Find an equation of the plane passing through the point (4.3.2) with a normal vector of the form n = (a,b,1) that slices off the region in the first octant with the least volume.
z = ...
Answers
y - (2/3)z = 2 is the equation of the plane passing through (4,3,2) with a normal vector of the form n = (0,-2/3,1) that slices off the region in the first octant with the least volume.
To find the equation of the plane passing through (4,3,2) with a normal vector of the form n = (a,b,1), we can use the point-normal form of the equation of a plane:
(a(x-4) + b(y-3) + 1(z-2) = 0
Simplifying this equation, we get:
ax + by + z = 4a + 3b + 2
To slice off the region in the first octant with the least volume, we want the plane to pass through the origin (0,0,0) and intersect the x, y, and z axes at positive values. This means that the equation of the plane must satisfy the following three conditions:
ax = 0 (plane intersects x-axis at x=0)
by = 0 (plane intersects y-axis at y=0)
z = 0 (plane intersects z-axis at z=0)
From the first two conditions, we can see that either a or b must be equal to 0. Without loss of generality, let's say a=0. Then the equation of the plane simplifies to:
by + z = 3b + 2
To satisfy the third condition, we need to set z=0:
by = 3b + 2
Solving for b, we get:
b = -2/3
Substituting this value of b back into the equation of the plane, we get:
y - (2/3)z = 2
This is the equation of the plane passing through (4,3,2) with a normal vector of the form n = (0,-2/3,1) that slices off the region in the first octant with the least volume.
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Related Questions
Suppose that X and Y are random variables and that X and Y are nonnegative for all points in a sample space S. Let Z be the random variable defined by Z(s) = max(X(s), Y (s)) for all elements s â S. Show that E(Z) ⤠E(X) + E(Y).
Answers
For the Random variables, X and Y where both are nonnegative for all points in a sample space S. The expected value of Z ( random variable) is, E(Z) = E(X) + E(Y).
We have X and Y are random variables and that X and Y are non-negative for all points in a sample space S. Let us consider X be another random variable defined as Z(s) = max( X(s), Y(s))
for all elements s belongs to S. We have to show that E( Z) = E(X) + E(Y)
Now, for simple way of representation let
Max ( X(s),Y(s)) = Max(X,Y)
We know that Max(X,Y) = [(X+Y) + |X-Y|]
So, Z = Max(X,Y) = [(X+Y) + |X-Y|]
Taking expectations E(Z) = E(Max(X,Y)
= E{[(X+Y) + |X-Y|]}
\(E(Z)= \frac{1}{2} [E(X+Y) + E|X-Y|] = [E(X) + E(Y) + E|X-Y|] \\ \) (Since E(X + Y) = E(X)+ E(Y))
\(E(Z) = \frac{1}{2}[E(X)+ E(Y) + E|X| + E|-Y|]\\ \)
\(= \frac{1}{2} [E(X) + E(Y) + E(X) + E(Y)]\) (Since X and Y are non negative so E|X| = E(X) and E(-Y)=E(Y) )
=> E(Z) = E(X)+ E(Y)
Hence, required results occurred.
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Jackson spends 2 1/4 hours painting 2 1/2 pictures. He spends the same amount of time painting each picture
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Is the relation shown in the arrow diagram a function? Explain
A)Yes: Ench input has only one output
B)Yes;Each output has only one input
C)No, Each input has only one output
D)No, The output 32 has two inputs
Answers
Answer: A yes. Each input has only one output.
Step-by-step explanation:
functions are determined by each input only having one output.
Answer:
Answer is A
Step-by-step explanation:
A pair of jeans cost $42.00. If the sales tax is 5%, how much will she pay altogether for the jeans?
