The most general antiderivative of f(x) = x^5 - x^3 + 6x is (1/6)x^6 - (1/4)x^4 + 3/2x^2 + C. The antiderivative of f(x) = 5x + 3cos(x) is (5/2)x^2 + 3sin(x) + C. The antiderivative of f(x) = 2ex - 9cosh(x) is 2ex - 9sinh(x) + C. The antiderivative of g(t) = 7 + t + t^2 is 7t + (1/2)t^2 + (1/3)t^3 + C.
The most general antiderivative of the function f(x) = x^5 - x^3 + 6x is F(x) = (1/6)x^6 - (1/4)x^4 + 3/2x^2 + C, where C is the constant of integration. To verify this antiderivative, we can differentiate F(x) and check if it equals f(x). Differentiating F(x) gives us f(x) = 6x^5 - 4x^3 + 3x, which matches the original function, confirming that F(x) is indeed the antiderivative of f(x). The most general antiderivative of the function f(x) = 5x + 3cos(x) is F(x) = (5/2)x^2 + 3sin(x) + C, where C is the constant of integration. To check if F(x) is the correct antiderivative, we can differentiate it and see if it matches the original function.
Differentiating F(x) gives us f(x) = 5x + 3cos(x), which is the same as the original function, confirming that F(x) is the antiderivative of f(x). The most general antiderivative of the function f(x) = 2ex - 9cosh(x) is F(x) = 2ex - 9sinh(x) + C, where C is the constant of integration. To verify this antiderivative, we can differentiate F(x) and see if it equals f(x). Differentiating F(x) gives us f(x) = 2ex - 9cosh(x), which matches the original function, confirming that F(x) is the antiderivative of f(x). The most general antiderivative of the function g(t) = 7 + t + t^2 is G(t) = 7t + (1/2)t^2 + (1/3)t^3 + C, where C is the constant of integration. We can check if G(t) is the correct antiderivative by differentiating it and verifying if it matches the original function. Differentiating G(t) gives us g(t) = 7 + t + t^2, which is the same as the original function, confirming that G(t) is the antiderivative of g(t).
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Solve the following functions for F(x): 4, -3, -2.7, -4.9 (show all your work) F(x)=2x2+4x F(x)= v=x+ 2 2 x+1 2. Solve the following function for f(x): P, R. (m+3) (show all your work) F(x) = 3x+5"
the following functions for F(x): 4, -3, -2.7, -4.9 (show all your work) F(x)=2x2+4x F(x)= v=x+ 2 2 x+1 2
F(x) = 3x + 5 a) For x = P:
F(P) = 3P + 5 .
To solve the given function for F(x), let's substitute the given values and evaluate the expressions step by step:
F(x) = 2x² + 4x a) For x = 4:
F(4) = 2(4)² + 4(4) = 2(16) + 16
= 32 + 16 = 48
b) For x = -3:
F(-3) = 2(-3)² + 4(-3) = 2(9) - 12
= 18 - 12 = 6
c) For x = -2.7:
F(-2.7) = 2(-2.7)² + 4(-2.7) = 2(7.29) - 10.8
= 14.58 - 10.8 = 3.78
d) For x = -4.9:
F(-4.9) = 2(-4.9)² + 4(-4.9) = 2(24.01) - 19.6
= 48.02 - 19.6
= 28.42
F(x) = √(x + 2) / (2x + 1) a) For x = 4:
F(4) = √(4 + 2) / (2(4) + 1) = √6 / (8 + 1)
= √6 / 9
b) For x = -3: F(-3) = √(-3 + 2) / (2(-3) + 1)
= √(-1) / (-6 + 1) = √(-1) / (-5)
c) For x = -2.7:
F(-2.7) = √(-2.7 + 2) / (2(-2.7) + 1)
= √(-0.7) / (-5.4 + 1) = √(-0.7) / (-4.4)
d) For x = -4.9:
F(-4.9) = √(-4.9 + 2) / (2(-4.9) + 1) = √(-2.9) / (-9.8 + 1)
= √(-2.9) / (-8.8)
b) For x = R: F(R) = 3R + 5
Please note that the given function F(x) = 3x + 5 does not involve the variable 'm,' so there is no need to solve for f(x) in this case.
there is no need to solve for f(x) in this case.
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how many ways are there to distribute six objects to five boxes if a) both the objects and boxes are labeled? b) the objects are labeled, but the boxes are unlabeled? c) the objects are unlabeled, but the boxes are labeled? d) both the objects and the boxes are unlabeled?
a) For labeled objects and boxes, there are 5⁶ = 15,625 possible distributions. b) For labeled objects and unlabeled boxes, there are 792 possible distributions. c) For unlabeled objects and labeled boxes, there are 5C6 = 5 possible distributions.d) There is only 1 possible distribution.
a) When both the objects and boxes are labeled, each object can be placed in any of the five labeled boxes, giving us 5 choices for each object. Since there are six objects in total, the total number of distributions is 5⁶ = 15,625.
b) When the objects are labeled but the boxes are unlabeled, we can use a technique called stars and bars. We have 6 objects (stars) and 5 boxes (bars). The objects can be distributed by placing the bars between the objects, so there are (6 + 5 - 1) choose (5 - 1) = 792 possible distributions.
c) When the objects are unlabeled but the boxes are labeled, we have 5 boxes, and we need to choose 6 objects to fill them. This can be thought of as choosing a subset of 6 objects out of 5, which can be done in 5C6 = 5 ways.
d) When both the objects and the boxes are unlabeled, there is only one possible distribution. Since the objects and boxes are indistinguishable, it does not matter which object goes into which box, resulting in a single distribution.
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A small amount of the trace element selenium, 50–200 micrograms (μg) per day, is considered essential to good health. Suppose that random samples of
n1 = n2 = 40 adults
were selected from two regions of Canada and that a day's intake of selenium, from both liquids and solids, was recorded for each person. The mean and standard deviation of the selenium daily intakes for the 40 adults from region 1 were
x1 = 167.8
and
s1 = 24.5 μg,
respectively. The corresponding statistics for the 40 adults from region 2 were
x2 = 140.9
and
s2 = 17.3 μg.
