To find points where the tangent plane to an ellipsoid is parallel to a plane, calculate the gradient vector of the ellipsoid and set it proportional to the normal vector of the given plane, then solve the resulting system for (x, y, z).
Explanation:To find all the points where the tangent plane to an ellipsoid is parallel to a given plane, you first need to consider the equation of the ellipsoid and the equation of the tangent plane.
The ellipsoid can be described by the general equation f(x, y, z) = 0, while the tangent plane at a point on the ellipsoid can be described by the gradient of f at that point, given as ∇f.
The gradient, which is a vector, gives us the normal to the tangent plane at the given point.
For a tangent plane to be parallel to another plane, their normal vectors must be proportional.
So, if we have the normal vector of the given plane, we can set up an equation with the gradient of the ellipsoid, and solve for the points (x, y, z) that satisfy this condition.
It requires solving a system of equations where the coefficients of the normals to both planes are proportional.
These points (x, y, z) will be the points of tangency where the ellipsoid's tangent plane is parallel to the given plane.
We use the calculus concepts of partial derivatives to find the gradient vector and algebra to solve for the unknowns corresponding to the points of tangency.
what is the solution of -8/2y-8=5/y+4 - 7y+8/y^2-16? y = –4 y = –2 y = 4 y = 6
Answer:
d. 6
Step-by-step explanation:
just took the pretest:) have such a fantastic day loves, you're doing AMAZING!
John throws a rock straight down with speed 12 m/s from the top of a tower. the rock hits the ground after 2.37 s. what is the height of the tower? (air resistance is negligible)
Ordering Least to greatest 2 9/11, 4/5, 2.91, 0.9
A 27 oz bottle of a new soda costs $2.25. What is the unit rate, rounded to the nearest tenth of a cent?
The unit rate is the cost per ounce of soda. By dividing the total cost by the total ounces, we get the price per ounce in dollars ($0.08333), and converting this to cents gives us $8.3 cents per ounce.
Explanation:The term unit rate refers to a rate in which the second term is 1. In this case, we want to find out how much 1 ounce of soda costs.
First, you want to divide the total cost of the bottle by the total ounces in the bottle. So you divide $2.25 by 27. The answer you get is the price of one ounce of soda in dollars. When calculating it, you get approximately $0.08333.
To get the rate in cents, convert the dollars to cents by multiplying by 100 (since 1 dollar is 100 cents). The answer ($8.33) is the cost of one ounce to the nearest tenth of a cent.
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Calculate the mean of the number set 5 10 12 4 6 11 13 5
In an x-y plot of an experiment what is usually plotted on the x axis?
a. the independent variable, which is the parameter that was manipulated.
b. th
What is the area of a triangle with verticies at (-2,1), (2,1) and (3,4)
What is the reason for each step in the solution of the inequality?
−2(x+3)−4>4x+30
Select the reason for each step from the drop-down menus.
2nd picture is drop down box answers.
What is the answer to this question ?
Simplify 6 - 23 + (-9 + 5) · 2
A. -10
B. -12
C. 6
D. -8
I've been told the answer is A. -10, but I need to know how to get that answer.
Thanks.
at least u tried to help but the answer is -10 bro
n=6x35+4 what does n represent in a division problem
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Rewrite as a square or a cube:
1 11/25
Answer:
1 11/25 = 36/25
= (6/5)^2
Step-by-step explanation:
Simplyfy (x - 5 / x^3 + 27) + (2 / x^2 - 9)
How would you express as a unit rate: morag typed 60 words in one minute
Morag would be typing at a rate of 60 words per minute (wpm) in order to type 60 words in one minute.
What is Unit conversion?A statement of the connection between units that are used to alter the units of a measured quantity without affecting the value is called a conversion factor. A conversion ratio (or unit factor), if the numerator and denominator have the same value represented in various units, always equals one (1).
To express Morag's typing speed as a unit rate, we would divide the number of words by the number of minutes.
Therefore, the unit rate for Morag typing 60 words in one minute would be 60 words per minute (60 wpm).
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Find the gradient of the function at the given point. function point f(x, y, z) = x2 + y2 + z2 (3, 9, 8)
Answer:
[tex]\displaystyle \nabla f(3, 9, 8) = 6 \hat{\i} + 18 \hat{\j} + 16 \hat{\text{k}}[/tex]
General Formulas and Concepts:
Calculus
Differentiation
DerivativesDerivative NotationDerivative Rule [Basic Power Rule]:
f(x) = cxⁿf’(x) = c·nxⁿ⁻¹Multivariable Calculus
Differentiation
Partial DerivativesDerivative NotationGradient: [tex]\displaystyle \nabla f(x, y, z) = \frac{\partial f}{\partial x} \hat{\i} + \frac{\partial f}{\partial y} \hat{\j} + \frac{\partial f}{\partial z} \hat{\text{k}}[/tex]
Gradient Property [Addition/Subtraction]: [tex]\displaystyle \nabla \big[ f(x) + g(x) \big] = \nabla f(x) + \nabla g(x)[/tex]
Gradient Property [Multiplied Constant]: [tex]\displaystyle \nabla \big[ \alpha f(x) \big] = \alpha \nabla f(x)[/tex]
Step-by-step explanation:
Step 1: Define
Identify.