Answers
Step-by-step explanation:
44.10 dollars after the sales tax
Answer: $44.10
Step-by-step explanation:
So to start take $42.00 and multiply it by 5% or 0.05
when you do that you will get 2.10 that is the amount you will add to the 42.00 to see how much it will be
Temp (in "C) S -40 30 20 10 A. 10 O C. D. Ch 20 Time (in minutes) The temperature graph of freshly baked cookies after they are removed from the oven is shown in the graph. With the help of the tangent shown in the graph, estimate the rate of change of the temperature of the cookies after 20 minutes. B. -0.33 °C/minute -0.55 °C/minute 0.33 °C/minute 0.55 °C/minute 30 40 X
Answers
From the graph, it looks like the tangent line at 20 minutes has a slope that is close to 0.55°C/minute. So, the estimate of the rate of change of the temperature of the cookies after 20 minutes is approximately 0.55°C/minute.
What is rate of change?With the help of the tangent shown in the graph, it is estimated that the rate of change of the temperature of the cookies after 20 minutes is 0.55 °C/minute.This indicates that the temperature of the cookies is dropping at a rapid rate. After 40 minutes, the temperature has dropped from 40 °C to 30 °C, which shows a decrease of 10 °C in temperature.This rate of decrease in temperature is significant and it ensures that the cookies are cooked properly while also retaining their texture and flavor.This rapid decrease in temperature is a testament to the quality of the freshly baked cookies.This is called a rate of change. It is the rate at which the temperature of the cookies changes over time after being removed from the oven. The graph shows the temperature of the cookies over time in the form of a curve. The tangent to the curve at the point 20 minutes shows the rate of change of the temperature of the cookies.The tangent is sloping downwards, which implies that the temperature of the cookies is decreasing at a rate of 0.55°C per minute. This rate of change is referred to as the negative rate of change since it is a decrease in temperature over time.To learn more about rate of change refer to:
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A concave shaving mirror has a radius of curvature of +31.5 cm. It is positioned so that the (upright) image of a man's face is 3.40 times the size of the face. How far is the mirror from the face? Number i Units
Answers
The data includes a concave mirror with a radius of curvature of +31.5 cm and magnification of m = 3.40. The formula for magnification is m = v/u, and the focal length is f = r/2. Substituting the values, we get u = v/m, and using the mirror formula, the distance of the object from the mirror is 10.15 cm.
Given data: Radius of curvature of a concave mirror, r = +31.5 cm Magnification produced by the mirror, m = 3.40
We know that the formula for magnification is given by:
m = v/u where, v = the distance of the image from the mirror u = the distance of the object from the mirror We also know that the formula for the focal length of the mirror is given by :
f = r/2where,f = focal length of the mirror
Using the mirror formula:1/f = 1/v - 1/u
We know that a concave mirror has a positive focal length, so we can replace f with r/2.
We can now simplify the equation to get:1/(r/2) = 1/v - 1/u2/r = 1/v - 1/u
Also, from the given data, we have :m = v/u
Substituting the value of v/u in terms of m, we get: u/v = 1/m
So, u = v/m Substituting the value of u in terms of v/m in the previous equation, we get:2/r = 1/v - m/v Substituting the given values of r and m in the above equation, we get:2/31.5 = 1/v - 3.4/v Solving for v, we get: v = 22.6 cm Now that we know the distance of the image from the mirror, we can use the mirror formula to find the distance of the object from the mirror.1/f = 1/v - 1/u
Substituting the given values of r and v, we get:1/(31.5/2) = 1/22.6 - 1/u Solving for u, we get :u = 10.15 cm
Therefore, the distance of the mirror from the face is 10.15 cm. The units are centimeters (cm).Answer: 10.15 cm.
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Help me pls idk how to do this
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Divide the number 520 into two parts in the ratio 5:8.
The parts of the number are ? and ?
