z - 9 = 10(x - 5) - 8(y - 4) this is the equation of the tangent plane at the point (5, 4, 9). (x - 5)/10 = (y - 4)/(-8) = (z - 9)/(-1). These are the symmetric equations for the normal line to the surface at the given point.
(a) To find the equation of the tangent plane to the surface z = x^2 - y^2 at the point (5, 4, 9), we first need to find the partial derivatives with respect to x and y:
∂z/∂x = 2x
∂z/∂y = -2y
Now, we evaluate these at the given points (5, 4, 9):
∂z/∂x(5, 4) = 2(5) = 10
∂z/∂y(5, 4) = -2(4) = -8
Using the tangent plane equation:
z - z₀ = ∂z/∂x (x - x₀) + ∂z/∂y (y - y₀)
Plugging in the values:
z - 9 = 10(x - 5) - 8(y - 4)
This is the equation of the tangent plane at the point (5, 4, 9).
(b) The normal vector to the surface at the given point is given by the gradient vector (∂z/∂x, ∂z/∂y, -1) = (10, -8, -1). To find the symmetric equations for the normal line, we use the point-normal form:
(x - x₀)/a = (y - y₀)/b = (z - z₀)/c
Plugging in the point (5, 4, 9) and the normal vector components (10, -8, -1):
(x - 5)/10 = (y - 4)/(-8) = (z - 9)/(-1)
These are the symmetric equations for the normal line to the surface at the given point.
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PLEASE HELP
What is the probability that
both events will occur?
Two dice are tossed.
Event A: The first die is a 1 or 2
Event B: The second die is 4 or less
P(A and B) = P(A) • P(B)
P(A and B) = [?]
Enter as a decimal rounded to the nearest hundredth.
The probability that both events will occur is 0.22.
What is probability?It is the chance of an event to occur from a total number of outcomes.
The formula for probability is given as:
Probability = Number of required events / Total number of outcomes.
Example:
The probability of getting a head in tossing a coin.
P(H) = 1/2
We have,
The probability of Event A is 2/6 or 1/3
(since there are two ways to get a 1 or 2 on a six-sided die).
The probability of Event B is 4/6 or 2/3
(since there are four ways to get a number 4 or less on a six-sided die).
Using the formula for the probability of the intersection of two independent events.
P(A and B)
= P(A) x P(B)
= (1/3) x (2/3)
= 2/9
Rounded to the nearest hundredth,
The probability that both events will occur is 0.22.
Thus,
The probability that both events will occur is 0.22.
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An experiment is conducted with a coin. The results of the coin being flipped twice 200 times is shown in the table.
Outcome Frequency
Heads, Heads 75
Heads, Tails 40
Tails, Tails 35
Tails, Heads 50
What is the P(No Heads)?
85%
75%
37.5%
17.5%
The probability of no heads is given as follows:
P(No Heads) = 17.5%.
How to calculate a probability?A probability is calculated as the division of the desired number of outcomes by the total number of outcomes in the context of a problem/experiment.
The total number of outcomes is given as follows:
200.
The desired outcomes, those without heads, are Tails, Tails, which happened 35 times, hence the probability is given as follows:
p = 35/200
p = 0.175
p = 17.5%.
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8-40.
For the triangle at right, write each of the following trigonometric ratios. The first one is done for you.
Answer:
tan A: BC/AB
cos A: AB/AC
sin C: AB/AC
cos C: BC/AC
sin A: BC/AC
Step-by-step explanation:
sin of an angle: opposite/hypotenuse
cosine of an angle: adjacent/hypotenuse
tangent of an angle: opposite/adjacent
Describe the specific sequence of transformations that would map triangle abc to triangle a'b'c'.
Translation, rotation, and reflection, By following these transformations in sequence, you can map triangle ABC to triangle A'B'C'.
To map triangle ABC to triangle A'B'C', you would need to follow a specific sequence of transformations, which may include translation, rotation, and reflection. Here's a step-by-step explanation:
Step 1: Translation
Translate triangle ABC by a specific vector (x, y) so that point A moves to point A'. The same vector will also move points B and C to their corresponding new positions.
