Exercise 2.6 Class 12 Math Solutions - Unit 2 Further Differentiation, Punjab Board
Step-by-step solutions to Exercise 2.6, Unit 2: Further Differentiation, for Class 12 Punjab Board students. Covers differentials of independent and dependent variables, using dy to approximate changes in y, and computing absolute, relative, and percentage errors in physical and geometric measurements, worked out question by question.
Exercise Overview
Exercise 2.6 of Class 12 Mathematics focuses on the theory and application of Differentials and Error Analysis under the Punjab Board PCTB / PECTAA curriculum. While derivatives represent instantaneous rates of change, differentials isolate the principal linear part of the change in a dependent variable. For a differentiable function (y = f(x)), the differential of the independent variable is defined as (dx = Δx), and the differential of the dependent variable is given by (dy = f'(x)·dx).
A major emphasis in this exercise is understanding the distinction between the exact increment (Δy = f(x + Δx) - f(x)) and the differential estimate (dy). Students practice using (dy) to approximate function changes and calculate numerical estimations for small increments (Δx). Furthermore, the exercise addresses practical error propagation in physical measurements-such as estimating errors in the surface area and volume of spheres, cubes, and circular cylinders resulting from small measurement errors in radius or side length-formulating absolute error (|Δy - dy|), relative error (dy / y), and percentage error ((dy / y) x 100%).
For Punjab Board HSSC Part 2 examinations across all nine educational divisions (Lahore, Rawalpindi, Gujranwala, Faisalabad, Multan, Bahawalpur, Sargodha, Sahiwal, and D.G. Khan), Exercise 2.6 yields crucial short questions, conceptual MCQs, and applied word problems. These step-by-step solutions provide clean variable identification, explicit derivative steps, and systematic error comparisons for every problem.