Answers
Answer:
The parts of the number are 200 and 320
Step-by-step explanation:
System of Equations
Suppose a and b are the two parts. They must meet two conditions:
\(a+b=520\qquad\qquad [1]\)
\(\displaystyle \frac{a}{b}=\frac{5}{8}\qquad\qquad [2]\)
From [1], we have:
\(b=520-a\)
Replacing in [2]:
\(\displaystyle \frac{a}{520-a}=\frac{5}{8}\)
Cross-multiplying:
\(8a=5*(520-a)\)
Operating:
\(8a=2600-5a\)
Joining like terms:
\(13a=2600\)
Solving:
\(a=2600/13=200\)
a=200
Since:
\(b=520-a\Rightarrow b=520-200\)
b=320
The parts of the number are 200 and 320
Answer:
200 and 320
Step-by-step explanation:
A model without any simplifying assumptions a.provides simplified solutions to complex problems. b.is highly complex and likely unworkable. c.excludes important predictive variables. d.is very helpful for solving tough, real-world problems.
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A model without any simplifying assumptions b.is highly complex and likely unworkable.
As their call implies, simplifying assumptions are assumptions that are protected within the version to simplify the evaluation as tons as feasible. While a simplified version now does not predict the conduct of the actual issue within perfect bounds, too many simplifying assumptions have been made.
While a scientific version enables us to make predictions, it's more valued. As clinical models are representations of simplified explanations, they do not are looking for to explain each scenario or every detail. This means that medical models often aren't equal to the 'real international' from which they are derived.
Mathematical models can help college students apprehend and discover the that means of equations or valuable relationships.
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11. Evaluate the following integrals: (a) (x + 2)(2² + 4x + 10)+ dz (b) (c) (d) COS √ [(In cos z) tan z dr t dt
Answers
The given integrals are evaluated using different methods. Finally, the values of these integrals are found.
Given integrals are: ∫(x + 2)(2² + 4x + 10) + dz(b) ∫cos4t sec²4t dt(c) ∫cosx/(1 + sin²x) dx(d) ∫cos √ [(In cos z) tan z] dr t dt
Here, let u = x + 2 so that du = dx;and v = (2² + 4x + 10) + z so that dv = 4dx + dz
Then the given integral is∫uv dv + ∫(v du) z∫(x + 2)(2² + 4x + 10) + dz= 1/2(uv² - ∫v du) z= 1/2(x + 2) [(2² + 4x + 10 + z)²/4 - (2² + 4x + 10)²/4] - 1/2 ∫[(2² + 4x + 10 + z)/2] dx= 1/2(x + 2) [(z²/4) + 2z (4x + 12) + 36x² + 80x + 196] - 1/4 (2² + 4x + 10 + z)² + 1/2 (2² + 4x + 10)z + C
Use substitution, u = 4t + π/2
Then the given integral is∫cosu du = sinu + C= sin(4t + π/2) + C(c)
Here, let u = sinx so that du = cosx dx
Then the given integral is∫du/(1 + u²)= tan⁻¹u + C= tan⁻¹sinx + C
Let u = ln(cosz) so that du = - tanz dz
Then the given integral is∫cos(√u) du= 2∫cos(√u) d√u= 2 sin(√u) + C= 2 sin(√ln(cosz)) + C.
In the first integral, we used substitution method and then we get a value. In the second integral, we used substitution u = 4t + π/2 to evaluate the integral. In the third integral, we used substitution u = sinx. Lastly, in the fourth integral, we used substitution u = ln(cosz) and then get the value. These integrals are solved by using different methods and get the respective values.
In this question, the given integrals are evaluated using different methods. Finally, the values of these integrals are found.
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Which of the following represents a function
1
X
6
-51 8 1
1 1723 171
10.1) (3,2), (8,3), (2,2), (34)
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use the method of variation of parameters to find a particular solution to the differential equation y" + 8y’ + 12y = 7e^x
having the form yp = u1y1 + u2y2
in this, (note that this asks for u1 and u2, not u1 and u2!) y1 = ___
y2 = ___
u1 = ___
u2 = ___
then use your solution to find the solution to the differential equation that satisfies the initial conditions y(0) = 0, y’(0) = 0.,
y = ___ .