Step 2: Rotation
If triangle A'B'C' is rotated compared to the translated triangle, rotate the translated triangle around point A' by a specific angle, either clockwise or counterclockwise, until point B aligns with point B'.
Step 3: Reflection
If triangle A'B'C' is a mirror image of the rotated triangle, reflect the rotated triangle across a line of symmetry (usually a line passing through A'). This will change the orientation of the triangle and align point C with point C'.
By following these transformations in sequence, you can map triangle ABC to triangle A'B'C'. Keep in mind that the specific details of translation, rotation, and reflection will depend on the coordinates and orientation of the given triangles.
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Which of the pair of linear equations has unique solution, no solution or infinitely many solutions. In case there is unique solution find it by using Substitution Method and Elimination Method
(i) x-3y -3=0, 3x-9y-2=0
(ii) 2x+y=5,3x+2y=8
(iii) 3x-5y=20,6x-10y=40
iv) x-3y-7 =0,3x-3y-15=0
v) 8x+5y=9,3x+2y=4
1. x-3y -3=0, 3x-9y-2=0 has no solution
2. 2x+y=5,3x+2y=8 has a unique solution
3. 3x-5y=20,6x-10y=40 has infinitely many solution
4. x-3y-7 =0,3x-3y-15=0 has No solution
5. 8x+5y=9,3x+2y=4 has a unique solution
How to solve the linear equations(i) To solve using substitution method, we can rearrange the first equation to x=3y+3 and substitute it into the second equation:
3(3y+3) - 9y - 2 = 0
9y + 9 - 9y - 2 = 0
7 = 0
This is a contradiction, so the pair of equations has no solution.
(ii) To solve using elimination method, we can multiply the first equation by 2 and subtract it from the second equation:
3x + 2y = 8
(4x + 2y = 10)
-x = -2
So, x = 2. Substituting this value into the first equation, we get:
2x + y = 5
2(2) + y = 5
y = 1
Therefore, the unique solution is (x,y) = (2,1).
(iii) To solve using elimination method, we can multiply the first equation by 2 and subtract it from the second equation:
6x - 10y = 40
(6x - 10y = 40)
0 = 0
This equation is true for any value of x and y, so the pair of equations has infinitely many solutions.
(iv) To solve using elimination method, we can subtract the first equation from the second equation:
3x - 3y - 15 - (x - 3y - 7) = 0
2x - 22 = 0
x = 11
Substituting this value into the first equation, we get:
11 - 3y - 7 = 0
-3y = -4
y = 4/3
Therefore, the unique solution is (x,y) = (11,4/3).
(v) To solve using elimination method, we can multiply the first equation by 2 and subtract it from the second equation:
3x + 2y = 4
(16x + 10y = 18)
-29x - 18y = -14
Solving for y, we get:
y = (29/18)x + (7/9)
Substituting this expression for y into the first equation, we get:
8x + 5((29/18)x + (7/9)) = 9
(143/18)x = 2/9
x = 2/13
Substituting this value into the expression for y, we get:
y = (29/18)(2/13) + (7/9) = 41/117
Therefore, the unique solution is (x,y) = (2/13,41/117).
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Do You Understand?
1. How can you find the volume of the
china cabinet?
1 ft,
7 ft
3 ft
4 ft
2ft
The volume of the china cabinet is 21 cubic feet.
To find the volume of the china cabinet, we need to multiply its length, width, and height.
Since the dimensions are given in feet, we will use cubic feet as the unit of volume.
The length of the china cabinet is given as 1 ft, the width as 7 ft, and the height as 3 ft.
The volume can be calculated as follows:
Volume = length * width * height
Volume = 1 ft * 7 ft * 3 ft
Volume = 21 cubic feet
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Consider the variable coefficient linear second order non-homogeneous ODE
x^2y^n - 2xy' + (x^2 +2)y =. 3x^3, for x > 0
Write down the associated homogeneous equation.
x²y' - 2xy' + (x² + 2)y = 0 is the associated homogeneous equation with non-homogeneous ODE x²yⁿ - 2xy' + (x² +2)y = 3x³.
