VTU 1BMATM101 Differential Calculus and Linear Algebra: Study Guide

1BMATM101 Differential Calculus and Linear Algebra is the Semester 1 mathematics course for the Mechanical Engineering stream at Visvesvaraya Technological University (VTU), Belagavi, under the 2025 scheme (effective from the academic year 2025-26). In the scheme it is the Applied Mathematics-I course, coded 1BMATx101, where the letter changes with your stream. This guide follows the official VTU syllabus PDF and maps every calculus topic to free practice.

Course at a glance

The syllabus gives 4 credits with a 3:2:0:0 teaching pattern: 3 lecture hours and 2 tutorial hours a week, 40 hours of theory plus 20 hours of tutorials in total. Assessment is 50 marks of Continuous Internal Evaluation (CIE) and 50 marks in a 3 hour Semester End Examination (SEE). No prerequisite is stated, so the course assumes PUC or Class 12 mathematics.

What 1BMATM101 covers

• Module 1, Polar curves and curvature: polar coordinates and polar curves, the angle between the radius vector and the tangent, the angle between two curves, pedal equations, and curvature and radius of curvature in Cartesian, parametric, polar and pedal forms.
• Module 2, Series expansion, indeterminate forms and multivariable calculus: Taylor's and Maclaurin's series in one variable, indeterminate forms and L'Hospital's rule, partial differentiation, the total derivative, differentiation of composite functions, Jacobians, and maxima and minima of functions of two variables.
• Module 3, First order ODEs: linear and Bernoulli equations, exact equations and equations reducible to exact form with an integrating factor, orthogonal trajectories, and the law of natural growth and decay.
• Modules 4 and 5, Linear algebra: row echelon form and rank, inverse by Jordan's method, Gauss elimination, LU decomposition, Gauss-Seidel, an application to traffic flow, eigenvalues and eigenvectors, Rayleigh's power method, diagonalisation, inverse by the Cayley-Hamilton theorem, and the Moore-Penrose pseudoinverse.

1BMATM101 topic map

PracticeCalculus focuses on calculus, so Modules 4 and 5 are not in the table.

Example problems at the 1BMATM101 level

These are original practice problems, not VTU question paper items.

1. Indeterminate form. \displaystyle\lim_{x \to 0}\frac{1 - \cos x}{x^2} = \lim_{x \to 0}\frac{\sin x}{2x} = \frac{1}{2}, applying L'Hospital's rule once and then the standard limit \frac{\sin x}{x} \to 1.

2. Maclaurin series. \ln(1 + x) = x - \frac{x^2}{2} + \frac{x^3}{3} - \cdots for -1 < x \le 1.

3. A linear ODE. Solve y' + y = e^{x}. The integrating factor is e^{x}, so (ye^{x})' = e^{2x} and y = \frac{1}{2}e^{x} + Ce^{-x}.

4. Radius of curvature. For y = x^2 at the origin, \rho = \frac{(1 + y'^2)^{3/2}}{|y''|} = \frac{1}{2}.

How to do well in 1BMATM101

• Use all 20 tutorial hours. The 3:2 pattern means almost half your contact time is problem solving. Bring attempted problems, not blank pages.
• Learn the polar formulas with pictures. Draw the radius vector and tangent each time; the angle formulas in Module 1 make sense once you see them.
• Drill partial derivatives until they are automatic, because Jacobians, total derivatives and maxima and minima all build on them. Try the partial derivatives practice set.
• Practise in short daily sessions on practice, and look ahead to 1BMATM201, which adds multiple integrals, higher order ODEs and vector calculus.