One variable calculus for non majors questions.

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#	Question
1	Using Newton`s Raphson method approximate to two decimal places, the root of equation $e^{x}=3x$ lying between 0 and 1. Mathematical Calculation
2	Determine the radius and interval of convergence of the $sum_{k=0}^{infty }frac{10^k}{k!}(3x+1)^{k+1}$ Mathematical Calculation
3	Determine whether the sequence is divergent or convergent $left { n sinleft ( frac{pi }{n} ight ) ight }$ Mathematical Calculation
4	Find the aarea of the region bounded by line y=x-1 and parabola $y^{2}=2x+6$ . Mathematical Calculation
5	The region bounded by thhe curve y=x and $y=x^{2}$ is rotated about the y-axis. Find the vvolume of the resuulting solid. Mathematical Calculation
6	Solve the differrential equation $frac{dy}{dx}=frac{2x+y-1}{4x+2y-4}$ . Mathematical Calculation
7	Use the method of Frobenius to detemine the inducial equation of $2x^{2}frac{d^{2}y}{dx^{2}}-xfrac{dy}{dx}+(1-x)y=0$ . Mathematical Calculation
8	Sketch and find the Fourier series expansion of the function $f(x)$ given by $f(x)=left{egin{matrix} -k ,&-pi <x<0 \ k, & 0<x<pi end{matrix} ight.$ With $f(x+2pi)=f(x)$ If in the Fourier series of the questions (a) above , $k>0$ deduce a series for $frac{pi}{4}$ at the point $x=frac{pi}{2}$ . Mathematical Calculation
9	Apply Euler’s modified method to solve $frac{dy}{dx}=x+3y$ subject to $y(0)=1$ and find the approximate value of y when x=0.3 , taking the interval 0.1. Mathematical Calculation
10	Find the root of the equation $2x-log_{10}x=7$ whcich lies between 3.5 and 4 correct to two decimal places, using method of false position. Mathematical Calculation