Q1 Expand \[\sinh x\] by using Maclaurin series
\[x+\frac{x^{3}}{3!}+\frac{x^{5}}{5!}+\frac{x^{7}}{7!}+\cdots\]
\[1+x+\frac{x^{2}}{2!}+\frac{x^{2}}{2!}+\frac{x^{3}}{3!}+\cdots\]
\[x-\frac{x^{3}}{3!}+\frac{x^{5}}{5!}-\frac{x^{7}}{7!}+\cdots\]
\[1-x-\frac{x^{2}}{2!}-\frac{x^{2}}{2!}-\frac{x^{3}}{3!}-\cdots\]

Q2 Expand \[\cos x\] by using Maclaurin series
\[1+x+\frac{x^{2}}{2!}+\frac{x^{2}}{2!}+\frac{x^{3}}{3!}+\cdots\]
\[1-x-\frac{x^{2}}{2!}-\frac{x^{2}}{2!}-\frac{x^{3}}{3!}-\cdots\]
\[1+\frac{x^{2}}{2!}+\frac{2x^{4}}{4!}+\frac{6x^{6}}{6!}+\cdots\]
\[1-\frac{x^{2}}{2!}+\frac{2x^{4}}{4!}-\frac{6x^{6}}{6!}+\cdots\]

Q3 Give the first few terms of \[\sin x\] using Maclaurin series
\[x+\frac{x^{3}}{3!}+\frac{x^{5}}{5!}+\frac{x^{7}}{7!}+\cdots\]
\[1+x+\frac{x^{2}}{2!}+\frac{x^{2}}{2!}+\frac{x^{3}}{3!}+\cdots\]
\[x-\frac{x^{3}}{3!}+\frac{x^{5}}{5!}-\frac{x^{7}}{7!}+\cdots\]
\[1-x-\frac{x^{2}}{2!}-\frac{x^{2}}{2!}-\frac{x^{3}}{3!}-\cdots\]

Q4 The product of \[e^{2x}\] and \[e^{-x}\] can be written as __________________
\[1+x+\frac{x^{2}}{2!}+\frac{x^{2}}{2!}+\frac{x^{3}}{3!}+\cdots\]
\[1-x-\frac{x^{2}}{2!}-\frac{x^{2}}{2!}+-frac{x^{3}}{3!}-\cdots\]
\[x+\frac{x^{3}}{3!}+\frac{x^{3}}{3!}+\frac{x^{4}}{4!}+\cdots\]
\[x-\frac{x^{3}}{3!}-\frac{x^{3}}{3!}-\frac{x^{4}}{4!}-\cdots\]

Q5 Find limit \[\lim_{(x, y, z)\rightarrow (1, 2, 5)} \sqrt(x+y+z)\]
2
3
\[2\sqrt(2)\]
\[2\sqrt(2)\]

Q6 Find the limit of \[\lim_{(x, y)\rightarrow(2,4)} \frac{x+y}{x-y}\]
1
2
6
-3

Q7 Find the limit of \[\lim_{(x, y)\rightarrow (2, 1)} x+3y^{2}\]
4
5
3
7

Q8 The gradient of the tangent at any point (x,y) of the conic \[f(x,y)=ax^{2}+2hxy+by^{2}+2gx+2fy+c=0\]
\[\frac{dy}{dx}=-\frac{2ax+2hy+2g}{2by+2hx+2f}\]
\[\frac{dy}{dx}=\frac{2ax+2hy+2g}{2by+2hx+2f}\]
\[\frac{dy}{dx}=-\frac{2ax-2hy-2g}{2by+2hx+2f}\]
\[\frac{dy}{dx}=--\frac{2ax+2hy+2g}{2by-2hx+2f}\]

Q9 Given the function \[f(x,y)=\tan^{-1} \frac{y}{x}\], find \[f_{yy}\]
\[f_{xy}=-\frac{2xy}{(x^{2}-y^{2})^{2}}\]
\[f_{xy}=-\frac{2xy}{(x^{2}+y^{2})^{2}}\]
\[f_{xy}=\frac{2xy}{(x^{2}+y^{2})^{2}}\]
\[f_{xy}=\frac{2xy}{(x^{2}-y^{2})^{2}}\]

Q10 Given the function \[f(x,y)=\tan^{-1} \frac{y}{x}\], find \[f_{xy}\]
\[f_{xy}=-\frac{2xy}{(x^{2}-y^{2})^{2}}\]
\[f_{xy}=-\frac{2xy}{(x^{2}+y^{2})^{2}}\]
\[f_{xy}=\frac{2xy}{(x^{2}+y^{2})^{2}}\]
\[f_{xy}=\frac{2xy}{(x^{2}-y^{2})^{2}}\]

