Find the area of the surface generated by revolving the curve $y = 2\sqrt{x}$, $1 \leq x \leq 2$, about x-axis. [5]

Determine the concavity and find the inflection point of the function $f(x) = x^3 - 3x^2 + 2$ [5]

Find $\frac{dy}{dx}$ for: $Y = 2u^3$, $u = 8x - 1$, $Y = \sin u$, $u = x - x\cos x$ [5]

Find the initial value problem in $\frac{dy}{dx} + 2y = 3$, $y(0) = 1$. [5]

What is even and odd function? Give example of each and write their symmetricity. Find the domain and range of the following functions.

$f(x) = \sqrt{5x + 10}$

$f(x) = \frac{x^2 - 3x - 4}{x+1}$

[6+4]

Sketch the graph of the function $f(x) = x^2$. Shifted vertically up to 1 and -2 units and horizontally up to 3 and -2 units. Find the $\delta$ algebraically for the following functions.

$\lim_{x \\to 5} \sqrt{x-1}, \text{and } L = 2, \epsilon = 1$

$\lim_{x \\to 2} (2x - 2), \text{and } L = 6, \epsilon = 0.02$

[5+5]

Find the Taylor's series generated by $f(x) = \frac{1}{x}$ at $a = 2$. Where, if anywhere, does the series converge to $\frac{1}{x}$? [10]

Determine the convergence or divergence of the series $\sum_{n=1}^{\infty} {n^2}{e^{-n}}$. [5]

Integrate the following: $\int_{0}^\frac{x}{4} \frac{dx}{1-\sin x}$ [5]

Show that $f(x) = \frac{x^2+x-6}{x^2-4}$, $x \neq 2$ has a continuous extension to $x = 2$ [5]

Evaluate the following: $\lim_{x \to \infty} (x-\sqrt{x^2 + 16})$ $\lim_{x \to 1} \left(\frac{\sqrt{6x+10}-5}{x^2}\right)$ [2.5+2.5]

Find the derivatives of the function $f(x, y) = x^3 - xy^2 + x^2y - y^3$ at the point $p_0(5, 5)$ in the direction of $\vec{u} = 4\vec{i} + 3\vec{j}$. [5]