), which is 0 / 0, an indeterminate form. This indicates that the left-sided limit does not exist.
Right-sided limit:
As x approaches 5 from the right side (x > 5), the expression |x - 5| / (x - 5) simplifies to (x - 5) / (x - 5). Taking the limit as x approaches 5 from the right side, we substitute x = 5 into the expression and get (5 - 5) / (5 - 5), which is 0 / 0, also an indeterminate form. This indicates that the right-sided limit does not exist.
Since the left-sided limit and the right-sided limit do not agree, the overall limit as x approaches 5 does not exist.
Geometrically, if we sketch the graph of the function y = |x - 5| / (x - 5), we would observe a vertical asymptote at x = 5, indicating that the function approaches positive and negative infinity as x approaches 5 from different sides. There is a discontinuity at x = 5, which prevents the existence of the limit at that point.
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Consider the series k=1∑[infinity](2k+1)(2k+3)4. (a) (4 pts) Do a partial fraction decomposition to rewrite the rule for ak, the k th term of the series. (b) (4 pts) Write the first 4 partial sums of the series, then determine a rule for the kth partial sum. (c) (4 pts) Find the sum of the original series, if it exists.
The series diverges, and there is no finite sum for the original series.
(a) To perform a partial fraction decomposition, we start by expressing the given series as a rational function:
ak = (2k + 1)(2k + 3)/4
Now, we'll decompose this expression into partial fractions. Let's assume that ak can be expressed as:
ak = A/(2k + 1) + B/(2k + 3)
To find the values of A and B, we'll find a common denominator on the right-hand side:
ak = [A(2k + 3) + B(2k + 1)] / [(2k + 1)(2k + 3)]
Expanding the numerator:
ak = (2Ak + 3A + 2Bk + B) / [(2k + 1)(2k + 3)]
Now, we can equate the numerators of the original expression and the partial fractions decomposition:
(2k + 1)(2k + 3)/4 = (2Ak + 3A + 2Bk + B) / [(2k + 1)(2k + 3)]
From this equation, we can equate the coefficients of like terms:
2Ak + 3A + 2Bk + B = 2k + 1
Matching the coefficients of k terms:
2A + 2B = 2
Matching the constant terms:
3A + B = 1
Now we have a system of equations to solve:
2A + 2B = 2
3A + B = 1
Solving this system, we find A = 1/2 and
B = 1/2.
Therefore, the partial fraction decomposition of ak is:
ak = 1/(2k + 1) + 1/(2k + 3)
(b) Let's write the first four partial sums of the series:
S1 = a1
= 1/(2(1) + 1) + 1/(2(1) + 3)
= 1/3 + 1/5
S2 = a1 + a2
= 1/3 + 1/5 + 1/(2(2) + 1) + 1/(2(2) + 3)
= 1/3 + 1/5 + 1/5 + 1/7
S3 = a1 + a2 + a3
= 1/3 + 1/5 + 1/5 + 1/7 + 1/(2(3) + 1) + 1/(2(3) + 3)
= 1/3 + 1/5 + 1/5 + 1/7 + 1/7 + 1/9
S4 = a1 + a2 + a3 + a4
= 1/3 + 1/5 + 1/5 + 1/7 + 1/7 + 1/9 + 1/(2(4) + 1) + 1/(2(4) + 3)
= 1/3 + 1/5 + 1/5 + 1/7 + 1/7 + 1/9 + 1/9 + 1/11
We can observe a pattern in the partial sums:
S1 = 1/3 + 1/5
S2 = 1/3 + 1/5 + 1/5 + 1/7
S3 = 1/3 + 1/5 + 1/5 + 1/7 + 1/7 + 1/9
S4 = 1/3 + 1/5 + 1/5 + 1/7 + 1/7 + 1/9 + 1/9 + 1/11
From this pattern, we can infer that the kth partial sum Sk can be expressed as:
Sk = 1/3 + 1/5 + 1/5 + 1/7 + 1/7 + 1/9 + ... + 1/(2k + 1) + 1/(2k + 3)
(c) To find the sum of the original series, we need to determine if it converges. Let's consider the behavior of the terms as k approaches infinity:
lim(k->∞) ak = lim(k->∞) (2k + 1)(2k + 3)/4
The term ak grows without bound as k approaches infinity. Therefore, the series diverges, and there is no finite sum for the original series.
