Define Newton's-Raphson method with their formula. An open top box is to be made by cutting small congruent squares from the corners of square sheet of tin having length 12 inch. and is bending up the sides. How large should the squares cut from the corners be to make the box hold as much as possible? [2+8]

State Mean value theorem. Verify the mean value theorem if $f(x) = x^2 + 2x - 1$ on $[0,1]$. [1+4]

Find the equations of tangent and normal to the curve $x^3 + y^3 - 9xy = 0$ at the point $(2,4)$. [5]

Find $\frac{dy}{dx}$ if $y^2 - x^2 = \sin x$. Find the slope of circle $x^2 + y^2 = 25$ at the point $(3,4)$. [2.5+2.5]

Solve the following differential equation: $x \frac{dy}{dx} = x^2 + 3y$, $x > 0$. [5]

Graph the following functions. Write their symmetricity and specify the interval over which the function is increasing and decreasing. $y = -x^3$ , $y = x^2$ [5]

Define integral test. Determine the convergence or divergence of the series $\sum_{n=1}^{\infty} \frac{1}{n^2+4}$. [1+4]

Define area between two curves. Find area of the region enclosed by the parabola $y=2-x^2$ and the line $y= -x$. Define volume integral. Find the volume of solid generated by revolving the region bounded by the curve $y^2=x$ and the line $y=1$, $x=4$ about the line $y=1$. [1+3+6]

Define integration. Evaluate the following integral. $\int \frac{dx}{(x-1)(x-2)}$, $\int_{-1}^{1} 3x^2 \sqrt{x^3 + 1} \,dx$ [1+4]

What do you mean by asymptotes? How many types of asymptotes define each? Find horizontal and vertical asymptotes of the following function: $f(x) = -\frac{-8}{x^2-4}$ Does there other asymptotes exist? [4+5+1]

What is L'Hospital's rule? Using this rule evaluate the following: $\lim_{x \to 0} (\sec x)^{\frac{1}{x^2}}$, $\lim_{x \to 0}\frac{x-\sin x}{x^3}$ [1+4]

Find the derivative of $f(x, y, z) = x^3 - xy^2 - z$ at point $P(1, 1, 0)$ in the direction of $v = 2i - 3j + 6k$. [5]