Find a 95% confidence interval for the difference
(μ1 − μ2)
in the mean selenium intakes for the two regions. (Round your answers to three decimal places.)
μg to μg
Interpret this interval.
In repeated sampling, 5% of all intervals constructed in this manner will enclose the difference in population means.There is a 95% chance that the difference between individual sample means will fall within the interval. 95% of all differences will fall within the interval.In repeated sampling, 95% of all intervals constructed in this manner will enclose the difference in population means.There is a 5% chance that the difference between individual sample means will fall within the interval.
We have come to find that confidence interval is (16.802, 37.998) μg
What is Micrograms?Micrograms: This is a unit for measuring the weight of an object. It is equal to one millionth of a gram.
To find a 95% confidence interval for the difference in mean selenium intakes between the two regions, we can use the following formula:
Confidence interval = (x1 - x2) ± t * SE
where:
x1 and x2 are the sample means for region 1 and region 2, respectively.
t is the critical value from the t-distribution for a 95% confidence level.
SE is the standard error of the difference, calculated as follows:
[tex]\rm SE = \sqrt{((s_1^2 / n_1) + (s_2^2 / n2))[/tex]
Let's calculate the confidence interval using the given values:
x₁ = 167.8
s₁ = 24.5 μg
n₁ = 40
x₂ = 140.9
s₂ = 17.3 μg
n₂ = 40
SE = √((24.5² / 40) + (17.3² / 40))
SE ≈ 4.982
Now, we need to determine the critical value from the t-distribution. Since both sample sizes are 40, we can assume that the degrees of freedom are approximately 40 - 1 = 39. Consulting a t-table or using a statistical software, the critical value for a 95% confidence level with 39 degrees of freedom is approximately 2.024.
Substituting the values into the confidence interval formula:
Confidence interval = (167.8 - 140.9) ± 2.024 * 4.982
Confidence interval = 26.9 ± 10.098
Rounded to three decimal places:
Confidence interval ≈ (16.802, 37.998) μg
Interpretation:
We are 95% confident that the true difference in mean selenium intakes between the two regions falls within the interval of 16.802 μg to 37.998 μg. This means that, on average, region 1 has a higher selenium intake than region 2 by at least 16.802 μg and up to 37.998 μg.
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0 The equation of the plane through the points -0 0-0 and can be written in the form Ax+By+Cz=1 2 doon What are A 220 B B 回回, and C=
The equation of the plane passing through the points (-0, 0, -0) and (1, 2) can be written in the form Ax + By + Cz = D, where A = 0, B = -1, C = 2, and D = -2.
To find the equation of a plane passing through two given points, we can use the point-normal form of the equation, which is given by:
Ax + By + Cz = D
We need to determine the values of A, B, C, and D. Let's first find the normal vector to the plane by taking the cross product of two vectors formed by the given points.
Vector AB = (1-0, 2-0, 0-(-0)) = (1, 2, 0)
Since the plane is perpendicular to the normal vector, we can use it to determine the values of A, B, and C. Let's normalize the normal vector:
||AB|| = sqrt(1^2 + 2^2 + 0^2) = sqrt(5)
Normal vector N = (1/sqrt(5), 2/sqrt(5), 0)
Comparing the coefficients of the normal vector with the equation form, we have A = 1/sqrt(5), B = 2/sqrt(5), and C = 0. However, we can multiply the equation by any non-zero constant without changing the plane itself. So, to simplify the equation, we can multiply all the coefficients by sqrt(5):
A = 1, B = 2, and C = 0.
Now, we need to determine D. We can substitute the coordinates of one of the given points into the equation:
11 + 22 + 0*D = D
5 = D
Therefore, D = 5. The final equation of the plane passing through the given points is:
x + 2y = 5
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The complete question is:
A Plane Passes Through The Points (-0,0,-0), And (1,2). Find An Equation For The Plane.
"Let u=
−2
12
4
and A=
4
−2
−3
5
1
1
. Is u in the plane in
ℝ3
spanned by the columns of A? Why or why not?
The answer is that u does not lie in the plane in [tex]$\mathbb{R}^3$[/tex] spanned by the columns of A.
Given that
[tex]$u = \begin{bmatrix} -2 \\ 12 \\ 4 \end{bmatrix}$ and $A = \begin{bmatrix} 4 & -2 & -3 \\ 5 & 1 & 1 \end{bmatrix}$[/tex].
We are required to determine whether $u$ lies in the plane in $\mathbb{R}^3$ spanned by the columns of $A$ or not.
A plane in [tex]$\mathbb{R}^3$[/tex] is formed by three non-collinear vectors. In this case, we can obtain two linearly independent vectors from the matrix A and then find a third non-collinear vector by taking the cross product of the two linearly independent vectors.
The resulting vector would then span the plane formed by the other two vectors.
Therefore,[tex]$$A = \begin{bmatrix} 4 & -2 & -3 \\ 5 & 1 & 1 \\ 0 & 0 & 0 \end{bmatrix}$$[/tex]
If we perform Gaussian elimination on A, we obtain
[tex]$$\begin{bmatrix} 1 & 0 & 1/2 \\ 0 & 1 & -7/3 \\ 0 & 0 & 0 \end{bmatrix}$$[/tex]
The matrix has rank 2, which means the columns of A are linearly independent. Therefore, A spans a plane in [tex]$\mathbb{R}^3$[/tex] .
We can now take the cross product of the two vectors [tex]$\begin{bmatrix} 4 \\ 5 \\ 0 \end{bmatrix}$ and $\begin{bmatrix} -2 \\ 1 \\ 0 \end{bmatrix}$[/tex] that form the plane. Doing this, we obtain
[tex]$$\begin{bmatrix} 0 \\ 0 \\ 13 \end{bmatrix}$$[/tex]
This vector is orthogonal to the plane. Therefore, if u lies in the plane in [tex]$\mathbb{R}^3$[/tex] spanned by the columns of A, then u must be orthogonal to this vector. But we can see that [tex]$\begin{bmatrix} -2 \\ 12 \\ 4 \end{bmatrix}$ is not orthogonal to $\begin{bmatrix} 0 \\ 0 \\ 13 \end{bmatrix}$[/tex].