It should be noted that the equation is same as the variable coefficient linear second order non-homogeneous ODE but just the right side zero.
The complementary solutions or homogeneous solutions to this homogeneous equation serve as the foundation for the space of all solutions to the non-homogeneous equation.
By assuming that y has the form y(x) = xr and substituting this into the homogeneous equation to create a characteristic equation, we can determine the complementary solutions. We may find the values of r that correspond to solutions of the type y(x) = xr by looking at the characteristic equation's roots.
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The length of a triangle is three times its width the perimeter of the rectangle is 24cmcalculate the area of the triangle
The area of the triangle is 6 cm².
Let's denote the width of the triangle as "w." According to the given information, the length of the triangle is three times its width, so the length can be expressed as "3w."
The perimeter of a rectangle is given by the formula: Perimeter = 2(length + width). In this case, the perimeter of the rectangle is given as 24 cm.
We can set up the following equation based on the given information:
24 = 2(3w + w)
Simplifying the equation:
24 = 2(4w)
12w = 24
w = 24/12
w = 2 cm
Now that we have the width of the triangle, we can find the length:
Length = 3w = 3 * 2 = 6 cm
The area of a triangle is given by the formula: Area = (base * height) / 2. In this case, the base of the triangle is the width (2 cm) and the height is the length (6 cm).
Area = (2 * 6) / 2
Area = 12 / 2
Area = 6 cm²
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(1 point) Use the linear approximation to estimate (-2.02)2(2.02)3 = Compare with the value given by a calculator and compute the percentage error: Error = %
the linear approximation, we estimated the value of (-2.02)^2 * (2.02)^3 as 31.68, and the percentage error compared to the calculator's value is approximately 0.1924%.
Let's break it down step-by-step:
1. Identify the function we want to approximate: f(x) = x^2 * (x+4)^3
2. Choose the point to approximate near Since we want to estimate f(-2.02), let's approximate near x = -2.
3. Compute the linear approximation (first-degree Taylor polynomial) at x = -2: f(-2) = (-2)^2 * (2)^3 = 4 * 8 = 32
4. Find the derivative of f(x): f'(x) = 2x(x+4)^3 + 3x^2(x+4)^2
5. Compute the derivative at x = -2: f'(-2) = 2(-2)(2)^3 + 3(-2)^2(2)^2 = -32 + 48 = 16
6. Use the linear approximation formula: f(-2.02) ≈ f(-2) + f'(-2)(-2.02 - (-2)) = 32 + 16(-0.02) = 32 - 0.32 = 31.68
Now, compare this approximation to the value given by a calculator: (-2.02)^2 * (2.02)^3 ≈ 31.741088. To compute the percentage error, use the formula:
Percentage Error = |(Approximate Value - Actual Value) / Actual Value| * 100%
Percentage Error = |(31.68 - 31.741088) / 31.741088| * 100% ≈ 0.1924%
So, using the linear approximation, we estimated the value of (-2.02)^2 * (2.02)^3 as 31.68, and the percentage error compared to the calculator's value is approximately 0.1924%.
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Please help asap
0 = pi/3 radians. identify the terminal point and tan 0
An angle of 0 radians is an angle along the positive x-axis of the unit circle. Its terminal point is (1, 0).
The tangent of 0 radians is defined as the ratio of the y-coordinate to the x-coordinate of the terminal point, which is 0/1 = 0.
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Kurts city took a survey about a plan for a new park. the city surveyed 3000 people. 53% of the people surveyed like the plan for the park. how many people like the plan?
The number of people who like the plan is 1,590 people out of the 3,000 surveyed.
To determine how many people liked the plan, we'll need to use the percentage given and apply it to the total number of people surveyed.
Percentage is a way of expressing a proportion or a fraction as a whole number out of 100. In this case, the percentage we're working with is 53%, which means 53 out of every 100 people surveyed liked the plan. To find the number of people who liked the plan, we can multiply the total number of people surveyed (3,000) by the percentage who liked the plan (53%).
To do this calculation, first convert the percentage to a decimal by dividing 53 by 100, which gives us 0.53. Next, multiply 3,000 by 0.53:
3,000 * 0.53 = 1,590
So, 1,590 people out of the 3,000 surveyed liked the plan for the new park.