Answers
The general solution to the differential equation y" + 8y' + 12y = 7e^x that satisfies the initial conditions y(0) = 0, y'(0) = 0 is:
y = y_h + yp
The characteristic equation for the homogeneous differential equation y" + 8y' + 12y = 0 is:
r^2 + 8r + 12 = 0
Solving this quadratic equation, we get:
r1 = -2 and r2 = -6
So the general solution to the homogeneous differential equation is:
y_h = c1e^(-2x) + c2e^(-6x)
To find a particular solution using the method of variation of parameters, we need to find the two linearly independent solutions y1 and y2.
Since the characteristic equation has real and distinct roots, we can take y1 = e^(-2x) and y2 = e^(-6x).
Using the formula for the variation of parameters, we have:
u1' = (g(x)y2)/W(y1,y2)
u2' = -(g(x)y1)/W(y1,y2)
where W(y1,y2) is the Wronskian of y1 and y2, which is given by:
W(y1,y2) = y1y2' - y1'y2
= e^(-2x)(-6e^(-6x)) - (-2e^(-2x))(e^(-6x))
= -4e^(-8x)
So we have:
u1' = -(7e^x e^(-6x))/(-4e^(-8x)) = (7/4)e^(3x)
u2' = (7e^x e^(-2x))/(-4e^(-8x)) = -(7/4)e^(-3x)
Integrating u1' and u2', we get:
u1 = (7/12)e^(3x) + C1
u2 = -(7/12)e^(-3x) + C2
where C1 and C2 are constants of integration.
Therefore, the particular solution is:
yp = u1y1 + u2y2
= [(7/12)e^(3x) + C1]e^(-2x) + [-(7/12)e^(-3x) + C2]e^(-6x)
To find the values of C1 and C2, we use the initial conditions y(0) = 0 and y'(0) = 0:
y(0) = yp(0) = [(7/12)e^(0) + C1]e^(0) + [-(7/12)e^(0) + C2]e^(0) = 0
y'(0) = yp'(0) = [7e^(0) + 3C1]e^(0) + [-7e^(0) - 2C2]e^(0) = 0
Simplifying these equations, we get:
C1 - C2 = 0
3C1 - 2C2 = -7
Solving for C1 and C2, we get:
C1 = -7/5
C2 = -7/5
Therefore, the particular solution that satisfies the initial conditions is:
yp = [(7/12)e^(3x) - (7/5)]e^(-2x) + [-(7/12)e^(-3x) - (7/5)]e^(-6x)
So the general solution to the differential equation y" + 8y' + 12y = 7e^x that satisfies the initial conditions y(0) = 0, y'(0) = 0 is:
y = y_h + yp
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HURRY UP Please answer this question
Answers
In response to the supplied query, we may state that 759 is the value decimal entered into the green box.
what is decimal?Both integer and non-integer values are frequently stated in decimal form. Non-integer values are now part of the enlarged Hindu-Arabic numeral system. The process of representing numbers in the decimal system is referred to as decimal notation. A decimal is a number that consists of both a whole and a fractional part. Whole and partially entire quantities are expressed numerically using decimal numbers, which fall between integers. A decimal number's whole number and its fractional component are separated by a decimal point. The decimal point is the dot that separates the full number from the fractions. For instance, 25.5 is a decimal number.
We may leverage the fact that 0.83 is a decimal representation of a rational number with a finite decimal portion to express 0.83 as a fraction. We may say it this way:
0.83 = 83/100
\(8.39 = 8 + 0.39\s= 8 + 39/100\s= 8 + 39/(10^2)\s= 8 + F/(10) (10)\)
Now that the decimal is gone from the right-hand side's numerator, we may multiply both sides by 10:
\(10 * 8.39 = 10 * (8 + F/10)\\839 = 80 + F\sF = 839 - 80 = 759\)
So, 8.39 may be expressed as the following fraction:
8.39 = 8 + 0.39 = 8 + 39/100 = 8 + 759/1000
759 is the value entered into the green box.
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explain the difference between solving 2x+15-3x=10 and 2x-4=3x+11
Answers
2nd one is x=-15
Pls mark brainliest
Caleb an investment banker sold his shares for $18,189.27 when there was a boom in the stock market. Calculate the amount he paid for the shares if his selling price was 130% of the amount he paid for the shares.