Therefore, u does not lie in the plane in [tex]$\mathbb{R}^3$[/tex] spanned by the columns of A.Hence, the answer is that u does not lie in the plane in [tex]$\mathbb{R}^3$[/tex] spanned by the columns of A.
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A stock analyst plots the price per share of a certain common stock as a function of time and finds that it can be approximated by the function S(t) = 42+ 18 e -0.06t, where t is the time (in years) s
The given function is S(t) = 42 + 18e^(-0.06t), where S(t) represents the price per share of a common stock as a function of time t in years.
To determine the price per share at different times, we can substitute specific values of t into the function.
a) To find the price per share after 5 years, we substitute t = 5 into the function:
S(5) = 42 + 18e^(-0.06(5))
S(5) = 42 + 18e^(-0.3)
Calculating this value will give you the price per share after 5 years.
b) To find the time when the price per share reaches $60, we set S(t) = 60 and solve for t:
60 = 42 + 18e^(-0.06t)
18e^(-0.06t) = 18
e^(-0.06t) = 1
Taking the natural logarithm of both sides, we have:
-0.06t = ln(1)
Since ln(1) = 0, we get:
-0.06t = 0
Solving for t will give you the time when the price per share reaches $60.
c) To find the maximum price per share, we can determine the value of t that maximizes the function S(t). This can be done by taking the derivative of S(t) with respect to t and setting it equal to 0:
dS(t)/dt = -0.06 * 18e^(-0.06t) = 0
Solving this equation will give you the value of t at which the maximum price per share occurs.
By evaluating the above calculations, you can determine the specific values requested based on the given function.
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Use cylindrical coordinates to evaluate W₁² xyz dv E where E is the solid in the first octant that lies under the paraboloid z = = 4-x² - y².
Evaluating the integral [tex]W_{1} ^{2}[/tex] xyz dv over the solid E in the first octant, which lies under the paraboloid [tex]z=4-x^{2} -y^{2}[/tex]. The integral can be expressed as an iterated integral in cylindrical coordinates.
In cylindrical coordinates, we express a point in three-dimensional space using the variables ([tex]p[/tex], θ, z). Here, [tex]p[/tex] represents the radial distance from the z-axis, θ is the azimuthal angle in the xy-plane, and z is the height.
To evaluate the given integral, we first need to determine the bounds for each variable in the cylindrical coordinate system.
The solid E lies in the first octant, which means [tex]p[/tex], θ, and z are all non-negative. The paraboloid [tex]z=4-x^{2} -y^{2}[/tex] can be expressed in cylindrical coordinates as [tex]z=4-p^{2}[/tex].
To find the bounds for [tex]p[/tex], we set z = 0 and solve for [tex]p[/tex]:
0 = 4 - [tex]p^{2}[/tex]
[tex]p^{2}[/tex] = 4
[tex]p[/tex] = 2
Since we are in the first octant, the bounds for θ are 0 to [tex]\frac{\pi }{2}[/tex].
For z, since the solid lies under the paraboloid, the bounds are 0 to [tex]4-[/tex][tex]p^{2}[/tex].
Now we can set up the iterated integral:
[tex]W_{1}^{2}[/tex] xyz dv = ∫∫∫E [tex]W_{1} ^{2}[/tex] xyz dV
∫[0, [tex]\frac{\pi }{2}[/tex]] ∫[0, 2] ∫[0, 4 - [tex]p^{2}[/tex]] W₁² ([tex]p[/tex] cosθ)([tex]p[/tex] sinθ)[tex]p[/tex] dz d[tex]p[/tex] dθ
Simplifying the integral, we have:
∫[0, [tex]\frac{\pi }{2}[/tex]] ∫[0, 2] ∫[0, 4 - [tex]p^{2}[/tex]] [tex]p^{3}[/tex] cosθ sinθ (4 - [tex]p^{2}[/tex]) dz d[tex]p[/tex] dθ
Evaluating this iterated integral will give the desired result.
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Aladder of length 6m rest against a Vertical wall and makes an angle 9 60°- with the ground. How far is the foot of the ladder from the wall?
The distance of the ladder to the foot of the war is 3 metres.
How to find the distance of the foot of the ladder to the wall?The ladder of length 6m rest against a vertical wall and makes an angle 60 degrees with the ground.
Therefore, the distance of the ladder from the foot of the wall can be calculated as follows:
Hence, using trigonometric ratios,
cos 60 = adjacent / hypotenuse
Therefore,
cos 60 = a / 6
cross multiply
a = 6 cos 60
a = 6 × 0.5
a = 3 metres
Therefore,
distance of the ladder to the foot of the war = 3 metres.
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I need help with this rq
a. The estimated probability of the spinner landing on orange is 0.42.
b. The best prediction for the number of times the arrow is expected to land on the orange section if it is spun 200 times is 84 times.
How to calculate the valuea. The estimated probability of the spinner landing on orange is:
= 168 / (49 + 168 + 183)
= 0.42.
Part B: The best prediction for the number of times the arrow is expected to land on the orange section if it is spun 200 times is:
= 200 * 0.42
= 84 times.
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Sketch the graph of the function y = 3 sin (2x+1). State the amplitude, the period, the phase shift (if any), and the vertical shift (if any). If there is no phase shift of there is no vertical shift, state none.
To sketch the graph of the function y = 3 sin(2x+1), we can analyze its components:
Amplitude:The amplitude of the function is the coefficient in front of the sine function.
this case, the amplitude is 3.
Period:
The period of the sine function is determined by the coefficient in front of the x. In this case, the coefficient is 2, so the period is given by 2π/2 = π.
Phase Shift:The phase shift of the function is determined by the constant inside the sine function. In this case, the constant is 1. To find the phase shift, we set the argument of the sine function equal to zero and solve for x:
2x + 1 = 0
2x = -1x = -1/2
So, the phase shift is -1/2.
Vertical Shift:
The vertical shift is determined by the constant term outside the sine function. In this case, there is no constant term, so there is no vertical shift.
Now, let's plot the graph based on these characteristics:- The amplitude is 3, which means the graph oscillates between -3 and 3.
- The period is π, so one full cycle of the graph occurs from x = 0 to x = π.- The phase shift is -1/2, which means the graph is shifted horizontally by -1/2 units.
- There is no vertical shift, so the graph passes through the origin (0, 0).
