Create a question paper of engineering mathematics containing questions from the topic of limits, continuity, and differentiability of two variables, Taylor series of one and two v... Create a question paper of engineering mathematics containing questions from the topic of limits, continuity, and differentiability of two variables, Taylor series of one and two variables, Jacobian, and maxima minima of two or three variables. The paper must contain 8 questions well skewed.
Understand the Problem
The question is asking for the creation of a question paper for engineering mathematics. It specifically requests questions from various topics such as limits, continuity, differentiability of two variables, Taylor series, Jacobians, and optimization (maxima and minima) for two or three variables. The user wants the paper to include a total of 8 questions that are well distributed among these topics.
Answer
A question paper with 8 well-skewed questions on limits, continuity, differentiability, Taylor series, Jacobian, and maxima-minima of functions of two or three variables.
[{"question":"Evaluate the limit of the function f(x, y) = (x^2y) / (x^2 + y^2) as (x, y) approaches (0,0).","topic":"Limits and Continuity"},{"question":"Verify the continuity of the function f(x, y) = sqrt(x^2 + y^2) at the point (1, 1).","topic":"Continuity"},{"question":"Determine whether the function f(x, y) = x^2y - y^3 is differentiable at the point (1,1).","topic":"Differentiability"},{"question":"Find the Taylor series expansion of the function f(x) = e^x at x = 0 up to the fourth term.","topic":"Taylor Series - One Variable"},{"question":"Obtain the Taylor series expansion of the function f(x, y) = sin(xy) at the point (0,0) up to the second degree terms.","topic":"Taylor Series - Two Variables"},{"question":"Calculate the Jacobian determinant of the transformation given by u = x + y, v = x - y.","topic":"Jacobian"},{"question":"Find the local maxima and minima of the function f(x, y) = x^3 - 3xy^2.","topic":"Maxima and Minima - Two Variables"},{"question":"Identify the critical points and determine their nature for the function f(x, y, z) = x^2 + y^2 + z^2 - 2xz - 4y.","topic":"Maxima and Minima - Three Variables"}]
Answer for screen readers
[{"question":"Evaluate the limit of the function f(x, y) = (x^2y) / (x^2 + y^2) as (x, y) approaches (0,0).","topic":"Limits and Continuity"},{"question":"Verify the continuity of the function f(x, y) = sqrt(x^2 + y^2) at the point (1, 1).","topic":"Continuity"},{"question":"Determine whether the function f(x, y) = x^2y - y^3 is differentiable at the point (1,1).","topic":"Differentiability"},{"question":"Find the Taylor series expansion of the function f(x) = e^x at x = 0 up to the fourth term.","topic":"Taylor Series - One Variable"},{"question":"Obtain the Taylor series expansion of the function f(x, y) = sin(xy) at the point (0,0) up to the second degree terms.","topic":"Taylor Series - Two Variables"},{"question":"Calculate the Jacobian determinant of the transformation given by u = x + y, v = x - y.","topic":"Jacobian"},{"question":"Find the local maxima and minima of the function f(x, y) = x^3 - 3xy^2.","topic":"Maxima and Minima - Two Variables"},{"question":"Identify the critical points and determine their nature for the function f(x, y, z) = x^2 + y^2 + z^2 - 2xz - 4y.","topic":"Maxima and Minima - Three Variables"}]
More Information
This sample paper covers critical areas in engineering mathematics, providing a well-rounded assessment of a student's understanding.
Tips
When finding Taylor series expansions, ensure the correct formula and calculate all necessary derivatives accurately.
Sources
- PDF on Jacobians, Taylor's Series, Maxima & Minima - dbscience.org
- Engineering Mathematics Notes - Vidyalankar Classes - vidyalankar.org
- Limits, Continuity and Differentiability | GeeksforGeeks - geeksforgeeks.org