Q11 If \[f(u)=\sin u\] and \[u=\sqrt(x^{2}+y^{2}\], then find \[f_{x}\]
\[f_{x}=\frac{x\cos \sqrt(x^{2}+y^{2})}{\sqrt (x^{2}+y^{2})}\]
\[f_{x}=\frac{x\cos \sqrt(x^{2}-y^{3})}{\sqrt (x^{2}-y^{2})}\]
\[f_{x}=\frac{y\cos \sqrt(x^{2}-y^{3})}{\sqrt (x^{2}-y^{2})}\]
\[f_{x}=\frac{y\cos \sqrt(x^{2}-y^{3})}{\sqrt (x^{3}-y^{3})}\]

Q12 If the function \[f(x,y)=\tan^{-1} \frac{y}{x}\], find \[f_{y}\]
\[f_{x}=\frac{x}{x^{2}-y^{2}}\]
\[f_{x}=\frac{y}{x^{2}+y^{2}}\]
\[f_{x}=\frac{y}{x^{2}-y^{2}}\]
\[f_{x}=\frac{x}{x^{2}+y^{2}}\]

Q13 If the function \[f(x,y)=\tan^{-1} \frac{y}{x}\], find \[f_{x}\]
\[f_{x}=\frac{x}{x^{2}-y^{2}}\]
\[f_{x}=\frac{y}{x^{2}+y^{2}}\]
\[f_{x}=\frac{y}{x^{2}-y^{2}}\]
\[f_{x}=\frac{x}{x^{2}+y^{2}}\]

Q14 Given that \[f(x,y)=\sin^{2} x\cos y+\frac{x}{y^{2}}\], find \[f_{y}\]
\[f_{y}=\sin^{2}x\sin y-\frac{x}{y^{2}}\]
\[f_{y}=-2\sin^{2} x\sin y-\frac{2x}{y^{3}}\]
\[f_{y}=2\sin^{2} x\sin y+\frac{2x}{y^{3}}\]
\[f_{y}=2\sin^{2} x\sin y-\frac{2x}{y^{3}}\]

Q15 Given that \[f(x,y)=\sin^{2} x\cos y+\frac{x}{y^{2}}\], find \[f_{x}\]
\[f_{x}=2\sin x\cos x\cos y+\frac{1}{y^{2}}\]
\[f_{x}=-2\sin x\cos x\cos y-\frac{1}{y^{2}}\]
\[f_{x}=2\cos x\cos x+\frac{x}{y^{2}}\]
\[f_{x}=\sin^{2} x\cos y+\frac{1}{y^{2}}\]

Q16 Find the total differential of the function \[f(x,y)=x^{2}+3xy\] wth respect to x, given that \[y=sin^{-1} x\].
\[x+2sin^{-1} x+\frac{x}{2-2x^{2}}^{\frac{1}{2}}\]
\[2x+3sin^{-1} x+\frac{3x}{(1-x^{2}}^{\frac{1}{2}}\]
\[x+sin^{-1} x+\frac{2x}{(1-x^{2}}^{\frac{1}{2}}\]
\[2x+sin^{-1} x+\frac{3x}{(1-x^{3}}^{\frac{1}{2}}\]

Q17 Find the total differential of the function \[f(x,y)=y e^{x+y}\]
\[d f=[y e^{x+y}]dx+[(1+y)e^{x+y}]dy\]
\[d f=[y e^{x+y}]dx-[(1+y)e^{x+y}]dy\]
\[d f=[y e^{x-y}]dx+[(1+y)e^{x-y}]dy\]
\[d f=[y e^{x-y}]dx-[(1+y)e^{x-y}]dy\]

Q18 Evaluate the second partial derivative of the functon \[f(x,y)=2x^{3}y^{2}+y^{3}\]
\[\frac{\partial^{2}f}{\partial x^{2}}=12xy, \frac{\partial^{2} f}{\partial y^{2}}=x^{3}+y, \frac{\partial^{2} f}{\partial x\partial y}=2x^{2}y \]
\[\frac{\partial^{2}f}{\partial x^{2}}=12x^{2}y^{2}, \frac{\partial^{2} f}{\partial y^{2}}=4x+6y, \frac{\partial^{2} f}{\partial x\partial y}=10x^{2}y \]
\[\frac{\partial^{2}f}{\partial x^{2}}=12xy^{2}, \frac{\partial^{2} f}{\partial y^{2}}=4x^{3}+6y, \frac{\partial^{2} f}{\partial x\partial y}=12x^{2}y \]
\[\frac{\partial^{2}f}{\partial x^{2}}=5x^{3}y^{2}, \frac{\partial^{2} f}{\partial y^{2}}=6x^{3}+6y, \frac{\partial^{2} f}{\partial x\partial y}=2x^{2}y^{2} \]