Based on these characteristics, we can sketch the graph of y = 3 sin(2x+1) as follows:
| 3 / \
/ \
0 / \ | |
-3 |------------|--------|--------------|--------| -π/2 0 π/2 π 3π/2
In summary:
- The amplitude is 3.- The period is π.
- There is a phase shift of -1/2.- There is no vertical shift.
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Use Green's Theorem to evaluate oint_c xy^2 dx + x^5 dy', where 'C' is the rectangle with vertices (0,0), (3,0), (3,5), and (0,5)
Find and classify the critical points of z=(x^2 - 4 x)(y^2 - 5 y) Lo
To evaluate the line integral using Green's Theorem, we need to find the curl of the vector field and then calculate the double integral over the region enclosed by the curve. Answer : the critical points of the function z = (x^2 - 4x)(y^2 - 5y) are (x, y) = (0, 0) and (x, y) = (0, 4)
Given the vector field F = (xy^2, x^5), we can find its curl as follows:
∇ × F = (∂Q/∂x - ∂P/∂y)
where P is the x-component of F (xy^2) and Q is the y-component of F (x^5).
∂Q/∂x = ∂/∂x (x^5) = 5x^4
∂P/∂y = ∂/∂y (xy^2) = 2xy
Therefore, the curl of F is:
∇ × F = (2xy - 5x^4)
Now, we can apply Green's Theorem:
∮C P dx + Q dy = ∬D (∇ × F) dA
where D is the region enclosed by the curve C.
In this case, C is the rectangle with vertices (0,0), (3,0), (3,5), and (0,5), and D is the region enclosed by this rectangle.
The line integral becomes:
∮C xy^2 dx + x^5 dy = ∬D (2xy - 5x^4) dA
To evaluate the double integral, we integrate with respect to x first and then with respect to y:
∬D (2xy - 5x^4) dA = ∫[0,5] ∫[0,3] (2xy - 5x^4) dx dy
Now, we can calculate the integral using these limits of integration and the given expression.
As for the second part of your question, to find the critical points of the function z = (x^2 - 4x)(y^2 - 5y), we need to find the points where the partial derivatives with respect to x and y are both zero.
Let's calculate these partial derivatives:
∂z/∂x = 2x(y^2 - 5y) - 4(y^2 - 5y)
= 2xy^2 - 10xy - 4y^2 + 20y
∂z/∂y = (x^2 - 4x)(2y - 5) - 5(x^2 - 4x)
= 2xy^2 - 10xy - 4y^2 + 20y
Setting both partial derivatives equal to zero:
2xy^2 - 10xy - 4y^2 + 20y = 0
Simplifying:
2y(xy - 5x - 2y + 10) = 0
This equation gives us two cases:
1) 2y = 0, which implies y = 0.
2) xy - 5x - 2y + 10 = 0
From the second equation, we can solve for x in terms of y:
x = (2y - 10)/(y - 1)
Now, substitute this expression for x back into the first equation:
2y(2y - 10)/(y - 1) - 10(2y - 10)/(y - 1) - 4y^2 + 20y = 0
Simplifying and combining like terms:
4y^3 - 32y^2 + 64y = 0
Factoring out 4y:
4y(y^2 - 8y +
16) = 0
Simplifying:
4y(y - 4)^2 = 0
This equation gives us two cases:
1) 4y = 0, which implies y = 0.
2) (y - 4)^2 = 0, which implies y = 4.
So, the critical points of the function z = (x^2 - 4x)(y^2 - 5y) are (x, y) = (0, 0) and (x, y) = (0, 4).
To classify these critical points, we can use the second partial derivative test or examine the behavior of the function in the vicinity of these points.
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What is 120% as a fraction?
Answer:
1 1/5
Step-by-step explanation:
Answer:
6/5 or [tex]1\frac{1}{5}[/tex]
Step-by-step explanation:
120% = 1.2 in decimal
1.2 = 120/100 in fraction
we can simplify by dividing by 20 so 6/5
400 students attend Ridgewood Junior High School. 5% of stuc bring their lunch to school everyday. How many students brou lunch to school on Thursday?
Answer:
20 students brought their lunch on Thursday.
Step-by-step explanation:
5% of 400 = 20 students
400 x .05 = 20
the number of typing errors per article typed typists follows a poisson distribution. a certain typing agency employs 2 typists. the average number of errors per article is 3 when typed by the first typist and 4.2 when typed by the second. if your article is equally likely to be typed by either typist, approximate the probability that it will have no errors.
The probability that the article will have no errors when typed by either typist is 0.03235, or about 3.24%.
To approximate the probability that an article typed by either typist will have no errors, we can use the concept of a mixed Poisson distribution.
Since the article is equally likely to be typed by either typist, we can consider the combined distribution of the two typists.
Let's denote X as the random variable representing the number of errors per article. The average number of errors per article when typed by the first typist (λ₁) is 3, and when typed by the second typist (λ₂) is 4.2.
For a Poisson distribution, the probability mass function (PMF) is given by:
P(X = k) = (e^(-λ) * λ^k) / k!
To calculate the probability of no errors (k = 0) in the mixed Poisson distribution, we can calculate the weighted average of the two Poisson distributions:
P(X = 0) = (1/2) * P₁(X = 0) + (1/2) * P₂(X = 0)
Where P₁(X = 0) is the probability of no errors when typed by the first typist (λ₁ = 3), and P₂(X = 0) is the probability of no errors when typed by the second typist (λ₂ = 4.2).
Using the PMF formula, we can calculate the probabilities:
P₁(X = 0) = (e^(-3) * 3^0) / 0! = e^(-3) ≈ 0.0498
P₂(X = 0) = (e^(-4.2) * 4.2^0) / 0! = e^(-4.2) ≈ 0.0149
Substituting these values into the weighted average formula:
P(X = 0) = (1/2) * 0.0498 + (1/2) * 0.0149
= 0.03235
Approximately, the probability that the article will have no errors when typed by either typist is 0.03235, or about 3.24%.
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9. Use an appropriate local linear approximation to estimate the value of √10. Recall that f'(a) [f(a+h)-f(a)] + h when h is very small. 10. A boat is pulled into a dock by means of a rope attached to a pulley on the dock. The rope is attached to the front of the boat, which is 7 feet below the level of the pulley. If the boat is approaching the dock at a rate of 18 ft/min, at what rate is the rope being pulled in when the boat is125 ft from the dock.