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Solve the initial value problem given by the differential equation: C1C2R2(Rc+R1)dt2d2qc2+[(Rc+R1)(C1+C2)+R2C2]dtdqc2+qc2=C2E and the initial conditions: qc2(0)dtdqc2(0)=0=0 Hereafter we will use the following values of the resistors, capacitances, and voltage: C1=10μFC2=100μFR1=100ΩR2=100ΩRc=1kΩE=5 V
To solve the given initial value problem, we will substitute the provided values of resistors (R1, R2, Rc), capacitances (C1, C2), and voltage (E) into the differential equation. Then, we will apply the initial conditions to determine the specific solution for qc2(t) and its derivative.
The initial value problem is described by the following differential equation:
C1C2R2(Rc+R1)d²qc²/dt² + [(Rc+R1)(C1+C2) + R2C2]dqc²/dt + qc² = C2E
By substituting the given values into the equation, we obtain:
10μF * 100μF * 100Ω * (1kΩ + 100Ω)d²qc²/dt² + [(1kΩ + 100Ω)(10μF + 100μF) + 100Ω * 100μF]dqc²/dt + qc² = 100μF * 5V
Simplifying the equation with these values, we can solve for qc²(t) by applying the initial conditions qc²(0) = 0 and dqc²/dt(0) = 0. The specific solution for qc²(t) will depend on the specific values obtained from the calculations.
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The equations for two planes are given at right π1: x+2 y+3 z=6
Show these two planes have the same x intercept. π2: 2 x-y+4z= 12
The x-intercept of plane π2 is also (6, 0, 0). Since both planes have the same x-coordinate for their x-intercepts, namely x = 6, we can conclude that they intersect the x-axis at the same point. Therefore, the two planes have the same x-intercept.
To determine if two planes have the same x-intercept, we need to find the x-coordinate where each plane intersects the x-axis. For a point to lie on the x-axis, its y and z coordinates must be zero.
For plane π1: x + 2y + 3z = 6, we set y = 0 and z = 0:
x + 2(0) + 3(0) = 6
x = 6
So, the x-intercept of plane π1 is (6, 0, 0).
For plane π2: 2x - y + 4z = 12, we again set y = 0 and z = 0:
2x - (0) + 4(0) = 12
2x = 12
x = 6
The x-intercept of plane π2 is also (6, 0, 0).
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In testing a certain kind of truck tire over rugged terrain, it is found that 25% of the frucks fail to complete the test run without a blowout. Of the next 15 trucks tested, find the probability that (a) from 3 to 6 have blowouts; (b) fewer than 4 have blowouts: (c) more than 5 have blowouts.
Probability that from 3 to 6 have blowouts is 0.4477 Probability that fewer than 4 have blowouts is 0.3615Probability that more than 5 have blowouts is 0.3973.
Given: It is found that 25% of the trucks fail to complete the test run without a blowout.Probability is the ratio of the number of favorable outcomes to the total number of possible outcomes.In order to find the probability of the given events, we will use Binomial Distribution.