Q19 Find the first partial derivative of the functon \[f(x,y)=2x^{3}y^{2}+y^{3}\]
\[\frac{\partial f}{\partial x}=6x^{2}y^{2}, \frac{\partial f}{\partial y}=4x^{3}y+y^{2}\]
\[\frac{\partial f}{\partial x}=6x^{3}y^{3}, \frac{\partial f}{\partial y}=4x^{4}y+y^{2}\]
\[\frac{\partial f}{\partial x}=x^{2}y, \frac{\partial f}{\partial y}=2x^{3}y+y\]
\[\frac{\partial f}{\partial x}=x^{2}y^{2}, \frac{\partial f}{\partial y}=x^{3}y+y^{2}\]

Q20 Evaluate the stationary points of the function \[f(x,y)=xy\left(x^{2}+y^{2}-1\right)\]
\[c=3\pm \sqrt(3)\]
\[(0,0), (0,0), (0, 0), \pm \left(0, \frac{1}{2}\right), \pm \left(0, -\frac{1}{2}\right)\]
\[(0,0), (0,0), (\pm 1, 0), \pm \left(\frac{1}{2}, \frac{1}{2}\right), \pm \left(\frac{1}{2}, 0\right)\]
\[(0,0), (0,\pm 1), (\pm 1, 0), \pm \left(\frac{1}{2}, \frac{1}{2}\right), \pm \left(\frac{1}{2}, -\frac{1}{2}\right)\]

Q21 Use Leibnitz theorem to evaluate the fourth derivative of \[\left(2x^{3}+3x^{2}+x+2\right)e^{2x}\]
\[16\left(2x^{3}+15x^{2}+31x+19\right)e^{2x}\]
\[8\left(x^{2}+5x^{2}+3x+14\right)e^{2x}\]
\[10\left(3x^{2}+10x^{2}+3x+15\right)e^{2x}\]
\[16\left(3x^{2}+5x^{2}+2x+3\right)e^{2x}\]

Q22 Compute the third derivative of \[\sin x In x\] using Leibnitz theorem
\[(2x^{-2}-3x^{-2})\cos x-(3x^{-3}+In 2x) \sin x\]
\[(x^{-3}-x^{-2})\cos x-(x^{-2}+In x) \cos x\]
\[(2x^{-3}-3x^{-1})\sin x-(3x^{-2}+In x) \cos x\]
\[(3x^{-3}-4x^{-1})\sin x-(3x^{-2}+In x) \sin x\]

Q23 Use Leibnitz theorem to find the second derivative of \[\cos x \sin 2x\]
\[2 \sin x (2-9\cos^{2} x)\]
\[2 \sin x (1-5\cos^{3} x)\]
\[3 \sin x (2-9\sin^{2} x)\]
\[2 \cos x (3-5\cos^{2} x)\]

Q24 Compute the n-th differential coefficient of \[y=x\log_{e}x\]
\[(-1)^{n-2}\frac{(n+2)!}{x^{n+1}}\left(n^{3}+2\right)\]
\[(-1)^{n-2}\frac{(n-2)!}{x^{n-1}}\left(n^{3}-2\right)\]
\[(-1)^{n-1}\frac{(n-1)!}{x^{n-2}}\left(n^{2}-2\right)\]
\[(-1)^{n+1}\frac{(n+1)!}{x^{n+2}}\left(n^{2}+2\right)\]

Q25 Obtain the n-th differential coefficient of \[y=(x^{2}+1)e^{2x}\]
\[2^{n-3}e^{4x}(x^{2x}+nx+n^{3}-n+4)\]
\[2^{n-2}e^{2x}(4x^{3x}+5nx+n^{3}-n+4)\]
\[2^{n-2}e^{2x}(4x^{2x}+4nx+n^{2}-n+4)\]
\[2^{n}e^{x}(4x^{2x}+4nx-n+4)\]

Q26 Expand the function \[f(x)=e^{3x}\] about x=0 using Maclaurin's series
\[e^{3x}=1+3x+\frac{(3x)^{2}}{2!}+\frac{(3x)^{3}}{3!}+\cdots+\frac{(3x)^{n}}{n!}\]
\[e^{3x}=1-3x-\frac{(3x)^{2}}{2!}-\frac{(3x)^{3}}{3!}-\cdots-\frac{(3x)^{n}}{n!}\]
\[e^{3x}=1+x+\frac{(x)^{2}}{2!}+\frac{(x)^{3}}{3!}+\cdots+\frac{(x)^{n}}{n!}\]
\[e^{3x}=1-x-\frac{(x)^{2}}{2!}-\frac{(x)^{3}}{3!}-\cdots-\frac{(x)^{n}}{n!}\]

Q27 Given \f(x)=3x(x-1)^{5}. Compute \[f'''(x)\]
\[2i-j\]
\[f'''(x)=80(2x-1)^{2}(x-1)\]
\[f'''(x)=100(x-1)^{2}(4x-1)\]
\[f'''(x)=180(x-1)^{2}(2x-1)\]