When the boat is 125 ft from the dock, the rope is being pulled in at a rate of approximately 178.57 ft/min.
How to estimate the value of √10?To estimate the value of √10 using local linear approximation, we can use the formula:
f(a + h) ≈ f(a) + f'(a) * h
where f(a) is the function value at a, f'(a) is the derivative of the function at a, and h is a small increment.
In this case, let's approximate √10 by choosing a = 9, which is close to 10. Taking the derivative of the function f(x) = √x with respect to x, we have:
f'(x) = 1 / (2√x)
Now, we can plug in a = 9, f(a) = √9 = 3, and h = 1:
√10 ≈ 3 + (1 / (2√9)) * 1
Simplifying the expression:
√10 ≈ 3 + (1 / (2 * 3)) * 1
≈ 3 + (1 / 6)
≈ 3 + 1/6
≈ 3 + 0.16667
≈ 3.16667
Therefore, using local linear approximation, we estimate that √10 is approximately 3.16667.
Moving on to the second part of the question regarding the rate at which the rope is being pulled in when the boat is 125 ft from the dock:
Let's denote the distance between the boat and the dock as x (in feet), and the rate at which the boat is approaching the dock as dx/dt = 18 ft/min. We want to find the rate at which the rope is being pulled in, which is dH/dt, where H represents the length of the rope.
Using the Pythagorean theorem, we have:
[tex]x^2 + (H - 7)^2 = H^2[/tex]
Simplifying the equation, we get:
[tex]x^2 + H^2 - 14H + 49 = H^2[/tex]
[tex]x^2 - 14H + 49 = 0[/tex]
Differentiating both sides of the equation with respect to time (t), we obtain:
2x * (dx/dt) - 14(dH/dt) = 0
Substituting x = 125 ft and dx/dt = 18 ft/min, we can solve for dH/dt:
2(125)(18) - 14(dH/dt) = 0
2500 - 14(dH/dt) = 0
14(dH/dt) = 2500
dH/dt = 2500/14
Simplifying the expression, we find:
dH/dt ≈ 178.57 ft/min
Therefore, when the boat is 125 ft from the dock, the rope is being pulled in at a rate of approximately 178.57 ft/min.
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Please box answers
Find each function value and limit. Use - or where appropriate 3x4 - 6x? f(x) = 12x + 6 (A) f(-6) (8) f(-12) (C) limf(0) 00 (A) f(- 6) = 0 (Round to the nearest thousandth as needed.) (B) f(- 12) = (R
Each function's value and limit is as:
(A) [tex]f(-6) = -66[/tex]
(B) [tex]f(8) = 102[/tex]
(C) [tex]f(-12) = -138[/tex]
(D) [tex]lim (x- > 0) (12x + 6) = 6[/tex]
What is a function value?
A function value refers to the output or result obtained when a specific input, known as the independent variable, is substituted into a function. In other words, it represents the value of the dependent variable corresponding to a given input.
In a mathematical function, the function value is determined by applying the input value to the function equation or expression and calculating the result. This allows us to associate each input value with a unique output value.
To find the function values and limit, let's substitute the given values into the function and evaluate them:
(A) f(-6):
Substituting x = -6 into the function
[tex]f(x) = 12x + 6:\\\\f(-6) = 12*(-6) + 6\\f(-6) = -72 + 6\\f(-6) = -66[/tex]
(B) f(8):
Substituting x = 8 into the function
[tex]f(x) = 12x + 6:\\f(8) = 12*8 + 6\\f(8) = 96 + 6\\f(8) = 102[/tex]
(C) f(-12):
Substituting x = -12 into the function
[tex]f(x) = 12x + 6:\\f(-12) = 12*(-12) + 6\\f(-12) = -144 + 6\\f(-12) = -138[/tex]
(D) lim f(x) as x approaches 0:
Taking the limit of [tex]f(x) = 12x + 6[/tex] as x approaches 0:
[tex]lim (x- > 0) (12x + 6) = 12(0) + 6\\\lim (x- > 0) (12x + 6) = 0 + 6\\lim (x- > 0) (12x + 6) = 6[/tex]
Therefore, the results are:
(A)[tex]f(-6) = -66[/tex]
(B) [tex]f(8) = 102[/tex]
(C)[tex]f(-12) = -138[/tex]
(D) [tex]lim (x- > 0) (12x + 6) = 6[/tex]
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What is 16/7+86. 8 and whoever answer's first, I will mark them the brainliest
Answer:
3118/35 or 89.0857142
Step-by-step explanation:
convert 86.8 to fraction form which is 86 4/5 or 434/5 and add 16/7 by making the denominator same.
(1 point) Use integration by parts to evaluate the definite integral l'te . te-' dt. Answer:
The result of the definite integral ∫ₗₜₑ t * e^(-t) dt obtained using integration by parts is: -te^(-t) - e^(-t) + C, where C is the constant of integration.
To evaluate the definite integral ∫ₗₜₑ t * e^(-t) dt using integration by parts, we apply the formula:
∫ u dv = uv - ∫ v du,
where u and v are functions of t. In this case, we choose u = t and dv = e^(-t) dt. Therefore, du = dt and v can be obtained by integrating dv. Integrating dv gives us v = -e^(-t).
Using the integration by parts formula, we have:
∫ₗₜₑ t * e^(-t) dt = -te^(-t) - ∫ₗₜₑ (-e^(-t)) dt.
Simplifying the integral on the right side, we get:
∫ₗₜₑ t * e^(-t) dt = -te^(-t) + e^(-t) + C,
where C is the constant of integration. This is the final result obtained using integration by parts.