Let’s find the probability of given events one by one:a) From 3 to 6 trucks have blowouts Number of trials = 15 (n)Number of success = trucks with blowouts (x)Number of failures = trucks without blowouts = 15 - xProbability of a truck with blowout = p = 0.25Probability of a truck without blowout = q = 1 - 0.25 = 0.75We need to find
P(3 ≤ x ≤ 6) = P(x = 3) + P(x = 4) + P(x = 5) + P(x = 6)P(x = r) = nCr * pr * q(n-r)
where nCr = n! / r!(n-r)!P(x = 3)
= 15C3 * (0.25)3 * (0.75)12
= 0.1859P(x = 4) = 15C4 * (0.25)4 * (0.75)11
= 0.1670P(x = 5)
= 15C5 * (0.25)5 * (0.75)10 = 0.0742P(x = 6)
= 15C6 * (0.25)6 * (0.75)9 = 0.0206P(3 ≤ x ≤ 6)
= 0.1859 + 0.1670 + 0.0742 + 0.0206
= 0.4477
Therefore, the probability that from 3 to 6 trucks have blowouts is 0.4477.b) Fewer than 4 trucks have blowoutsNumber of trials = 15 (n)Number of success = trucks with blowouts (x)Number of failures = trucks without blowouts = 15 - xProbability of a truck with blowout = p = 0.25Probability of a truck without blowout = q = 1 - 0.25 = 0.75We need to find P(x < 4) = P(x = 0) + P(x = 1) + P(x = 2) + P(x = 3)P(x = r) = nCr * pr * q(n-r)where nCr = n! / r!(n-r)!P(x = 0) = 15C0 * (0.25)0 * (0.75)15 = 0.0059P(x = 1) = 15C1 * (0.25)1 * (0.75)14 = 0.0407P(x = 2) = 15C2 * (0.25)2 * (0.75)13 = 0.1290P(x = 3) = 15C3 * (0.25)3 * (0.75)12 = 0.1859P(x < 4) = 0.0059 + 0.0407 + 0.1290 + 0.1859= 0.3615Therefore, the probability that fewer than 4 trucks have blowouts is 0.3615.c) More than 5 trucks have blowoutsNumber of trials = 15 (n)Number of success = trucks with blowouts (x)Number of failures = trucks without blowouts = 15 - xProbability of a truck with blowout = p = 0.25Probability of a truck without blowout = q = 1 - 0.25 = 0.75
We need to find P(x > 5)P(x > 5) = P(x = 6) + P(x = 7) + ... + P(x = 15)P(x = r) = nCr * pr * q(n-r)
where nCr = n! / r!(n-r)!
P(x > 5) = 1 - [P(x ≤ 5)]P(x ≤ 5) = P(x = 0) + P(x = 1) + P(x = 2) + P(x = 3) + P(x = 4) + P(x = 5)P(x = 0) = 15C0 * (0.25)0 * (0.75)15
= 0.0059P(x = 1) = 15C1 * (0.25)1 * (0.75)14 = 0.0407P(x = 2)
= 15C2 * (0.25)2 * (0.75)13 = 0.1290P(x = 3)
= 15C3 * (0.25)3 * (0.75)12 = 0.1859P(x = 4)
= 15C4 * (0.25)4 * (0.75)11 = 0.1670P(x = 5)
= 15C5 * (0.25)5 * (0.75)10
= 0.0742P(x ≤ 5)
= 0.0059 + 0.0407 + 0.1290 + 0.1859 + 0.1670 + 0.0742
= 0.6027P(x > 5) = 1 - 0.6027= 0.3973
Therefore, the probability that more than 5 trucks have blowouts is 0.3973.Answer:Probability that from 3 to 6 have blowouts is 0.4477Probability that fewer than 4 have blowouts is 0.3615Probability that more than 5 have blowouts is 0.3973.
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Use algebra to evaluate the limit. limh→0 (4+h)2−(4−h)2/2h = ___
In order to evaluate the given limit, we need to use algebra.
Here's how to evaluate the limit:
We are given the expression:
limh→0 (4+h)² - (4-h)²/2h
To simplify the given expression, we need to use the identity:
a² - b² = (a+b)(a-b)
Using this identity, we can write the given expression as:
limh→0 [(4+h) + (4-h)][(4+h) - (4-h)]/2h
Simplifying this expression further, we get:
limh→0 [8h]/2h
Cancelling out the common factor of h in the numerator and denominator, we get:
limh→0 8/2= 4
Therefore, the value of the given limit is 4.
Hence, the required blank is 4.
What we have used here is the identity of difference of squares, which states that a² - b² = (a+b)(a-b).
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The tables show the ratios of black socks to white socks in the women’s and men’s departments of a store. If each department has the same number of black socks, which department stocks more white socks?
