(a) If \(f(x) = \frac{1}{x+3}\) and \(g(x) = \frac{x}{x-2}\), then compute \(f \circ g\) and \(g \circ f\). Also, find the domains of \(f \circ g\) and \(g \circ f\).
(b) State Mean Value Theorem with geometrical interpretation. Verify the Mean Value Theorem for the function \(f(x) = 1 - x^2\) on \([0, 2]\).
(a) Find the area of the region that lies in the plane enclosed by the cardioid \(r = 2(1 + \cos \theta)\).
(b) Show that every member of the family of functions \(y = \frac{1+ce^t}{1-ce^t}\) is a solution of the differential equation \(y' = \frac{1}{2}(y^2 - 1)\).
(a) Find the Maclaurin series for the function \(f(x) = \sin x\) and prove that the series represents \(\sin x\) for all \(x\).
(b) Find unit tangent, unit normal and bi-normal vectors for the circular helix: \(\vec r(t) = \cos t\,\vec i + \sin t\,\vec j + t\,\vec k\).
Attempt any Eight questions
[8×5=40]If \(f(x) = 2x^2 - 5x + 1\) and \(h \ne 0\), then evaluate: \(\frac{f(a+h)-f(a)}{h}\).
Compute: \(\lim_{x \to 0+}(1 + \sin 4x)^{\cot x}\).
Find \(d^2y/dx^2\) if \(x^4 + y^4 = 16\).
A particle moves along the line so that its velocity at a time \(t\) is \(v(t) = t^2 - t - 6\) (measured in per second).
(a) Find the displacement of the particle during the time period \(1 \le t \le 4\).
(b) Find distance traveled during this time period.
Solve the initial value problem: \(y'' + y' - 6y = 0\), \(y(0) = 1\), \(y'(0) = 0\).
Find the sum of the series: \(\sum_{n=1}^{\infty}(\frac{3}{n(n+1)} + \frac{1}{2^n})\).
Find the equation of a plane that passes through the points: \(P(1,3,2)\), \(Q(3,-1,6)\) and \(R(5,2,0)\).
Find absolute maximum and minimum values of the function: \(f(x,y) = x^2 - 2xy + 2y\) on the rectangle \(R = \{(x,y): 0 \le x \le 2, 0 \le y \le 2\}\).
Evaluate: \(\int_1^{\log 8}\int_0^{\log x} e^{x+y}\,dy\,dx\).