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\frac{3m}{2m-5}-\frac{7}{3m+1}=\frac{3}{2}
[tex] \sf \longrightarrow \: \frac{3m}{2m-5}-\frac{7}{3m+1}=\frac{3}{2} \\ [/tex]
[tex] \sf \longrightarrow \: \frac{3m(3m + 1) - 7(2m-5)}{(2m-5)(3m+1)}=\frac{3}{2} \\ [/tex]
[tex] \sf \longrightarrow \: \frac{9 {m}^{2} + 3m \: - 14m + 35}{(2m-5)(3m+1)}=\frac{3}{2} \\ [/tex]
[tex] \sf \longrightarrow \: \frac{9 {m}^{2} + 3m \: - 14m + 35}{6 {m}^{2} + 2m - 15m - 5 }=\frac{3}{2} \\ [/tex]
[tex] \sf \longrightarrow \: 2(9 {m}^{2} + 3m \: - 14m + 35) = 3(6 {m}^{2} + 2m - 15m - 5 )\\ [/tex]
[tex] \sf \longrightarrow \: 18 {m}^{2} + 6m - 28m + 70 \: = 3(6 {m}^{2} + 2m - 15m - 5 )\\ [/tex]
[tex] \sf \longrightarrow \: 18 {m}^{2} + 6m - 28m + 70 \: =18 {m}^{2} + 6m - 45m - 15 \\ [/tex]
[tex] \sf \longrightarrow \: 18 {m}^{2} + 6m - 28m + 70 \: - 18 {m}^{2} - 6m + 45m + 15 = 0 \\ [/tex]
[tex] \sf \longrightarrow \: \cancel{18 }{m}^{2} + \cancel{ 6m} - 28m + 70 \: - \cancel{18 {m}^{2} } - \cancel{ 6m } + 45m + 15 = 0 \\ [/tex]
[tex] \sf \longrightarrow \: - 28m + 70 \: + 45m + 15 = 0 \\ [/tex]
[tex] \sf \longrightarrow \: 17m + 85 = 0 \\ [/tex]
[tex] \sf \longrightarrow \: 17m = - 85\\ [/tex]
[tex] \sf \longrightarrow \: m = - \frac{ 85}{17}\\ [/tex]
[tex] \sf \longrightarrow \: m = - 5 \\ [/tex]
2. Find the derivative. a) g(t) = (tº - 5)3/2 b) y = x ln(x² +1)
a) The derivative of the function g(t) = (tº - 5)^(3/2) is (3/2)(t^2 - 5)^(1/2) because it follows the chain rule.
b) The derivative of the function y = x ln(x² + 1) is y' = ln(x² + 1) + (2x^2)/(x² + 1).
a) The derivative of a function measures the rate at which the function changes with respect to its independent variable. In the case of g(t) = (tº - 5)^(3/2), we can differentiate it using the chain rule. The chain rule states that if we have a composition of functions, such as (f(g(t)))^n, the derivative is given by n(f(g(t)))^(n-1) * f'(g(t)) * g'(t).
In this case, we have (tº - 5)^(3/2), which can be rewritten as (f(g(t)))^(3/2) with f(u) = u^3/2 and g(t) = t^2 - 5. Taking the derivative of f(u) = u^3/2 gives us f'(u) = (3/2)u^(1/2). The derivative of g(t) = t^2 - 5 is g'(t) = 2t. Applying the chain rule, we multiply these derivatives together and obtain the final result: (3/2)(t^2 - 5)^(1/2).
b) To differentiate the function y = x ln(x² + 1), we apply the product rule, which states that if we have a product of two functions u(x) and v(x), the derivative of the product is given by u'(x)v(x) + u(x)v'(x). In this case, u(x) = x and v(x) = ln(x² + 1).
The derivative of u(x) = x is u'(x) = 1. To find v'(x), we apply the chain rule since v(x) = ln(u(x)) and u(x) = x² + 1. The chain rule states that the derivative of ln(u(x)) is (1/u(x)) * u'(x). In this case, u'(x) = 2x, so v'(x) = (1/(x² + 1)) * 2x.
Applying the product rule, we multiply u'(x)v(x) and u(x)v'(x) together and obtain the derivative of y = x ln(x² + 1): y' = ln(x² + 1) + (2x^2)/(x² + 1).
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Liquid leaked from a damaged tank at a rate of r(t) liters per hour. The rate decreased as time passed and values of the rate at five-hour time intervals are shown in the table. t (hr) r(t) (L/h) 0 10.6 5 9.5 10 8.6 15 7.7 20 6.9 25 6.2 Find lower and upper estimates for the total amount of liquid that leaked out. lower estimate liters upper estimate liters
The total amount of liquid that leaked out is 102.75 liters, and the upper estimate is 108.75 liters.
How to find the lower and upper estimates for the total amount of liquid that leaked out?To find the lower and upper estimates for the total amount of liquid that leaked out, we can use the trapezoidal rule to approximate the integral of the leakage rate over the given time intervals.
t (hr) r(t) (L/h)
0 10.6
5 9.5
10 8.6
15 7.7
20 6.9
25 6.2
Calculate the time intervals and average the rates
To calculate the lower and upper estimates, we divide the given time period into subintervals. Since the intervals are 5 hours, we have 5 subintervals: [0, 5], [5, 10], [10, 15], [15, 20], [20, 25].
For each subinterval, we calculate the average rate using the given values:
Average rate for [0, 5] = (10.6 + 9.5) / 2 = 10.05 L/h
Average rate for [5, 10] = (9.5 + 8.6) / 2 = 9.05 L/h
Average rate for [10, 15] = (8.6 + 7.7) / 2 = 8.15 L/h
Average rate for [15, 20] = (7.7 + 6.9) / 2 = 7.3 L/h
Average rate for [20, 25] = (6.9 + 6.2) / 2 = 6.55 L/h
Calculate the lower and upper estimates using the trapezoidal rule
The lower estimate is obtained by approximating the integral as a sum of areas of trapezoids, where the height of each trapezoid is the average rate and the width is the time interval.
Lower estimate = (5/2) * [(10.05) + (9.05) + (8.15) + (7.3) + (6.55)]
= (5/2) * [41.1]
= 102.75 L
The upper estimate is obtained by using the average rate of the previous interval as the height of the first trapezoid and the average rate of the current interval as the height of the second trapezoid.
Upper estimate = (5/2) * [(10.6) + (9.5) + (8.6) + (7.7) + (6.9)]
= (5/2) * [43.5]
= 108.75 L
Therefore, the lower estimate for the total amount of liquid that leaked out is 102.75 liters, and the upper estimate is 108.75 liters.
