Define the area under a curve and explain how to calculate it using integration. Find the area enclosed between the curve y equals x squared and the x-axis from x equal to 0 to x equal to 3.
Explain the difference between indefinite and definite integrals. Evaluate the integral of x squared dx from 0 to 2.
Describe the process of finding the maximum or minimum of a function using derivatives. Find the maximum or minimum value of f of x equal to 3x squared minus 6x plus 7.
Explain the concept of continuity of a function at a point. Is the function f of x equal to x squared plus 2x plus 1 continuous at x equal to 1?
Define the determinant of a 2×2 matrix and explain its significance. Find the determinant of a matrix with elements 2, 3 in the first row and 4, 5 in the second row.
Fill in the blanks:
(a) Growers of woad in Europe saw __________
as a crop which would provide competition
to their earnings.
(b) The demand for indigo increased in late
eighteenth-century Britain because of
__________.
(c) The international demand for indigo was
affected by the discovery of __________.
(d) The Champaran movement was against
__________.
Define a square matrix and explain the concept of the transpose of a matrix. Show that the transpose of a symmetric matrix is itself.
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