Practice • Improve • Repeat
Welcome to Part 22, the finale of Module 5, updated for the UPSC ISS 2026 to 2027 cycle. Across the last four parts we built the entire integration toolkit, from straight lines to parabolas to Weddle’s precision. Knowing the formulas is half the battle. The other half: knowing when you never need them. A quadrature … Read more
Welcome back to Part 21 of Module 5, updated for the UPSC ISS 2026 to 2027 cycle. Simpson’s 1/3 Rule upgraded straight lines to parabolas, but it locked us into an even number of intervals. What if the data arrives in groups of 3 or 6? Two formulas answer: Simpson’s 3/8 Rule and the precision … Read more
Welcome back to Part 20 of Module 5, the single most important part of this module, updated for the UPSC ISS 2026 to 2027 cycle. The Trapezoidal Rule draws straight lines and pays for it with an O(h2)O(h^2) error. The upgrade: replace straight lines with parabolas. That upgrade is Simpson’s 1/3 Rule, and hidden inside … Read more
Welcome back to Part 19 of Module 5, updated for the UPSC ISS 2026 to 2027 cycle. In Part 18 we met the mother of all integration rules and watched n=1n = 1 give birth to its first child. Today we study that child in full depth: the Trapezoidal Rule, and more importantly, the concept … Read more
Welcome back to Part 18 of Module 5, updated for the UPSC ISS 2026 to 2027 cycle. Part 17 taught us rates of change from tables. Today we learn accumulated areas from tables: Numerical Integration, and it all begins with one master equation, the General Quadrature Formula, also called the Newton Cotes Quadrature Formula. Numerical … Read more
Welcome to Part 17 and the opening of Module 5, updated for the UPSC ISS 2026 to 2027 cycle. With Modules 1 to 4 behind you, interpolation holds no fear. Now we enter the highest weightage block of the Numerical Analysis unit: calculus without calculators, beginning with Numerical Differentiation. Numerical differentiation evaluates the derivative of … Read more
Welcome to Part 16, the final part of Module 4, updated for the UPSC ISS 2026 to 2027 cycle. So far the problem was always the same: given the arguments xx and their entries yy, predict a missing yy. That is direct interpolation. Today the examiner flips the script: given a value of yy, find … Read more
Welcome back to Part 15 of Module 4, updated for the UPSC ISS 2026 to 2027 cycle. In Part 14 we met Newton’s Divided Difference formula. Powerful, but it still needs a table. Today we go table free with the formula the syllabus names as “Lagrange’s formula for unequal intervals”: Lagrange’s Interpolation Formula, the examiner’s … Read more
Welcome back to Part 14 of Module 4, updated for the UPSC ISS 2026 to 2027 cycle. In Part 13 we defined divided differences and discovered their symmetry. But a single difference is not a prediction. To actually build a polynomial and estimate a missing value from scattered data, we need the most versatile weapon … Read more
Welcome to Part 13 and the opening of Module 4, updated for the UPSC ISS 2026 to 2027 cycle. In Modules 1 to 3 we lived in a perfect world where every formula, from Newton Gregory to Stirling and Bessel, demanded equally spaced arguments. Real data is rarely that polite. When the intervals turn unequal, … Read more