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Wite the point-slope form of the line satisfying the given conditions Then use the point-stope form of the equation to write the slope-ntercept form of the equation Passing through (714) and (8.16) Ty
The slope-intercept form of the equation is y = 2x.
To find the point-slope form of a line, we use the formula:
y - y₁ = m(x - x₁),
where (x₁, y₁) represents a point on the line, and m is the slope of the line. Given two points, (7,14) and (8,16), we can calculate the slope (m) using the formula: m = (y₂ - y₁) / (x₂ - x₁),
where (x₂, y₂) represents the second point. Plugging in the values, we get:
m = (16 - 14) / (8 - 7) = 2.
Now we can use the point-slope form with either of the two points. Let's use (7,14):
y - 14 = 2(x - 7).
To convert this to the slope-intercept form (y = mx + b), we simplify:
y - 14 = 2x - 14,
y = 2x.
Therefore, the slope-intercept form of the equation is y = 2x.
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Consider the following 2nd order ODE fory (where the independent variable is t): 2y" + 3y' = 0 1) Find the general solution to the above ODE. 2) Use the initial conditions y(0)-6, y 10)-0 to find the
The solution to the given ODE with the initial conditions y(0) = -6 and y'(0) = 0 is y(t) = -6.
To solve the given second-order ordinary differential equation (ODE) 2y" + 3y' = 0, we can proceed as follows:
Find the general solution to the ODE:
Let's assume y = e^(rt) as a trial solution. Taking the derivatives with respect to t, we have:
y' = re^(rt)
y" = r^2e^(rt)
Substituting these derivatives into the ODE, we get:
2(r^2e^(rt)) + 3(re^(rt)) = 0
Dividing through by e^(rt) (which is nonzero), we have:
2r^2 + 3r = 0
Factoring out r, we get:
r(2r + 3) = 0
So we have two possible solutions for r:
r1 = 0 and r2 = -3/2
The general solution to the ODE is a linear combination of these solutions:
y(t) = C1e^(r1t) + C2e^(r2t)
Substituting the values of r1 and r2, the general solution becomes:
y(t) = C1e^(0t) + C2e^(-3/2t)
y(t) = C1 + C2e^(-3/2t)
Use the initial conditions y(0) = -6 and y'(0) = 0 to find the particular solution:
Given y(0) = -6, we can substitute t = 0 into the general solution:
-6 = C1 + C2e^(0)
-6 = C1 + C2
Given y'(0) = 0, we differentiate the general solution with respect to t and substitute t = 0:
0 = C2(-3/2)e^(-3/2(0))
0 = -3/2C2
C2 = 0
Substituting C2 = 0 back into the first equation, we get:
-6 = C1 + 0
C1 = -6
Therefore, the particular solution to the ODE with the given initial conditions is:
y(t) = -6 + 0e^(-3/2t)
y(t) = -6
So, the solution to the given ODE with the initial conditions y(0) = -6 and y'(0) = 0 is y(t) = -6.
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II. Find the slope of the tan gent line to Vy + y + x = 10 at (1,8). y х III. Find the equation of the tan gent line to x² – 3xy + y2 =-1 at (2,1). -
ii. The slope of the tangent line at (1,8) is -1/2.
iii. The equation of the tangent line to x² - 3xy + y² = -1 at (2,1) is y = (1/3)x + 1/3.
II. To find the slope of the tangent line to the equation Vy + y + x = 10 at the point (1,8), we need to find the derivative of the equation and evaluate it at x = 1 and y = 8.
Differentiating the equation with respect to x, we get:
dy/dx + dy/dx + 1 = 0
Simplifying, we have:
2(dy/dx) = -1
dy/dx = -1/2
Therefore, the slope of the tangent line at (1,8) is -1/2.
III. To find the equation of the tangent line to the equation x² - 3xy + y² = -1 at the point (2,1), we need to find the derivative of the equation and evaluate it at x = 2 and y = 1.
Differentiating the equation with respect to x, we get:
2x - 3y - 3xdy/dx + 2ydy/dx = 0
Rearranging the terms, we have:
(2x - 3y) - 3(dy/dx)(x - y) = 0
At the point (2,1), we substitute x = 2 and y = 1 into the equation:
(2(2) - 3(1)) - 3(dy/dx)(2 - 1) = 0
4 - 3 - 3(dy/dx) = 0
-3(dy/dx) = -1
dy/dx = 1/3
Therefore, the slope of the tangent line at (2,1) is 1/3.
Using the point-slope form of the equation of a line, we can write the equation of the tangent line at (2,1) as:
y - 1 = (1/3)(x - 2)
Simplifying, we have:
y - 1 = (1/3)x - 2/3
y = (1/3)x + 1/3
Therefore, the equation of the tangent line to x² - 3xy + y² = -1 at (2,1) is y = (1/3)x + 1/3.
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Find the Taylor or Maclaurin polynomial P(x) for the function with the given values of cand n. Then give a bound on the error that is incurred if P(x) is used to approximate f(x) on the given interval
The Taylor or Maclaurin polynomial P(x) for a function f(x) is ∣f(x)−P(x)∣≤ M/(n+1)! ∣x−c∣ n+1
What is the polynomial equation?
A polynomial equation is an equation in which the variable is raised to a power, and the coefficients are constants. A polynomial equation can have one or more terms, and the degree of the polynomial is determined by the highest power of the variable in the equation.
To find the Taylor or Maclaurin polynomial P(x) for a function f(x) with a given value of c and degree n, we need to calculate the derivatives of f(x) and evaluate them at x=c.
The Taylor polynomial P(x) is given by the formula:
P(x)=f(c)+f′(c)(x−c)+ 2! f′′(c)(x−c)2 + 3! f′′′(c)(x−c) 3 +⋯+ n! f(n)(c) (x−c)n
To give a bound on the error incurred when using P(x) to approximate f(x) on the given interval, we can use the error formula for Taylor polynomials:
∣f(x)−P(x)∣≤ M/(n+1)! ∣x−c∣ n+1
where, M is an upper bound for the absolute value of the n+1st derivative of f on the interval.
Without specific information about the function f(x), the value of c, and the degree n, it is not possible to determine the exact Taylor or Maclaurin polynomial P(x) or provide a bound on the error.
