Podcast
Questions and Answers
What method can be used to find the dimension of the box that requires the least material for construction?
What method can be used to find the dimension of the box that requires the least material for construction?
- Simplex Method
- Newton's Method
- Gradient Descent
- Lagrange’s Method of Multipliers (correct)
How can the Jacobian be defined in the coordinate transformation from spherical to Cartesian coordinates?
How can the Jacobian be defined in the coordinate transformation from spherical to Cartesian coordinates?
- It is irrelevant in the transformation process.
- It relates only to cylindrical coordinates.
- It is derived from the derivatives of the transformation equations. (correct)
- It is constant across all transformations.
What is the relationship between the variables u and v given by the equation x, y, z?
What is the relationship between the variables u and v given by the equation x, y, z?
- u and v can be derived from independent variables.
- u is a function of v only.
- u and v are functionally independent.
- u and v are not functionally independent. (correct)
Given the equation ( a + k, b + k, c + k ) resulting in roots ( \lambda, \mu ), what does the equation signify?
Given the equation ( a + k, b + k, c + k ) resulting in roots ( \lambda, \mu ), what does the equation signify?
If the radius of a balloon is increased by 0.01 m and the length by 0.05 m, how would this affect the percentage change in volume?
If the radius of a balloon is increased by 0.01 m and the length by 0.05 m, how would this affect the percentage change in volume?
What is the second degree term of the expansion of $e^{a ext{sin } x}$ using Maclaurin’s theorem?
What is the second degree term of the expansion of $e^{a ext{sin } x}$ using Maclaurin’s theorem?
What is the method used to find the maximum and minimum value of the function $x^3 + y^3 - 3axy$?
What is the method used to find the maximum and minimum value of the function $x^3 + y^3 - 3axy$?
For the function $tan^{-1}(x)$, at what value of $x$ is $f(1.1, 0.9)$ calculated when expanded up to second degree terms?
For the function $tan^{-1}(x)$, at what value of $x$ is $f(1.1, 0.9)$ calculated when expanded up to second degree terms?
Which statement is true about the shape of a rectangular solid inscribed in a sphere for maximum volume?
Which statement is true about the shape of a rectangular solid inscribed in a sphere for maximum volume?
What is the shortest distance from the point $(1, 2, -1)$ to the surface described by $x^2 + y^2 + z^2 = 24$?
What is the shortest distance from the point $(1, 2, -1)$ to the surface described by $x^2 + y^2 + z^2 = 24$?
Which mathematical concept is primarily applied to solve the problem of finding the maximum volume rectangular solid?
Which mathematical concept is primarily applied to solve the problem of finding the maximum volume rectangular solid?
What does the term 'given capacity' refer to in the context of the problem about the rectangular box?
What does the term 'given capacity' refer to in the context of the problem about the rectangular box?
Which theorem is applicable for expanding functions like $tan^{-1}(x)$ in a neighborhood?
Which theorem is applicable for expanding functions like $tan^{-1}(x)$ in a neighborhood?
Flashcards
Lagrange's multipliers
Lagrange's multipliers
A method used to find the maximum or minimum values of a function subject to constraints.
Jacobian
Jacobian
A matrix of all first-order partial derivatives of a vector-valued function.
Functionally independent
Functionally independent
Two or more functions are functionally independent if none of them can be expressed as a function of the others.
Approximation
Approximation
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Percentage change
Percentage change
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Maclaurin's Theorem
Maclaurin's Theorem
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Expand e^(−1/x)sin(x)/y
Expand e^(−1/x)sin(x)/y
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Expand tan⁻¹(x)
Expand tan⁻¹(x)
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Find f(1.1, 0.9)
Find f(1.1, 0.9)
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Max/Min of x³ + y³ - 3axy
Max/Min of x³ + y³ - 3axy
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Rectangular Solid in Sphere
Rectangular Solid in Sphere
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Shortest Distance from point to surface
Shortest Distance from point to surface
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Rectangular Box Capacity
Rectangular Box Capacity
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Study Notes
Engineering Mathematics I - Tutorial Sheet T3
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Question 1: Expand e^(x sin⁻¹ x) using Maclaurin's theorem.
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Question 2: Expand tan⁻¹(x/y) in the neighborhood of (1,1) up to second degree terms. Calculate f(1.1, 0.9).
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Question 3: Find the maximum and minimum values of x³ + y³ - 3axy.
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Question 4: Prove that a rectangular solid with maximum volume inscribed within a sphere is a cube.
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Question 5: Find the shortest distance from point (1, 2, -1) to the surface x² + y² + z² = 24.
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Question 6: A rectangular box, open at the top, has a given capacity. Determine its dimensions using Lagrange multipliers to minimize material usage.
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Question 7: Calculate the Jacobian (∂(x, y, z)/∂(r, θ, φ)) given x = r sin θ cos φ, y = r sin θ sin φ, z = r cos θ.
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Question 8: Show that u = (x-y)/(x+z), v = (x+z)/(y+z) are not independent and find the relation between them.
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Question 9: If λ, μ, ν are the roots of the equation x³/a + y²/b + z²/c = 1, then prove that (x,y,z) = (λ-μ)(μ-ν)(ν-λ) / (a-b)(b-c)(c-a).
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Question 10a: Calculate an approximate value of [(3.82)² +2(2.1)]³.
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Question 10b: A balloon has a right circular cylindrical body with hemispherical ends. If the radius is 1.5 m and length 4 m, and the radius increases by 0.01 m and length by 0.05 m, determine the percentage change in volume.
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