The men's department stocks more white socks. The ratio of black socks to white socks in the women's department is 3:4, while in the men's department it is 1:3.
Since the number of black socks is the same in both departments, the department with the smaller ratio of black to white socks will have more white socks. In the women's department, for every 3 black socks, there are 4 white socks, resulting in a total of 7 socks. In the men's department, for every 1 black sock, there are 3 white socks, making a total of 4 socks. Since the number of black socks is the same in both departments, the women's department has a higher total number of socks (7) compared to the men's department (4). Therefore, the men's department stocks more white socks.
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THTONFOUR [10 MARKS] Helean Algebra Theorems 41. Write the two De-Morgans theorems [2 MARKS] 42.Use the two theorems to simplify the following expressions 4.2.1. \( X=\overline{\overline{A(B+\bar{A})
The simplified expression is \(X = A \cdot \overline{B}\).
41. De Morgan's Theorems state the following:
a) De Morgan's First Theorem: The complement of the union of two sets is equal to the intersection of their complements. In terms of Boolean algebra, it can be expressed as:
\(\overline{A \cup B} = \overline{A} \cap \overline{B}\)
b) De Morgan's Second Theorem: The complement of the intersection of two sets is equal to the union of their complements. In terms of Boolean algebra, it can be expressed as:
\(\overline{A \cap B} = \overline{A} \cup \overline{B}\)
42. Now, let's use the two De Morgan's theorems to simplify the given expression:
\(X = \overline{\overline{A(B + \bar{A})}}\)
Using De Morgan's Second Theorem, we can distribute the complement over the sum:
\(X = \overline{\overline{A} \cdot \overline{(B + \bar{A})}}\)
Now, applying De Morgan's First Theorem, we can distribute the complement over the sum inside the brackets:
\(X = \overline{\overline{A} \cdot (\overline{B} \cap \overline{\bar{A}})}\)
Since \(\overline{\bar{A}}\) is equal to \(A\), we can simplify further:
\(X = \overline{\overline{A} \cdot (\overline{B} \cap A)}\)
Applying De Morgan's First Theorem again, we can distribute the complement over the intersection:
\(X = \overline{\overline{A} \cdot \overline{B} \cup \overline{A} \cdot A}\)
Since \(A \cdot \overline{A}\) is always equal to 0, we can simplify further:
\(X = \overline{\overline{A} \cdot \overline{B} \cup 0}\)
The union of any set with 0 is equal to the set itself:
\(X = \overline{\overline{A} \cdot \overline{B}}\)
Finally, applying the double complement law (\(\overline{\overline{X}} = X\)), we get:
\(X = A \cdot \overline{B}\)
Therefore, the simplified expression is \(X = A \cdot \overline{B}\).
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pleade solve
A card is drawn from a well-shuffled deck of 52 cards. What is the probability of drawing a black 10 or a red 7?
The probability of drawing a black 10 or a red 7 from a well-shuffled deck of 52 cards is 3/52, which can also be expressed as approximately 0.0577 or about 5.77%.
To find the probability of drawing a black 10 or a red 7 from a well-shuffled deck of 52 cards, we need to determine the number of favorable outcomes (black 10 or red 7) and the total number of possible outcomes (all cards in the deck).
Let's first calculate the number of black 10 cards in the deck. In a standard deck, there is only one black 10, which is the 10 of clubs or the 10 of spades.
Next, let's calculate the number of red 7 cards in the deck. In a standard deck, there are two red 7s, namely the 7 of hearts and the 7 of diamonds.
Therefore, the total number of favorable outcomes is 1 (black 10) + 2 (red 7s) = 3.
Now, let's calculate the total number of possible outcomes, which is the total number of cards in the deck, 52.
The probability of drawing a black 10 or a red 7 can be calculated as:
Probability = Number of favorable outcomes / Total number of possible outcomes
Probability = 3 / 52
Simplifying the fraction, we get:
Probability = 3/52
So, the probability of drawing a black 10 or a red 7 from a well-shuffled deck of 52 cards is 3/52, which can also be expressed as approximately 0.0577 or about 5.77%.