Hence, the Taylor or Maclaurin polynomial P(x) for a function f(x) is ∣f(x)−P(x)∣≤ M/(n+1)! ∣x−c∣ n+1
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A function is of the form y = sin(kx), where x is in units of radians. If the period of the function
is 70 radians, what is the value of k
The period of a sine function is given by the formula: Period = 2π / |k| where k is the coefficient of x in the function. In this case, we are given that the period is 70 radians.
Plugging this value into the formula, we have: 70 = 2π / |k|
To solve for k, we can rearrange the equation as follows: |k| = 2π / 70
|k| = π / 35
Since k represents the coefficient of x, which determines the rate at which the function oscillates, we are only interested in the positive value of k. Therefore: k = π / 35. So, the value of k is π / 35.
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Monthly sales of a particular personal computer are expected to decline at a rate of S'(t) = -5t e 0.2t computers per month, where t is time in months, and S(t) is the number of computers sold each mo
The number of computers sold each month, S(t), is given by:
S(t) = -125te^(0.2t) + 625e^(0.2t)/0.2 + C.
To determine the number of computers sold each month, we need to integrate the rate of decline function S'(t) = -5te^(0.2t) with respect to t.
Let's integrate S'(t):
[tex]∫S'(t) dt = ∫-5te^(0.2t) dt[/tex]
To solve this integral, we can use integration by parts. The formula for integration by parts is:
[tex]∫u dv = uv - ∫v du[/tex]
Let's assign u and dv:
[tex]u = tdv = -5e^(0.2t) dt[/tex]
Taking the derivatives:
[tex]du = dtv = -∫5e^(0.2t) dt[/tex]
To find v, we can integrate dv:
[tex]v = -∫5e^(0.2t) dtv = -∫5e^(0.2t) dt = -∫5 * (1/0.2)e^(0.2t) dt = -25e^(0.2t)/0.2 + C[/tex]
Now, let's apply the integration by parts formula:
[tex]∫S'(t) dt = -t * (25e^(0.2t)/0.2) + ∫(25e^(0.2t)/0.2) dt[/tex]
Simplifying:
[tex]∫S'(t) dt = -5t * (25e^(0.2t)/0.2) + 125∫e^(0.2t) dt∫S'(t) dt = -125te^(0.2t) + 125(5e^(0.2t))/0.2 + C[/tex]
Combining terms:
[tex]∫S'(t) dt = -125te^(0.2t) + 625e^(0.2t)/0.2 + C[/tex]
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one in every 9 people in a town vote for party a. all others vote for party b. how many people vote for party b in a town of 810?
If one in every 9 people in the town vote for party A, then the remaining 8 out of 9 people would vote for party B. Therefore, we can calculate the number of people who vote for party B by multiplying the total number of people in the town by 8/9.
So, in a town of 810 people, 720 people would vote for party B, while the remaining 90 people would vote for party A.
In a town of 810 people, one in every 9 people votes for party A, and all others vote for party B. To find the number of people voting for party B, first, calculate the number of people voting for party A: 810 / 9 = 90 people. Since the remaining people vote for party B, subtract the number of party A voters from the total population: 810 - 90 = 720 people. or 810 x (8/9) = 720. Therefore, 720 people in the town vote for Party B.
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a random sample of 80 high school students consists of 30 students taking the sat. form a 95% confidence interval for the true proportion of students taking the sat. what is the lower tail of this interval? pick the closest answer.
The lower tail of the 95% confidence interval for the true proportion of high school students taking the SAT depends on the specific values obtained from the sample. Without the sample data, it is not possible to determine the exact lower tail value.
To calculate a confidence interval, the sample proportion and sample size are needed. In this case, the sample proportion of students taking the SAT is 30 out of 80, which is 30/80 = 0.375.
Using this sample proportion, along with the sample size of 80, the confidence interval can be calculated. The lower and upper bounds of the interval depend on the chosen level of confidence (in this case, 95%).
Since the lower tail value is not specified, it cannot be determined without the actual sample data. The lower tail value will be determined by the sample proportion, sample size, and the specific calculations based on the confidence interval formula. Therefore, without the sample data, it is not possible to determine the exact lower tail value.
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The cumulative distribution function of continuous random variable X is given by F(x) = 0, x < 0 23,0 1 (a) Find P (0.1 < X < 0.6). (b) Find f(x), the probability density function of X. (c) Find X0.6, the 60th percentile of the distribution of X.
A. P(0.1 < X < 0.6) = F(0.6) - F(0.1) = 1 - 0.23 = 0.77.
B. the PDF of X is given by:
f(x) = 0 for x < 0
f(x) = 23 for 0 ≤ x < 1
f(x) = 0 for x ≥ 1
C. X0.6, the 60th percentile of the distribution of X, is equal to 1.
How did we get these values?To answer these questions, use the given cumulative distribution function (CDF) and perform the necessary calculations.
(a) To find P(0.1 < X < 0.6), calculate the difference between the CDF values at those points. The CDF is defined as F(x):
P(0.1 < X < 0.6) = F(0.6) - F(0.1)
Since the CDF is given as a piecewise function, evaluate it at the specified points:
F(0.6) = 1
F(0.1) = 0.23
Therefore, P(0.1 < X < 0.6) = F(0.6) - F(0.1) = 1 - 0.23 = 0.77.
(b) To find the probability density function (PDF) f(x), we can differentiate the CDF. The PDF is the derivative of the CDF:
f(x) = d/dx [F(x)]
Differentiating each part of the piecewise CDF function:
For x < 0, f(x) = 0 (since F(x) is constant in this interval).
For 0 ≤ x < 1, f(x) = d/dx [23x] = 23.
For x ≥ 1, f(x) = 0 (since F(x) is constant in this interval).
Therefore, the PDF of X is given by:
f(x) = 0 for x < 0
f(x) = 23 for 0 ≤ x < 1
f(x) = 0 for x ≥ 1
(c) To find X0.6, the 60th percentile of the distribution of X, we need to find the value of x for which F(x) = 0.6. From the given CDF, we know that F(x) = 0.6 for x = 1. So X0.6 = 1.
Therefore, X0.6, the 60th percentile of the distribution of X, is equal to 1.
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