Section A
Attempt any TWO questions.
(a) Define composition of two functions. Let \(f(x) = \sqrt{x}\) and \(g(x) = \sqrt{3 - x}\). Find \((f \circ f)(x)\), \((g \circ g)(x)\), \((g \circ f)(x)\) and \((f \circ g)(x)\).
(b) Evaluate: \(\lim_{x \to \infty} \sqrt{x}(\sqrt{x} - \sqrt{x - 5a})\).
(a) Find the local maximum and local minimum value of the function \(f(x) = 4x^3 - 6x^2 - 9x + 17\).
(b) Examine the continuity of the function \(f(x) = \sqrt{9 - x^2}\) on the interval \([-3, 3]\).
(a) Using method of integration, find the area enclosed by the circle \(x^2 + y^2 = 16\).
(b) Evaluate: \(\int_0^1 x^3\sqrt{1 + 2x^4}\, dx\).
Attempt any Eight questions
[8×5=40]Section B
Attempt any EIGHT questions.
Sketch the function \(y = x - x^2\) indicating its different characteristics. Also, find the range of function.
Verify Mean value theorem for the function \(f(x) = (x-1)(x-2)(x-3)\) in \([1, 4]\).
Using L'Hospital's rule, evaluate: \(\lim_{x \to 1}\left(\dfrac{x}{x-1} - \dfrac{1}{\ln x}\right)\).
Find \(\dfrac{d^2y}{dx^2}\) as a function of \(t\) when \(x = t - \dfrac{1}{t}\), \(y = t + \dfrac{1}{t}\).
Find the area of surface swept out by revolving the circle \(x^2 + y^2 = 1\) about x-axis.
Examine the convergence of \(\displaystyle\sum_{n=1}^{\infty} \dfrac{2n^2 + 3n}{\sqrt{5 + n^5}}\).
Solve: \(xy' + 2y = 3\), \(y(1) = 1\).
Find the Maclaurin series of the function \(f(x) = e^x\) and its radius of convergence.
Find all second order partial derivatives when \(f(x, y) = x^4 - 5x^3y^2 + 7x^2y^3 + 4xy^5\).