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A company sells multiple of a half foot. It has found that it can sell 200 carpets in a week when the carpets are 3ft by 3ft, the minimum size. Beyond this, for each additional foot of length and width, the number sold goes down by 5 . What size carpets should the company sell to maximize its revenue? What is the maximum weekly revenue? Write the equation for the revenue, R, the company will earn as function of the length, x, of the carpet squares sold. R(x)=___
That the length of the DFT affects the number of samples in the output sequence.
a) To compute y[n] = x[n] * h[n] using a 5-point DFT, we first need to extend both x[n] and h[n] to length N = 5 by zero-padding:
x[n] = {1, 2, 3, 4, 5}
h[n] = {1, 3, 5, 0, 0}
Next, we take the DFT of both x[n] and h[n]. Let X[k] and H[k] denote the DFT coefficients of x[n] and h[n], respectively.
X[k] = DFT{x[n]} = {X[0], X[1], X[2], X[3], X[4]}
H[k] = DFT{h[n]} = {H[0], H[1], H[2], H[3], H[4]}
Now, we can compute the element-wise product of X[k] and H[k]:
Y[k] = X[k] * H[k] = {X[0]*H[0], X[1]*H[1], X[2]*H[2], X[3]*H[3], X[4]*H[4]}
Finally, we take the inverse DFT (IDFT) of Y[k] to obtain y[n]:
y[n] = IDFT{Y[k]} = {y[0], y[1], y[2], y[3], y[4]}
b) To compute the convolution of x[n] and h[n] using a 10-point DFT, we first extend both x[n] and h[n] to length N = 10 by zero-padding:
x[n] = {1, 2, 3, 4, 5, 0, 0, 0, 0, 0}
h[n] = {1, 3, 5, 0, 0, 0, 0, 0, 0, 0}
Next, we take the DFT of both x[n] and h[n]. Let X[k] and H[k] denote the DFT coefficients of x[n] and h[n], respectively.
X[k] = DFT{x[n]} = {X[0], X[1], X[2], X[3], X[4], X[5], X[6], X[7], X[8], X[9]}
H[k] = DFT{h[n]} = {H[0], H[1], H[2], H[3], H[4], H[5], H[6], H[7], H[8], H[9]}
Now, we can compute the element-wise product of X[k] and H[k]:
Y[k] = X[k] * H[k] = {X[0]*H[0], X[1]*H[1], X[2]*H[2], X[3]*H[3], X[4]*H[4], X[5]*H[5], X[6]*H[6], X[7]*H[7], X[8]*H[8], X[9]*H[9]}
Finally, we take the inverse DFT (IDFT) of Y[k] to obtain y[n]:
y[n] = IDFT{Y[k]} = {y[0], y[1], y[2], y[3], y[4], y[5], y[6], y[7], y[8], y[9]}
By comparing the results from parts (a) and (b), we can observe
that the length of the DFT affects the number of samples in the output sequence.
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Find the midpoint of the line segment with the given endpoints. 5) \( (-4,0),(3,5) \) 6) \( (9,-2),(8,-4) \) Find the midpoint of each line segment. 8
5) The midpoint of points (-4,0), and (3,5) is, (- 1/2, 5/2)
6) The midpoint of points (9,-2), and (8,-4) is, (17/2, - 6/2)
We have to given that,
To find the midpoint of the line segment with the given endpoints.
5) (-4,0), and (3,5)
6) (9,-2), and (8,-4)
Now, We get;
5) The midpoint of points (-4,0), and (3,5) is,
(- 4 + 3)/2, (0 + 5)/2
(- 1/2, 5/2)
6) The midpoint of points (9,-2), and (8,-4) is,
(9 + 8)/2, (- 2 - 4)/2
(17/2, - 6/2)
Thus, We get;
5) The midpoint of points (-4,0), and (3,5) is, (- 1/2, 5/2)
6) The midpoint of points (9,-2), and (8,-4) is, (17/2, - 6/2)
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Direction: Read each statement and decide whether the answer is correct or not. If the statement is correct write true, if the statement is incorrect write false and write the correct statement (5 X 2 Mark= 10 Marks)
1. PESTLE framework categorizes environmental influences into six main types.
2. PESTLE framework analysis the micro-environment of organizations.
3. Economic forces are one of the types included in PESTLE framework.
4. An organization’s strength is part of the types studied in PESTLE framework.
5. PESTLE framework provides a comprehensive list of influences on the possible success or failure of strategies.
1. True. The PESTLE framework categorizes environmental influences into six main types: Political, Economic, Sociocultural, Technological, Legal, and Environmental factors.