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6. (2.5 pts) at the beginning of week 5, they broke up. jack wanted to run off to the city with
diane, but diane said he was crazy. unfortunately, their relationship ended. both were
angry with each other. suppose we could somehow quantify and measure anger. let's
call the units "anger units". on the day of the break-up, jack had 100 anger units. every
week he lost 5% of his anger. recall that the growth factor needs to be the amount that
"stays on" jack (not the 5% that "comes off" jack). for example, after 1 week, he had 95
anger units. after 2 weeks he had 90.25 anger units, and so on. write an equation that
models jack's anger (let that be )) after t weeks.
We'll model Jack's anger in anger units after t weeks using an exponential decay equation, as he loses 5% of his anger every week.
To write an equation that models Jack's anger (let that be A(t)) after t weeks, we need to follow these steps:
1. Identify the initial amount of anger units (A0): Jack had 100 anger units at the beginning (t=0).
2. Determine the growth factor (1 - decay rate): Since Jack loses 5% of his anger every week, the growth factor is 1 - 0.05 = 0.95.
3. Set up the exponential decay equation: A(t) = A0 * (growth factor)^t.
By following these steps, the equation modeling Jack's anger after t weeks is:
A(t) = 100 * (0.95)^t
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The spinner has 8 congurent sections it is spun 24 times what is a reasonable prediction for the number of times the spinner will land on the number 3.
A reasonable prediction for the number of times the spinner will land on the number 3 is 3 times.
Since the spinner has 8 congruent sections and is spun 24 times, we can use probability to make a reasonable prediction for the number of times it will land on the number 3.
1. Calculate the probability of landing on the number 3 for a single spin:
Since there are 8 congruent sections, the probability of landing on the number 3 is 1/8.
2. Determine the expected number of times the spinner will land on the number 3:
To do this, multiply the probability of landing on the number 3 (1/8) by the total number of spins (24).
Expected number of times = (1/8) * 24
3. Simplify the expression:
Expected number of times = 3
So, a reasonable prediction for the number of times the spinner will land on the number 3 is 3 times.
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Assume base is 2.a b c
a = 5
b = 4
c= 0
Therefore, the equation for graph C is Y = a ^b + c
Y = 5 ^4 + 0
What is a graph?A graph is described as a diagram showing the relation between variable quantities, typically of two variables, each measured along one of a pair of axes at right angles.
Graphs are a popular tool for graphically illuminating data relationships. A graph serves the purpose of presenting data that are either too numerous or complex to be properly described in the text while taking up less room.
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Your name is Galileo Galilei, and you toss a weight upward at 16 feet per second from the top of the Leaning Tower of Pisa (height 186 ft). (a) Neglecting air resistance, find the weight's velocity as a function of time t in seconds. v(t) = Correct: Your answer is correct. ft/s (b) Find the height (in feet) of the weight above the ground as a function of time. s(t) =
(a) The weight's velocity as a function of time t in seconds is v(t) = 16 - 32.2t
(b) The height (in feet) of the weight above the ground as a function of time is s(t) = 186 + 16t - (1/2)(32.2)t^2
To find the weight's velocity and height as a function of time:
(a) The equation for velocity as a function of time is v(t) = v0 - gt,
where v0 is the initial velocity (in this case, 16 ft/s) and g is the acceleration due to gravity (32.2 ft/s^2).
Using this equation, we can find the weight's velocity as it travels upward:
v(t) = 16 - 32.2t
(b) The equation for height as a function of time is s(t) = s0 + v0t - (1/2)gt^2,
where s0 is the initial height (in this case, 186 ft).
Using this equation, we can find the height of the weight above the ground at any point in time:
s(t) = 186 + 16t - (1/2)(32.2)t^2
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The coiling dragon cliff skywalk in china is $128$ feet longer than the length $x$ (in feet) of the tianmen skywalk in china. The world's longest glass-bottom bridge, located in china's zhangjiaji national park, is about $4. 3$ times longer than the coiling dragon cliff skywalk. Write and simplify an expression that represents the length (in feet) of the world's longest glass-bottom bridge
The expression that represents the length (in feet) of the world's longest glass-bottom bridge is 4.3x+550.4.
