Mathematics I TU Board 2074 question paper

A function is defined by f(x) = $\{\begin{array}{c}x+2ifx<0\\ 1–xifx>0\end{array}$, Concluate f(-1), f(3) and sketch graph.

Estimate the area between the curve y^2 = x and the lines x = 0 and x = 2.

Find the Maclaurin series for e^x and prove that it represents e^x for all x.

For What values of x does the series ${\sum }_{n=1}^{\infty }\frac{(x–3{)}^n}{x}$ converge?

Prove that ${\lim }_{x\to 0}\frac{|x|}{x}$ the does not exist.

Define Initial Value Problem. Solve that value problem of $y^2+5y=1$, y(0) = 2

Find the volume of a sphere of radius r

Calculate ${\int }_R\int f(x,y)dA$ for f(x,y) = 100 – 6x^2y and $R:0\le x\le 2,-1\le y\le 1$

If $f(x)=\sqrt{x}$ and $g(x)=\sqrt{3-x}$, find gof and gog.

Use Continuity to evaluate the limit, ${\lim }_{x\to 4}\left(\frac{5+\sqrt{x}}{\sqrt{5+x}}\right)$

Verify Mean Value Theorem by f(x) = x^3 – 3x + 3 for [-1, 2]

Sketch the curve y = x^3 + x

Determine whether the integral ${\int }_1^{\infty }\left(\frac{1}{x}\right)dx$ is convergent or divergent

Find the length of the arc of the semi cubical parabola y^2 = x^2 between the points (1,1) and (4,8)

Find the solution of y"+6y'+9=0, y(0)=2, y'(0)=1

Test the convergence of the series ${\sum }_{n=1}^{\infty }\left(\frac{n^n}{n!}\right)$

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

Define limit of a function. Find limit ${\lim }_{x\to \infty }(x–\sqrt{x})$

Find the extremes values of f(x, y) = y^2 – x^2