These factors help analyze the external macro-environmental forces that can impact an organization's strategies and operations. 2. False. The PESTLE framework analyzes the macro-environmental factors and not the micro-environment of organizations. The micro-environment is examined through other frameworks like Porter's Five Forces, which focus on specific industry dynamics and competitive factors.
3. True. Economic forces, such as inflation, interest rates, exchange rates, and economic growth, are one of the types included in the PESTLE framework. Economic factors play a significant role in shaping business decisions and strategies.
4. False. An organization's strengths are not part of the types studied in the PESTLE framework. Strengths, weaknesses, opportunities, and threats (SWOT) analysis is a separate framework used to assess internal strengths and weaknesses of an organization.
5. True. The PESTLE framework provides a comprehensive list of influences on the possible success or failure of strategies. By considering the political, economic, sociocultural, technological, legal, and environmental factors, organizations can gain insights into the external forces that may impact their strategies and make informed decisions.
The PESTLE framework categorizes environmental influences into six main types, including political, economic, sociocultural, technological, legal, and environmental factors. It analyzes the macro-environmental forces, not the micro-environment of organizations. Economic forces are one of the types studied in the framework, while an organization's strengths are not included. The framework provides a comprehensive list of influences on the success or failure of strategies, allowing organizations to consider various external factors in their decision-making process.
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Using the substitution: u=2x−10x2−4. Re-write the indefinite integral then evaluate in terms of u.
∫(−10x+1)e²ˣ−¹⁰ˣ²−⁴dx=∫
To evaluate the indefinite integral ∫(−10x+1)e²ˣ−¹⁰ˣ²−⁴dx, we can rewrite it in terms of the substitution u=2x−10x²−4 and then integrate with respect to u.
Let's rewrite the integral using the substitution u=2x−10x²−4. To do this, we need to express dx in terms of du. Differentiating u with respect to x gives du/dx=2−20x, which implies dx=du/(2−20x). We can substitute these expressions into the original integral to obtain ∫(−10x+1)e²ˣ−¹⁰ˣ²−⁴dx = ∫(-10x+1)e²ˣ−¹⁰ˣ²−⁴(du/(2−20x)).
Simplifying this expression, we have ∫(-10x+1)e²ˣ−¹⁰ˣ²−⁴(du/(2−20x)) = ∫(-10x+1)e²ˣ−¹⁰ˣ²−⁴du/(2−20x). Now, we can factor out the common term (2−20x) from the numerator, resulting in ∫(-10x+1)e²ˣ−¹⁰ˣ²−⁴du/(2−20x) = ∫(-10x+1)e²ˣ−¹⁰ˣ²−⁴du/2(1−10x).
Now, the integral can be evaluated easily with respect to u, as the expression inside the integral no longer contains x. The resulting integral is ∫(-10x+1)e²ˣ−¹⁰ˣ²−⁴du/2(1−10x). Finally, we integrate with respect to u and replace u with the original expression 2x−10x²−4, giving the final result in terms of u: ∫(-10x+1)e²ˣ−¹⁰ˣ²−⁴dx = ∫(-10x+1)e²ˣ−¹⁰ˣ²−⁴du/2(1−10x).