Let's denote the length of the Coiling Dragon Cliff Skywalk as y (in feet). According to the given information, we have:
y = x + 128
The length of the world's longest glass-bottom bridge is 4.3 times longer than the Coiling Dragon Cliff Skywalk, so we can write an expression for it as:
Length of the longest glass-bottom bridge = 4.3 * y
Now, we can substitute the expression for y from the first equation:
Length of the longest glass-bottom bridge = 4.3 * (x + 128)
To simplify, distribute the 4.3:
Length of the longest glass-bottom bridge = 4.3x + 550.4
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This is part of a city map.
City map with First Street and Second Street as two lines equal distance apart that never meet. Main Street is intersecting Arch, First, Second, and Elm. Elm Street is intersecting Main and First Street.
Which streets are parallel to each other?
A.
First Street and Second Street
B.
First Street and Arch Street
C.
None of the streets are parallel to one another.
D.
Elm Street and Main Street
PLEASE HELP
Based on data taken from airline fares and distances flown, it is determined that the equation of the least-squares regression line is ŷ = 102. 50 + 0. 65x, where ŷ is the predicted fare and x is the distance, in miles. One of the flights was 500 miles and its residual was 115. 0.
What was the fare for this flight?
102. 50
312. 50
427. 50
542. 50
The fare for this flight was $542.50 which is calculated using least-squares regression line equation. Therefore, the correct answer 542.50
To find the fare for this flight, we will first use the provided least-squares regression line equation to predict the fare and then account for the residual.
Step 1: Use the least-squares regression line equation to predict the fare.
ŷ = 102.50 + 0.65x, where ŷ is the predicted fare and x is the distance in miles.
Step 2: Substitute the given distance (x = 500 miles) into the equation.
ŷ = 102.50 + 0.65(500)
Step 3: Calculate the predicted fare.
ŷ = 102.50 + 325
ŷ = 427.50
The predicted fare for a 500-mile flight is $427.50.
Step 4: Adjust for the residual.
The residual for this flight is 115.0, which means the actual fare is $115 higher than the predicted fare.
Step 5: Add the residual to the predicted fare to find the actual fare.
Actual fare = Predicted fare + Residual
Actual fare = 427.50 + 115
Actual fare = 542.50
The fare for this flight was $542.50.
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π × 4 to the second power × 3 ( with steps !! )
a decimal number that is larger than 0.0467 but smaller than 0.0468
Answer: .04671 - 0.04679
Step-by-step explanation:
Answer:
0.04675
Step-by-step explanation:
0.04675 > 0.0467
0.04675 < 0.0468
The start of an arithmetic sequence is 29, 17, 21, 25, The rule for the sequence can be written in the form xn=cn+d, where c and d are numbers. a) By first calculating the values of c and d, work out the rule for the sequence. b) What is the value of x11?
Answer:
Step-by-step explanation:
The value of 2s in 43,290 is 2,000, while the value of 2s in 32,865 is 20.
B. 200 is 1/20 the value of 2,000.
This statement is correct, as 200 is 1/10 of 2,000, and there are two 0s in the value of 2s in 43,290 compared to one 0 in the value of 2s in 32,865.
A bag contains five red socks and eight blue socks. Lucky reaches into the bag and randomly selects two socks without replacement. What is the probability that Lucky will get different colored socks? Express your answer as a common fraction. I will give brainliest if you give a full explanation, I have the answer but I need to know HOW to solve the problem!!!
A bag contains five red socks and eight blue socks. Lucky reaches into the bag and randomly selects two socks without replacement, the probability that Lucky will get different colored socks is 10/39.
We can divide the issue into two distinct possibilities and multiply them together to find a solution.
Let's start by thinking about the likelihood of choosing a red sock during the initial draw.
The likelihood of choosing a red sock on the first draw is 5/13 due to the fact that there are only five red socks among the total of thirteen socks (five red plus eight blue).
There are now twelve socks left in the bag after the first one is drawn, with four red and eight blue.
On the second draw, there is an 8/12 chance of choosing a blue sock, which is a different colour.