Answers
Therefore, Caleb paid approximately $14,067.90 for the shares.
Let's assume the amount Caleb paid for the shares is represented by the variable "x". According to the given information, his selling price was 130% of the amount he paid.
Selling price = 130% of the amount paid
$18,189.27 = 1.3 * x
To find the amount he paid for the shares, we can solve the equation for "x" by dividing both sides by 1.3:
x = $18,189.27 / 1.3
Calculating this, we find:
x ≈ $14,067.90
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Find the reflection of the point (25, 25) across the y-axis.
Answers
Answer:
(-25, 25)
Step-by-step explanation:
let me know if you want an explanation!
find each value of k for which the lines y=9kx-1 and kx 4y=12 are perpendicular
Answers
The value of k in the equations is 2/3
How to determine the value of kFrom the question, we have the following parameters that can be used in our computation:
y = 9kx - 1
kx + 4y = 12
This can be expressed as
y = 9kx - 1
y = -kx/4 + 3
The slopes of perpendicular lines are opposite reciprocal
This means that
9k * -k/4 = -1
So, we have
k^2 = 4/9
Evaluate
k = 2/3
Hence, the value of k is 2/3
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Let f(x)=x2+3x−10. Enter the x-intercepts of the quadratic function in the boxes. x = and x =
Answers
Answer:
X= 2.
X= -5.
Step-by-step explanation:
Hope it was helpful ;)
A certain recipe calls for 3 tablespoons of an ingredient in 22 fl oz of liquid. How
many tablespoons would be in 56 fl oz of liquid?
Answers
Answer:
-11x-7=-3x^2-11x-7=-3x^2-11x-7=-3x^2-11x-7=-3x^2-11x-7=-3x^2-11x-7=-3x^2-11x-7=-3x^2-11x-7=-3x^2-11x-7=-3x^2-11x-7=-3x^2-11x-7=-3x^2
Step-by-step explanation:
-11x-7=-3x^2-11x-7=-3x^2-11x-7=-3x^2-11x-7=-3x^2-11x-7=-3x^2
Two teams, a and b, are about to play a game. previous records show that a has an 80% chance of winning, while b has a 20% of winning. spectators can either choose to buy regular tickets or special tickets. a spectator with a special ticket will receive a refund of $2 if team a wins and a refund of $5 if team b wins. how much extra, minimally, should the special ticket cost to cover the expected value of the refunds offered?
Answers
The special ticket cost to cover the expected value of the refunds would be $2.60
How to solve for the minimal extra cost.Probability of A winning = 80%
Probability of B winning = 20%
We have to solve for the rebate of the bet.
80% = 0.8, 20% = 0.2
0.8 x 2 = 1.6
0.2 x 5 = 1
The extra minimal cost of the special ticket would be calculated as: 1 + 1.6
= $2.6
Therefore the answer to the question is $2.6
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Which choices are equivalent to the expression below? Check all that apply. (3/8)^3
A. (3^3/8^3)
B. 27/512
C. 24
D. 3^18
E. 3^5
F. 9/7
Please answer I would really appreciate it.
Answers
Answer:
A. (3^3/8^3)
B. 27/512
Step-by-step explanation:
(3/8)^3 =
3/8 * 3/8 * 3/8 =
27/512
Answer:
A & B
Step-by-step explanation:
if you put the (3/8)^3 in desmos then put all the answer choices in desmos calculator you can see which ones match up to the problem.
What are the solutions to the system of conics?
y^2/4−x^2/2=1
y^2=8(x+1)
Drag the ordered pairs into the box to correctly complete the table.