Q28 Evaluate the \[\frac{d ^{3}f}{d x^{3}}\] of \[f(x)= sin (x) cos (x)\]
\[\frac{d ^{3}f}{d x^{3}}=-4\left(cos^{2} (x)-sin^{2} (x)\right)\]
\[\frac{d ^{3}f}{d x^{3}}=-2\left(Cos^{2} (x)+sin^{2} (x)\right)\]
\[\frac{d ^{3}f}{d x^{3}}=-4\left(tan^{2} (x)-cos^{2} (x)\right)\]
\[f'(x)=5x^{4}-\frac{1}{2}x^{\frac{1}{2}}+ \frac{1}{2x^{2}}, 20x^{3}-\frac{3}{4}x^{-\frac{1}{2}}- \frac{1}{x^{1}}, 100x^{2}-\frac{3}{8}x^{-\frac{3}{2}}+ \frac{3}{x^{4}}\]

Q29 Compute the first thrre derivatives of \[f(x)=2x^{5}+x^{\frac{3}{2}}-\frac{1}{2x}\]
\[f'(x)=10x^{3}-\frac{2}{2}x^{\frac{1}{2}}+ \frac{1}{2x^{2}}, 20x^{3}-\frac{3}{4}x^{-\frac{1}{2}}- \frac{1}{x^{3}}, 10x^{2}-\frac{1}{8}x^{-\frac{3}{2}}+ \frac{3}{x^{4}}\]
\[f'(x)=10x^{4}-\frac{3}{2}x^{\frac{1}{2}}+ \frac{1}{2x^{2}}, 40x^{3}-\frac{3}{4}x^{-\frac{1}{2}}- \frac{1}{x^{3}}, 120x^{2}-\frac{3}{8}x^{-\frac{3}{2}}+ \frac{3}{x^{4}}\]
\[f'(x)=10x^{4}-\frac{3}{2}x^{\frac{2}{2}}-\frac{1}{2x^{2}}, 40x^{3}\frac{3}{4}x^{-\frac{1}{2}}- \frac{1}{x^{3}}, 120x^{2}-\frac{3}{8}x^{-\frac{1}{2}}+ \frac{3}{x^{4}}\]
\[f'(x)=5x^{4}-\frac{1}{2}x^{\frac{1}{2}}+ \frac{1}{2x^{2}}, 20x^{3}-\frac{3}{4}x^{-\frac{1}{2}}- \frac{1}{x^{1}}, 100x^{2}-\frac{3}{8}x^{-\frac{3}{2}}+ \frac{3}{x^{4}}\]

Q30 For \[g(x)=\frac{x-4}{x-3}, we can use the mean value theorem on [4, 6], Hence determine \[c\]
\[c=3\pm \sqrt(3)\]
\[\sqrt (112) \]
\[c=2\pm \sqrt(3)\]
\[c=-2\pm \sqrt(5)\]

Q31 Find the number \[c\] guaranteed by the mean value theorem for derivatives for \[f(x)=(x+1)^{3}, [-1, 1] \]
\[c=\frac{-\sqrt (3) \pm 2}{\sqrt(3)}\]
\[c=\frac{-\sqrt (2) \pm 1}{\sqrt(3)}\]
\[c=1\pm \sqrt(5)\]
\[c=\frac{-\sqrt (5) \pm 2}{\sqrt(5)}\]

Q32 Determine whether the Rolle's theorem can be applied to \[f\] on the closed interval [a, b] . If can be applied, Find the values of \[c\] in open interval (a, b) such that \[f'( c) = 0\], \[f(x)=\frac{x^{2}-2x-3}{x+2}, [-1, 3]\]
\[c=-2\pm\sqrt(5)\]
\[c=-1\pm\sqrt(5)\]
\[c=-2\pm 2\sqrt(5)\]
\[c=-2\pm\sqrt(5)\]

Q33 Determine whether the mean value theorem can be applied to \[f\] on the closed interval [a, b] . If can be applied, Find the value of \[c\] in open interval (a, b) such that \[f(x)=x(x^{2}-x-2), [-1, 1]\]
\[c=\frac{-1}{2}\]
\[c=\frac{-1}{3}\]
\[c=\frac{-2}{3}\]
\[c=\frac{-2}{5}\]

Q34 Find the two x-intercept of \[f(x)=x^{2}-3x+2\]
x=1, 3
x=1, 1
x=-2, 2
x= 1, 2

Q35 Let \[f(x)=x^{4}-2x^{2}\]. Find the all \[c\] (where \[c\] is the interception on the x-axis ) in the interval (-2, 2) such that \[f'(x)=0\]. ( Hint use Rolle's theorem )
(-1, 0, 1)
(-1, 1, 1)
(-1, 2, 1)
(-1, 0, 2)