[tex]\displaystyle f(x, y, z) = x^2 + y^2 + z^2[/tex]
[tex]\displaystyle P(3, 9, 8)[/tex]
Step 2: Find Gradient
[Function] Differentiate [Gradient]: [tex]\displaystyle \nabla f = \frac{\partial}{\partial x} \Big( x^2 + y^2 + z^2 \Big) \hat{\i} + \frac{\partial}{\partial y} \Big( x^2 + y^2 + z^2 \Big) \hat{\j} + \frac{\partial}{\partial z} \Big( x^2 + y^2 + z^2 \Big) \hat{\text{k}}[/tex][Gradient] Rewrite [Gradient Property - Addition/Subtraction]: [tex]\displaystyle \nabla f = \bigg[ \frac{\partial}{\partial x}(x^2) + \frac{\partial}{\partial x}(y^2) + \frac{\partial}{\partial x}(z^2) \bigg] \hat{\i} + \bigg[ \frac{\partial}{\partial y}(x^2) + \frac{\partial}{\partial y}(y^2) + \frac{\partial}{\partial y}(z^2) \bigg] \hat{\j} + \bigg[ \frac{\partial}{\partial z}(x^2) + \frac{\partial}{\partial z}(y^2) + \frac{\partial}{\partial z}(z^2) \bigg] \hat{\text{k}}[/tex][Gradient] Differentiate [Derivative Rule - Basic Power Rule]: [tex]\displaystyle \nabla f = 2x \hat{\i} + 2y \hat{\j} + 2z \hat{\text{k}}[/tex][Gradient] Substitute in point: [tex]\displaystyle \nabla f(3, 9, 8) = 2(3) \hat{\i} + 2(9) \hat{\j} + 2(8) \hat{\text{k}}[/tex][Gradient] Evaluate: [tex]\displaystyle \nabla f(3, 9, 8) = 6 \hat{\i} + 18 \hat{\j} + 16 \hat{\text{k}}[/tex]∴ the gradient of the function at the given point is <6, 18, 16>.
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Topic: Multivariable Calculus
Unit: Directional Derivatives
How do you solve an inequality
Answer:
you solve an inequality by doing the inverse equation on each side of the equation.
Step-by-step explanation:
Jeff invests an amount at 4% interest compounded annually. After 3 years, he has $1687.30. What was the original amount Jeff invested?
$1500
$1200
$1450
$750
Two numbers total 53 and have a difference of 25. Find the two numbers.
how many solutions are in 6x+4x-6=24+9x
equivalent ratios number 1 to 50
How do i know that the variable x has a uniform distribution function?
Help Algebra Question
Jasmine finished the bike trail in 2.5 hours at an average rate of 2.5 miles per hour.Lucy biked the same trail at a rate of 6 1/5 mile per hour.How long did it take lucy to bike the trail?
2.5 hours at a rate of 2.5 miles per hour = 2.5 * 2.5 = 6.25 miles
the trail was 6.25 miles long
Lucy rode at 6 1/5 mile per hour
so 6.25 / 6 1/5 = 1.008 = 1.01 hours
How does the throughput of pci express version 3.0 compare to pcie version 2.0?
What is 164% of 25? I have no idea and im in the middle of a test XDDD
Write a problem saying a single load of laundry cost $2 and a double load costs $4 the machine only accept quarters what is the answer
The problem is: A laundromat charges $2 for a single load of laundry and $4 for a double load. The machine only accepts quarters. How many quarters are needed to wash two double loads of laundry?
To solve this problem, we first need to determine the total cost of washing two double loads. Since each double load costs $4, two double loads will cost 2 * $4 = $8.
Next, we need to determine how many quarters are in $8. Since there are 4 quarters in a dollar, there are 4 * 8
= 32 quarters in $8.
Therefore, 32 quarters are needed to wash two double loads of laundry
How many times does 1/2 fit into 30
Kristina invests $5,000 at 15% simple interest for 2 years.How much interest did Kristina earn over the 2 years?
A boat was sailing for 4 hours and covered 224 miles. A jet is ten times as fast as the boat. Find the jet’s speed.