Answers
The cοmplete table with the οrdered pairs is:
(2-√6,2√(6-2√6)) (2+√6,2√(6-2√6)) (2+√6,-2√(6-2√6))
(2-√6,2√(6+2√6)) (2+√6,-2√(6+2√6)) (2+√6,2√(6+2√6))
What is quadratic equatiοn?A quadratic equatiοn is a secοnd-degree pοlynοmial equatiοn in οne variable οf the fοrm: \(ax^2 + bx + c = 0.\)
Tο find the sοlutiοns tο the system οf cοnics, we can use substitutiοn tο eliminate οne variable and οbtain a quadratic equatiοn in the οther variable. Then, we can sοlve this quadratic equatiοn tο find the pοssible values οf the remaining variable.
Frοm the given system οf cοnics:
\(y^2/4 - x^2/2 = 1 ...(1)\)
\(y^2 = 8(x + 1) ...(2)\)
We can eliminate \(y^2\) frοm equatiοn (1) by multiplying bοth sides by 4:
\(y^2 - 2x^2 = 4\)
Substituting the value οf \(y^2\) frοm equatiοn (2), we get:
\(8(x + 1) - 2x^2 = 4\)
Simplifying and rearranging, we get:
\(x^2 + 2x - 2 = 0\)
We can sοlve this quadratic equatiοn using the quadratic fοrmula:
\(x = (-2 \± \sqrt{(2^2 - 4(1)(-2))}) / (2(1))\)
x = (-2 ± √12) / 2
x = -1 ± √3
Substituting these values οf x in equatiοn (2), we can find the cοrrespοnding values οf y:
When x = -1 + √3, we get:
\(y^2\) = 8((-1 + √3) + 1) = 8√3
y = ±2√(2√3)
Therefοre, the sοlutiοns are:
(-1 + √3, 2√(2√3)) and (-1 + √3, -2√(2√3))
When x = -1 - √3, we get:
\(y^2\) = 8((-1 - √3) + 1) = -8√3
This equatiοn has nο real sοlutiοns fοr y.
Therefοre, the cοmplete table with the οrdered pairs is:
(2-√6,2√(6-2√6)) (2+√6,2√(6-2√6)) (2+√6,-2√(6-2√6))
(2-√6,2√(6+2√6)) (2+√6,-2√(6+2√6)) (2+√6,2√(6+2√6))
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(a) select one of the pairs of systems and write down its number. (b) for the pair selected, identify each system and state one function of that system. (c) explain how the two systems work together to help maintain homeostasis in an individual.
Answers
a. Pair 2 is selected.
b. Respiratory system, its function is to breathe. Digestive system, its function is to break down food into nutrients.
c. The two systems work together by sharing the region of the mouth and upper throat.
First picture in Pair 2 is Respiratory system, which its function is to breathe. Second picture in Pair 2 is Digestive system, which its function is to break down food into nutrients such as carbohydrates, fats and proteins. The respiratory system takes in oxygen from the air, and also gets rid of carbon dioxide. The digestive system absorbs water and nutrients from the food we consume.
How does the respiratory system and the digestive system work together?The respiratory system is mainly used to transport air, whilst the digestive system is used to transport fluids and solids, from water and food we eat. The respiratory and the digestive systems share the area of the mouth and upper throat, in which air, fluids, and solids can be mixed.
The digestive system does homeostasis by ensuring that the stomach area has the right pH balance. The body uses both positive and negative mechanisms to perform homeostasis. If the body detects an imbalance, the other systems work together to counterbalance and restore the right equilibrium.
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Solve using the quadratic formula
Answers
By using the quadratic formula, the solution to this quadratic equation is equal to -0.5251 or -3.8081.
What is a quadratic equation?In Mathematics, a quadratic equation can be defined as a mathematical expression that can be used to define and represent the relationship that exists between two or more variable on a graph. In Mathematics, the standard form of a quadratic equation is given by;
ax² + bx + c = 0
Mathematically, the quadratic formula is modeled or represented by this mathematical expression:
\(x = \frac{-b\; \pm \;\sqrt{b^2 - 4ac}}{2a}\)
From the information provided, we have the following values;
3x² + 13x + 4 = 0
Where:
a = 3
b = 13
c = 4
Substituting the values into the quadratic formula, we have the following;
\(x = \frac{-(13)\; \pm \;\sqrt{(13)^2 - 4(3)(13)}}{2(3)}\\\\x = \frac{-13\; \pm \;\sqrt{169 - 72}}{6}\)
x = (-13 + √97)/6
x = -0.5251 or -3.8081.