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Find a vector equation and parametric equations for the line. (Use the parameter t.)
the line through the point (7,0,−4) and parallel to the line x=4−4t, y=−1+2t, z=6+8t
r(t) = ______
(x(t), y(t), z(t)) = _______
A vector equation for the line through the point (7, 0, -4) and parallel to the line x = 4 - 4t, y = -1 + 2t, z = 6 + 8t is r(t) = (7, 0, -4) + t(-4, 2, 8). Parametric equations for the line are: x(t) = 7 - 4t, y(t) = 2t, z(t) = -4 + 8t
To find the vector equation and parametric equations for the line, we need a point on the line and a vector parallel to the line.
Given that the line is parallel to the line x = 4 - 4t, y = -1 + 2t, z = 6 + 8t, we can observe that the direction vector of the line is (-4, 2, 8). This vector represents the change in x, y, and z as the parameter t changes.
Since we are given a point (7, 0, -4) on the line, we can use it to determine the position vector of any point on the line. Therefore, the vector equation for the line is r(t) = (7, 0, -4) + t(-4, 2, 8), where t is the parameter.
To obtain the parametric equations, we separate the vector equation into its components:
x(t) = 7 - 4t
y(t) = 2t
z(t) = -4 + 8t
These equations represent the coordinates of a point on the line as t varies. By plugging in different values of t, we can obtain different points on the line.
Overall, the vector equation and parametric equations allow us to describe the line through the given point and parallel to the given line using the parameter t, enabling us to express any point on the line as a function of t.
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These tables represent a quadratic function with a vertex at (0, -1). What is
the average rate of change for the interval from x = 9 to x = 10?
A. -82
B. -2
C. -101
D. -19
X
0
1
2345
6
y
-1
-2
-5
-10
-17
-26
-37
Interval
0
to 1
1 to 2
2 to 3
3 to 4
4 to 5
5 to 6
Average rate
of change
-1
-3
-5
-7
-9
-11
1-2
J-2
J-2
3-2
1-2
The average rate of change for the interval from x = 9 to x = 10 is -19
How to determine the average rate of change for the intervalFrom the question, we have the following parameters that can be used in our computation:
The table of values
From the table of values, we have
Rate from 5 to 6 = -11
Also, we have
Common difference = -2
This means that
Rate from 8 to 9 = -11 - 2 * 2 * 2
Evaluate
Rate from 8 to 9 = -19
Hence, the rate is -19
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The partial fraction decomposition of (x^2+20/x^3+20)/(x^3+2x^2)
can be written in the form of f(x)/x + g(x)/x^2 + h(x)/x+2,
where
f(x)=
g(x)=
h(x)=
The partial fraction decomposition of (x^2 + 20) / (x^3 + 2x^2) can be written in the form of f(x)/x + g(x)/x^2 + h(x)/(x + 2), where f(x), g(x), and h(x) are yet to be determined.
f(x) =
g(x) =
h(x) =
To find the values of f(x), g(x), and h(x), we need to decompose the given rational function into partial fractions.
We start by factoring the denominator: x^3 + 2x^2 = x^2(x + 2).
The partial fraction decomposition will have three terms corresponding to the factors in the denominator: f(x)/x + g(x)/x^2 + h(x)/(x + 2).
To find the values of f(x), g(x), and h(x), we clear the denominators by multiplying both sides of the equation by x^2(x + 2):
(x^2 + 20) = f(x)(x + 2) + g(x)x(x + 2) + h(x)x^2.
Expanding and simplifying, we have:
x^2 + 20 = f(x)(x + 2) + g(x)(x^2 + 2x) + h(x)x^2.
Now, we equate the coefficients of the like terms on both sides to determine the values of f(x), g(x), and h(x).
For the constant term: 20 = 2f(x).
For the x term: 0 = g(x) + 2h(x).
For the x^2 term: 1 = f(x) + g(x).
Solving this system of equations, we find:
f(x) = 10,
g(x) = 1 - f(x) = -9,
h(x) = (0 - g(x)) / 2 = 9/2.
Therefore, the partial fraction decomposition of (x^2 + 20) / (x^3 + 2x^2) can be written as:
(x^2 + 20) / (x^3 + 2x^2) = 10/x - 9/x^2 + (9/2)/(x + 2).
Hence, f(x) = 10, g(x) = -9, and h(x) = 9/2.
Learn more about partial fraction :
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