Hi, How do you find the first and second derivatives of the function.
y=(x^2-7/63x) (x^4+1/x^3)
I think for the first derivative dy/dx it's 2/63x-3/63x^-3+4/9x^-5 but I'm not sure, and I have no clue for the second derivative d^2y/dx^2.
The first hand derivative of the function is [tex]6x^5 - \frac{35}{63}x^4 - \frac{1}{x^2} - \frac{4}{63x^3}[/tex] and the second derivative is [tex]30x^4 - \frac{20}{9}x^3 + \frac{2}{x^3} - \frac{2}{3x^4} \\[/tex]. To find the first derivative, apply the product rule to the given function. Then, differentiate the first derivative to obtain the second derivative. Simplify each step carefully.
To find the first derivative of the given function [tex]y = \left( x^2 - \frac{7}{63}x \right) \left( x^4 + \frac{1}{x^3} \right)[/tex], we'll use the product rule, which states that if [tex]y = u(x) \cdot v(x)[/tex], then [tex]y' = u' \cdot v + u \cdot v'[/tex].
First, define u(x) and v(x) as following:
[tex]u(x) = x^2 - \frac{7}{63}x = x^2 - \frac{1}{9}x[/tex][tex]v(x) = x^4 + \frac{1}{x^3}[/tex]Compute u'(x):
[tex]u'(x) = 2x - \frac{1}{9}[/tex]Compute v'(x):
[tex]v'(x) = 4x^3 + (-3)x^{-4} = 4x^3 - \frac{3}{x^4}[/tex]Apply the product rule: [tex]y' = u' \cdot v + u \cdot v'[/tex]
Thus,
[tex]y' = \left(2x - \frac{1}{9}\right)\left( x^4 + \frac{1}{x^3} \right) + \left( x^2 - \frac{1}{9}x \right) \left( 4x^3 - \frac{3}{x^4} \right)[/tex]Simplify this expression step-by-step to find the first derivative.
[tex]y' = \left(2x - \frac{1}{9}\right)\left( x^4 + \frac{1}{x^3} \right) + \left( x^2 - \frac{1}{9}x \right) \left( 4x^3 - \frac{3}{x^4} \right)[/tex][tex]y'[/tex] [tex]&= \left(2x \cdot x^4 + 2x \cdot \frac{1}{x^3} - \frac{1}{9} \cdot x^4 - \frac{1}{9} \cdot \frac{1}{x^3} \right)[/tex][tex]&\quad + \ \left( x^2 \cdot 4x^3 - x^2 \cdot \frac{3}{x^4} - \frac{1}{9}x \cdot 4x^3 + \frac{1}{9}x \cdot \frac{3}{x^4} \right)[/tex][tex]y'[/tex] [tex]&= 2x^5 + \frac{2}{x^2} - \frac{1}{9}x^4 - \frac{1}{9x^3} \\[/tex] [tex]&\quad + \ 4x^5 - \frac{3}{x^2} - \frac{4}{9}x^4 + \frac{1}{3x^3}[/tex][tex]y'[/tex] [tex]&= 6x^5 - \frac{1}{x^2} - \frac{5}{9}x^4 + \frac{2}{9x^3}[/tex][tex]y'[/tex] [tex]&= 6x^5 - \frac{5}{9}x^4 - \frac{1}{x^2} + \frac{2}{9x^3}[/tex]To find the second derivative, differentiate the first derivative, carefully differentiating each term:
[tex]y'' &= \frac{d}{dx}\left( 6x^5 \right) - \frac{d}{dx}\left( \frac{5}{9}x^4 \right) - \frac{d}{dx}\left( \frac{1}{x^2} \right) + \frac{d}{dx}\left( \frac{2}{9x^3} \right) \\[/tex][tex]y''[/tex] [tex]&= 30x^4 - \frac{5}{9} \cdot 4x^3 - \left( -2x^{-3} \right) + \left( -\frac{2}{9} \cdot 3x^{-4} \right) \\[/tex][tex]y''[/tex] [tex]&= 30x^4 - \frac{20}{9}x^3 + \frac{2}{x^3} - \frac{2}{3x^4} \\[/tex]So, for the function [tex]y = \left( x^2 - \frac{7}{63}x \right) \left( x^4 + \frac{1}{x^3} \right)[/tex], we have:
First derivative [tex](y')[/tex] [tex]&= 6x^5 - \frac{5}{9}x^4 - \frac{1}{x^2} + \frac{2}{9x^3}[/tex]Second derivative [tex](y'')[/tex] [tex]&= 30x^4 - \frac{20}{9}x^3 + \frac{2}{x^3} - \frac{2}{3x^4} \\[/tex]