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For problem 2 - 25, what is the slope of line ?
Answers
What is the sum of -4/5 and -3/5
Answers
Answer:
-7/5
Step-by-step explanation:
It is given that,
→ (-4/5) + (-3/5) = ?
Let's solve the problem,
→ (-4/5) + (-3/5)
→ (-4/5) - (3/5)
→ (-4 - 3)/5 = -7/5
Thus, the answer is -7/5.
.Evaluate the following integral over the Region D
. (Answer accurate to 2 decimal places).
∬ D 5(r^2⋅sin(θ))rdrdθ
D={(r,θ)∣0≤r≤1+cos(θ),0π≤θ≤1π}
Hint: The integral and region is defined in polar coordinates.
Answers
The double integral in polar coordinates evaluates to (5/4)∫π0 [(1+cos(θ))^3(1-cos^2(θ))]dθ, which simplifies to (4/3)(2^4 - 1) = 85.33 when evaluated.
We start by evaluating the integral in polar coordinates:
∬ D 5(r^2⋅sin(θ))rdrdθ = ∫π0 ∫1+cos(θ)0 5r^3sin(θ)drdθ
Integrating with respect to r first, we get:
∫π0 ∫1+cos(θ)0 5r^3sin(θ)drdθ = ∫π0 [(5/4)(1+cos(θ))^4sin(θ)]dθ
Using a trigonometric identity, we can simplify this expression:
(5/4)∫π0 [(1+cos(θ))^4sin(θ)]dθ = (5/4)∫π0 [(1+cos(θ))^3(1-cos^2(θ))]dθ
We can then use a substitution u = 1 + cos(θ) to simplify the integral further:
u = 1 + cos(θ), du/dθ = -sin(θ), dθ = -du/sin(θ)
When θ = 0, u = 1 + cos(0) = 2, and when θ = π, u = 1 + cos(π) = 0. Therefore, the limits of integration become:
∫π0 [(1+cos(θ))^3(1-cos^2(θ))]dθ = ∫20 -u^3du = (4/3)(2^4 - 1) = 85.33
Rounding to two decimal places, the answer is approximately 85.33.
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The second YES
The third NO
The forth YES
express the limit limn→[infinity]∑i=1n(4(x∗i)2−2(x∗i))δx over [−1,1] as an integral.
Answers
The answer is 16/3, which is obtained by evaluating the integral of (8x² - 4x) over the interval [-1,1].
How to express limit as integral?To express the limit of limn→[infinity]∑i=1n(4(x∗i)2−2(x∗i))δx over [−1,1] as an integral, we can use the definition of a Riemann sum.
First, we note that delta x, or the width of each subinterval, is given by (b-a)/n, where a=-1 and b=1. Therefore, delta x = 2/n.
Next, we can express each term in the sum as a function evaluated at a point within the ith subinterval. Specifically, let xi be the right endpoint of the ith subinterval. Then, we have:
4(xi)² - 2(xi) = 2(2(xi)² - xi)
We can rewrite this expression in terms of the midpoint of the ith subinterval, mi, using the formula:
mi = (xi + xi-1)/2
Thus, we have:
2(2(xi)² - xi) = 2(2(mi + delta x/2)² - (mi + delta x/2))
Simplifying this expression gives:
8(mi)² - 4(mi)delta x
Now, we can express the original limit as the integral of this function over the interval [-1,1]:
limn→[infinity]∑i=1n(4(x∗i)2−2(x∗i))δx = ∫[-1,1] (8x² - 4x) dx
Evaluating this integral gives:
[8x³/3 - 2x²] from -1 to 1
= 16/3
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whats the mean and median and range for 12 , 4 , -2 , 0 , 9 , -2 , 1 , 7 , 8 , 2
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Mean: 4.7
Median: 3
Range: 14