Rolles Theorem ---**Document Header Information:** * **Document ID:** RBBT2AEM20723 * **University Name:** ALLIANCE UNIVERSITY * *Annotation:* Private University established in Karnataka State by Act No 34 of year 2010 Recognised by the University Grants Commission (UGC), New Delhi * **College Name:** Alliance College of Engineering and Design * **Degree:** Bachelor of Technology * **Examination:** Semester - II, Re-Examination, July 2025 * **Course Code & Title:** 2BS1404, Engineering Mathematics-II * **Branches:** AE/ CE/ ME/ ECE/ EEE * **Date:** 23.07.2025 * **Time Allowed:** 2 Hrs. * **Maximum Marks:** 50 * **Registration Number:** Reg No.: .................... **Instructions to Students:** 1. This Question paper contains Part A, Part B, and Part C. 2. For the Part-A questions, an explanation to choose the answer should be recorded. 3. All questions are compulsory. 4. DO NOT use a pencil to mark your responses except for drawings/diagrams/sketches. 5. Attach question paper with the Answer Sheet. 6. Except registration number, students must not write anything on the question paper. 7. Assume suitable data if missing. --- **PART A (Multiple Choice Questions) - 5 questions x 2 marks each = 10 marks** **Question 1:** * **Question Stem:** Define singularity of an analytic function. * **(CO1, BTL-2)** **Question 2:** * **Question Stem:** Evaluate ∫₀^(π/2) cos⁶x dx * **(CO2, BTL-1)** **Question 3:** * **Question Stem:** If Φ = x²y + 2xz - 4, find ∇Φ at (2, -2, 3) * **(CO3, BTL-2)** **Question 4:** * **Question Stem:** Comment on the statement. "Orthogonal curvilinear coordinates are orthogonal". * **(CO4, BTL-2)** **Question 5:** * **Question Stem:** Define Hermitian matrix. * **(CO5, BTL-1)** --- **PART B (Short Answer Questions) - 4 questions x 5 marks each = 20 marks** **Question 6:** * **Question Stem:** Find the directional derivative of Φ = 4xz³ - 3x²y²z at (2, -1, 2) along 2î - 3ĵ + 6k * **(CO3, BTL3)** **Question 7:** * **Question Stem:** Show that F⃗ = (xî + yĵ)/(x² + y²) is solenoidal. * **(CO3, BTL3)** **Question 8:** * **Question Stem:** Find the scale factors of spherical polar coordinates. * **(CO4, BTL4)** **Question 9:** * **Question Stem:** Find the rank of the matrix A = ``` ⎡ 1 2 1 0 ⎤ ⎢-2 4 3 0 ⎥ ⎣ 1 0 2 8 ⎦ ``` * **(CO5, BTL3)** --- **PART C (Long Answer Questions) - 2 questions x 10 marks each = 20 marks** **Question 10:** * **Question Stem:** (a) Show that the cylindrical polar coordinates system is orthogonal. Express gradψ and divA⃗ in cylindrical polar coordinates where ψ and A⃗ are scalar and vectors respectively. * **(CO4, BTL -4)** --- **Footer:** * Page 1 of 2

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Rolle's Theorem is a fundamental result in calculus that establishes the existence of horizontal tangent lines under specific conditions. The theorem states that if a function f is continuous on a closed interval from a to b, differentiable on the open interval from a to b, and the function values at the endpoints are equal, then there exists at least one point c in the open interval where the derivative equals zero. This means there is at least one horizontal tangent line between the endpoints. Rolle's Theorem requires three specific conditions to be satisfied. First, the function must be continuous on the closed interval from a to b. Second, the function must be differentiable on the open interval from a to b. Third, the function values at the endpoints must be equal. Let's examine examples of functions that satisfy and violate these conditions. The green curve satisfies all conditions. The red curve shows a discontinuous function that violates the first condition. The orange curve has a corner point where it's not differentiable, violating the second condition. The purple curve has unequal endpoint values, violating the third condition. The geometric interpretation of Rolle's Theorem provides intuitive understanding of why the theorem works. When a continuous curve starts and ends at the same height, there must be at least one point where the tangent line is horizontal, meaning the slope equals zero. Let's examine different types of curves. The blue parabola has one horizontal tangent at its peak. The purple curve shows that multiple horizontal tangents are possible - this curve has two critical points. The pink sinusoidal curve demonstrates even more horizontal tangents, with three critical points between the endpoints. This geometric insight helps us understand that Rolle's Theorem guarantees at least one horizontal tangent, but there can be many more depending on the curve's shape. Let's work through a complete example using the function f of x equals x squared minus 4x plus 3 on the interval from 1 to 3. First, we check continuity. Since polynomial functions are continuous everywhere, f is continuous on the closed interval from 1 to 3. Second, we check differentiability. Polynomial functions are also differentiable everywhere, so f is differentiable on the open interval from 1 to 3. Third, we verify equal endpoints. f of 1 equals 1 minus 4 plus 3, which equals 0. f of 3 equals 9 minus 12 plus 3, which also equals 0. So f of 1 equals f of 3. Finally, we find the critical point. The derivative f prime of x equals 2x minus 4. Setting this equal to zero gives us 2c minus 4 equals 0, so c equals 2. The graph shows our parabola with the horizontal tangent line at x equals 2, confirming Rolle's Theorem. Rolle's Theorem serves as a foundation for many important mathematical concepts and real-world applications. It connects directly to the Mean Value Theorem, where Rolle's Theorem is actually a special case when the function values at the endpoints are equal. In practical applications, we use Rolle's Theorem in optimization problems to find maximum and minimum values, such as determining the highest point of a projectile's trajectory. The red curve shows projectile motion where the maximum height occurs where the derivative equals zero. Root-finding algorithms also rely on this theorem to locate solutions between known points. In engineering design, we use these principles to optimize performance and efficiency. The theorem extends to more advanced topics like Taylor's Theorem, the Intermediate Value Theorem, and the Fundamental Theorem of Calculus, making it a cornerstone of mathematical analysis.