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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?
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?
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?
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?
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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?
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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?
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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 $x^3 + y^3 - 3axy$?
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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?
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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?
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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 $x^2 + y^2 + z^2 = 24$?
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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?
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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?
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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?
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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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Description
This quiz covers various topics in Engineering Mathematics I, including Maclaurin expansions, optimization problems, and the Jacobian calculation. Dive into concepts like maximum volume for inscribed solids and the application of Lagrange multipliers. Test your skills with problems designed to challenge your understanding of calculus and multivariable functions.
