Determine the concavity of y = 3 + sin x on [0, 2$\pi$]. [5]

Define implicit differentiation and find the slope of the circle $ x^2 + y^2 = 25$ at the point (3, -4). [5]

State Rolle's Theorem and show that $x^3 + 3x + 1 = 0$ has exactly one real solution. Find the area of the region enclosed by the parabola $y = 2 - x^2$ and the line y = -x. [5+5]

Define absolute value function and Sketch the graph of absolute value. [5]

Find the Taylors Series generated by $f(x) = \frac{1}{x}$ at a = 2. where, if anywhere, does the series converge to $\frac{1}{x}$? [5]

Test for convergence of the series $\sum_{n=1}^{\infty}\left(\frac{1}{n+1}\right)^n$. [5]

Evaluate $\int_{0}^{\frac{\pi}{4}} \frac{dx}{1-\sin x}$, Evaluate $\int \ x^2 sin x dx$. Solve the differential equation $\frac{dy}{dx} - \frac{3y}{x} = x$, x > 0. [5+5]

State integral test and apply it to test the convergence of the series $\sum_{n=1}^{\infty}\frac{1}{n^2+1}$. [5]

Find the limit of $\lim_{h\to\infty} \frac { \sqrt {6h+25-5} }{h^2}$. [5]

Define gradient of vector function of f(x,y,z) and find the derivative of f(x,y,z) = $x^3 - xy^2 + z$ at p(1,0,0) in the direction of $\vec{v}$ = 2$\vec{i}$ - j + $\vec{k}$. Define Volume of the solid and find the volume of the solid generated by revolving the region bounded by $y = \sqrt{x}$ and the line y = 1, x = 4 about the line y = 1. [5+5]

Define partial derivative and find the value of $\frac{\partial f}{\partial x} \& \frac{\partial f}{\partial y}$ at the point (4, -5) if $f(x, y) = x^3 + 3xy + y - 1$. [5]

Evaluate $\int_0^{\frac{\pi}{2}}(\sin 2x\cos 3x + \cos 2x\sin 3x)dx$ and $\int_{0}^{\frac{\pi}{4}} \frac{dx}{1 - \sin x}$. [5]