We add the probabilities together to determine the likelihood that both events (drawing a red sock first and a blue sock second) will occur:
(5/13) * (8/12) = 40/156 = 10/39
Therefore, the probability that Lucky will get different colored socks is 10/39.
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Noel borrows $800
and is charged compound interest at 60%
per year. How much will he have to pay back in total after 8
years?
Answer:
Step-by-step explanation:
The data set is 12, 46, 32, 18, 26, 41, 46. the mean is 31.6 and the median is 32. if we add another 12, what affect does this have on the mean and median?
Adding another 12 to the data set would increase the sum of the values by 12, resulting in a new sum of 239. To find the new mean, we divide the new sum by the total number of values in the set, which is now 8. So the new mean would be 29.875, which is slightly lower than the original mean of 31.6.
To find the new median, we first need to rearrange the values in ascending order: 12, 18, 26, 32, 41, 46, 46, 12. Since there are now an even number of values, we take the average of the middle two, which in this case is (26 + 32) / 2 = 29. So the new median would be 29, which is lower than the original median of 32.
In summary, adding another 12 to the data set would slightly decrease the mean and lower the median.
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Let R(x). C(x), and P(x) be, respectively, the revenue, cost, and profit, in dollars, tomi the production and sale of x items. I R(%) = 6x and C(X) = 0.001x^2 + 1 8x + 40.
find each of the following
a) P(x)
b) R(200). C(200), and P(200)
c) R'(. C't and P'(x)
d) R' (200). C'(200), and P' (200)
a) P(x) = R(x) - C(x) = 6x - (0.001x^2 + 18x + 40) = -0.001x^2 - 12x - 40
b) R(200) = 6(200) = 1200
C(200) = 0.001(200)^2 + 18(200) + 40 = 4000
P(200) = R(200) - C(200) = 1200 - 4000 = -2800
c) R'(x) = 6
C'(x) = 0.002x + 18
P'(x) = R'(x) - C'(x) = 6 - (0.002x + 18) = -0.002x - 12
d) R'(200) = 6
C'(200) = 0.002(200) + 18 = 18.4
P'(200) = -0.002(200) - 12 = -12.4
Here are the answers to each part:
a) P(x) is the profit function, which is calculated as the difference between the revenue function and the cost function: P(x) = R(x) - C(x). In this case, P(x) = 6x - (0.001x^2 + 18x + 40).
b) To find R(200), C(200), and P(200), plug x = 200 into each function:
R(200) = 6(200) = 1200
C(200) = 0.001(200^2) + 18(200) + 40 = 7600
P(200) = 1200 - 7600 = -6400
c) To find R'(x), C'(x), and P'(x), we need to find the derivative of each function with respect to x:
R'(x) = d(6x)/dx = 6
C'(x) = d(0.001x^2 + 18x + 40)/dx = 0.002x + 18
P'(x) = R'(x) - C'(x) = 6 - (0.002x + 18)
d) To find R'(200), C'(200), and P'(200), plug x = 200 into each derivative function:
R'(200) = 6
C'(200) = 0.002(200) + 18 = 18.4
P'(200) = 6 - 18.4 = -12.4
I hope this helps! Let me know if you have any further questions.
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Suppose f'(x) = 8x³ + 12x + 2 and f(1) = -4. Then f(-1) equals (Enter a number for your answer.)
If f'(x) = 8x³ + 12x + 2 and f(1) = -4, f(-1) is equal to -18.
Given that f'(x) = 8x³ + 12x + 2, we can find the original function f(x) by integrating f'(x) with respect to x:
f(x) = 2x⁴ + 6x² + 2x + C, where C is an arbitrary constant.
We can then use the given initial condition f(1) = -4 to solve for C:
f(1) = 2(1)⁴ + 6(1)² + 2(1) + C = -4
Simplifying, we get:
C = -16
Therefore, the function f(x) is:
f(x) = 2x⁴ + 6x² + 2x - 16
To find f(-1), we substitute x = -1 into the expression for f(x):
f(-1) = 2(-1)⁴ + 6(-1)² + 2(-1) - 16 = -18
Thus, f(-1) equals -18.
