Find the Maclaurin series for $e^x$ and prove that it represents $e^x$ for all x. Define initial value problem. Solve that initial value problem of $y' + 5y = 1$, $y(0) = 2$. Find the volume of a sphere of radius r. [4+4+2]

Verify Mean value theorem of $f(x) = x^3 - 3x + 3$ for [-1,2]. [5]

Sketch the curve $y = x^3 + x$. [5]

Find the length f the arc of the semicubical $y^2 = x^2$ between the points (1,1) and (4,8). [5]

Find the extreme values of $f(x, y) = y^2 - x^2$. [5]

Find the derivative of State the domain of f. Estimate the area between the curve and the line x=0 and x=2 where curve is

$y^2 = x$

[3+2+5]

A function is defined by Calculate f(-1), f(3), and sketch the graph. Prove that the limit does not exist.

$\lim_{x \to 0} \frac{|x|}{x}$

[5+0+5]

If $f(x) = \sqrt{x}$ and $g(x) = \sqrt{3-x}$, find $gof$ and $fog$. [5]

Define limit of a function. [5]

For what values of x does the series converge? Calculate $\iint_{R} f(x, y) dA$, for $f(x, y) = 100 - 6x^2y$, and $R: 0 \leq x \leq 2, -1 \leq y \leq 1$. [5+5]

Determine whether the integral is convergent or divergent. [5]

Test the convergence of the series [5]

Use continuity to evaluate the limit, [5]

Find the solution of $y'' + 6y' + 9 = 0$, $y(0) = 2$, $y'(0) = 1$. [5]

Define cross product of two vectors. If $\vec{a} = \hat{i} + 3\hat{j} + 4\hat{k}$ and $\vec{b} = 2\hat{i} + 7\hat{j} - 5\hat{k}$, find the vector $\vec{a} \times \vec{b}$ and $\vec{b} \times \vec{a}$. [5]