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100 POINTS IF HELP
what is the average rate of change for the function g(x) for the interval [4,9]?
SHOW ALL WORK
g(x)=4x^2+3x-2
Answer:
Step-by-step explanation:
Find the Riemann sum S₅ for the following information. Round your answer to the nearest hundredth. f(x) = 64 - x²; [a, b] = (-8, -3]; n = 5.c₁ = -7.5.c² = -6.5.c₃ = -5.5.c₄ = - 4.5.c₅ = -3.5
The Rounding to nearest hundredth, we get S₅ ≈ -12.25
How to find the Riemann sum S₅?The formula for a Riemann sum with n subintervals is:
[tex]S_n[/tex]= ∑ᵢ₌₁ⁿ f(cᵢ) Δx,
where Δx = (b - a)/n is the width of each subinterval and cᵢ is a point in the i-th subinterval. The value of cᵢ can be chosen arbitrarily, but here we are given specific values for c₁, c₂, c₃, c₄, and c₅.
In this problem, we have:
f(x) = 64 - x²
[a, b] = (-8, -3]
n = 5
Δx = (b - a)/n = (-3 - (-8))/5 = 1
Therefore, the width of each subinterval is 1.
The Riemann sum S₅ is:
S₅ = f(c₁) Δx + f(c₂) Δx + f(c₃) Δx + f(c₄) Δx + f(c₅) Δx
Substituting the given values for c₁, c₂, c₃, c₄, and c₅, we get:
S₅ = f(-7.5) + f(-6.5) + f(-5.5) + f(-4.5) + f(-3.5)
where f(x) = 64 - x².
Evaluating each term, we get:
f(-7.5) = 64 - (-7.5)² = 17.75
f(-6.5) = 64 - (-6.5)² = 5.75
f(-5.5) = 64 - (-5.5)² = -2.75
f(-4.5) = 64 - (-4.5)² = -12.25
f(-3.5) = 64 - (-3.5)² = -20.75
Therefore,
S₅ = 17.75(1) + 5.75(1) - 2.75(1) - 12.25(1) - 20.75(1) = -12.25.
Rounding to the nearest hundredth, we get S₅ ≈ -12.25.
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Offering brainiest to whoever can give me the answer fastest, a nice explanation, and the correct answer!
Box C has the smallest volume, followed by Box A, and Box B has the largest volume.
Explanation on how to get the least volumeFirst, we need to find the volume of each box.
Recall that the formula for volume of a box is given as:
V = length x height x width
For Box A,
V = 3 cm x 2 cm x 4 1/2 cm = 27 cm³
For Box B,
V = 2 1/3 cm x 3 cm x 5 cm = 7/3 cm x 3 cm x 5 cm = 35 cm³
For Box C,
V = 4 cm x 3 cm x 1 1/4 cm = 4 cm x 3 cm x 5/4 cm = 15 cm³
So, the order of the boxes by volume from least to greatest is: Box C, Box A, and Box B.
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What is the osmotic pressure for a 4. 50% by a mass aqueous solution of glucose (C6H12O6) at 300 K?
The osmotic pressure for a 4.50% by mass aqueous solution of glucose (C6H12O6) at 300 K is 0.616 atm.
To calculate the osmotic pressure of a solutionWe can use the equation:
π = MRT
where:
π = osmotic pressure
M = molarity of the solution
R = gas constant
T = temperature in Kelvin
We must translate the proportion by mass to molarity in order to determine the molarity of the glucose solution. Glucose (C6H12O6) has a molecular weight of 180 g/mol.
So, for a 4.50% by mass solution of glucose, we have:
4.50 g glucose / 100 g solution = (4.50 g glucose / 180 g/mol) / (Molarity of solution)
Solving for molarity, we get:
Molarity of solution = 0.025 mol/L
Now we can plug in the values into the equation for osmotic pressure:
π = (0.025 mol/L) * (0.0821 L atm / mol K) * (300 K)
π = 0.616 atm
Therefore, the osmotic pressure for a 4.50% by mass aqueous solution of glucose (C6H12O6) at 300 K is 0.616 atm.
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