# MTH281 TMA

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 )….

Compute the first thrre derivatives of $f(x)=2x^{5}+x^{\frac{3}{2}}-\frac{1}{2x}$…

Given \f(x)=3x(x-1)^{5}. Compute $f”'(x)$….

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]…. Evaluate the \[\frac{d ^{3}f}{d x^{3}}$ of $f(x)= sin (x) cos (x)$…

Given$f(x)=\sqrt(9-x^{2})$….

Find the two x-intercept of $f(x)=x^{2}-3x+2$….

For $g(x)=\frac{x-4}{x-3}$ we can use the mean value theorem on [4, 6], Hence determine $c$…

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 $f(x)=x(x^{2}-x-2), [-1, 1]$….

Find the number $c$ guaranteed by the mean value theorem for derivatives for $f(x)=(x+1)^{3}, [-1, 1]$….

Find the number $c$ guaranteed by the mean value theorem for derivatives for $f(x)=(x+1)^{3}, [-1, 1]$

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 )….

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 $f(x)=x(x^{2}-x-2), [-1, 1]$…

Given$f(x)=\sqrt(9-x^{2})$…

Given \f(x)=3x(x-1)^{5}. Compute $f”'(x)$…

For $g(x)=\frac{x-4}{x-3}$ we can use the mean value theorem on [4, 6], Hence determine $c$…

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]… Compute the first thrre derivatives of \[f(x)=2x^{5}+x^{\frac{3}{2}}-\frac{1}{2x}$…

Evaluate the $\frac{d ^{3}f}{d x^{3}}$ of $f(x)= sin (x) cos (x)$…

Find the two x-intercept of $f(x)=x^{2}-3x+2$…

…………………………………………
Q1 Expand the function $f(x)=e^{3x}$ about x=0 using Maclaurin’s series
$e^{3x}=1+3x+\frac{(3x)^{2}}{2!}+\frac{(3x)^{3}}{3!}+\cdots+\frac{(3x)^{n}}{n!}$
$e^{3x}=1-3x-\frac{(3x)^{2}}{2!}-\frac{(3x)^{3}}{3!}-\cdots-\frac{(3x)^{n}}{n!}$
$e^{3x}=1+x+\frac{(x)^{2}}{2!}+\frac{(x)^{3}}{3!}+\cdots+\frac{(x)^{n}}{n!}$
$e^{3x}=1-x-\frac{(x)^{2}}{2!}-\frac{(x)^{3}}{3!}-\cdots-\frac{(x)^{n}}{n!}$

Q2 Given$f(x)=3x(x-1)^{5}$. Compute $f”'(x)$
$f”'(x)=8(2x-1)^{3}(x-1)$
$f”'(x)=80(2x-1)^{2}(x-1)$
$f”'(x)=100(x-1)^{2}(4x-1)$
$f”'(x)=180(x-1)^{2}(2x-1)$

Q3 Evaluate the $\frac{d ^{3}f}{d x^{3}}$ of $f(x)= sin (x) cos (x)$
$\frac{d ^{3}f}{d x^{3}}=-4\left(cos^{2} (x)-sin^{2} (x)\right)$
$\frac{d ^{3}f}{d x^{3}}=-2\left(Cos^{2} (x)+sin^{2} (x)\right)$
$\frac{d ^{3}f}{d x^{3}}=-4\left(tan^{2} (x)-cos^{2} (x)\right)$
$f'(x)=5x^{4}-\frac{1}{2}x^{\frac{1}{2}}+ \frac{1}{2x^{2}}, 20x^{3}-\frac{3}{4}x^{-\frac{1}{2}}- \frac{1}{x^{1}}, 100x^{2}-\frac{3}{8}x^{-\frac{3}{2}}+ \frac{3}{x^{4}}$

Q4 Compute the first thrre derivatives of $f(x)=2x^{5}+x^{\frac{3}{2}}-\frac{1}{2x}$
$f'(x)=10x^{3}-\frac{2}{2}x^{\frac{1}{2}}+ \frac{1}{2x^{2}}, 20x^{3}-\frac{3}{4}x^{-\frac{1}{2}}- \frac{1}{x^{3}}, 10x^{2}-\frac{1}{8}x^{-\frac{3}{2}}+ \frac{3}{x^{4}}$
$f'(x)=10x^{4}-\frac{3}{2}x^{\frac{1}{2}}+ \frac{1}{2x^{2}}, 40x^{3}-\frac{3}{4}x^{-\frac{1}{2}}- \frac{1}{x^{3}}, 120x^{2}-\frac{3}{8}x^{-\frac{3}{2}}+ \frac{3}{x^{4}}$
$f'(x)=10x^{4}-\frac{3}{2}x^{\frac{2}{2}}-\frac{1}{2x^{2}}, 40x^{3}\frac{3}{4}x^{-\frac{1}{2}}- \frac{1}{x^{3}}, 120x^{2}-\frac{3}{8}x^{-\frac{1}{2}}+ \frac{3}{x^{4}}$
$f'(x)=5x^{4}-\frac{1}{2}x^{\frac{1}{2}}+ \frac{1}{2x^{2}}, 20x^{3}-\frac{3}{4}x^{-\frac{1}{2}}- \frac{1}{x^{1}}, 100x^{2}-\frac{3}{8}x^{-\frac{3}{2}}+ \frac{3}{x^{4}}$

Q5 For $g(x)=\frac{x-4}{x-3}$, we can use the mean value theorem on [4, 6], Hence determine $c$
$c=3\pm \sqrt(3)$
$\sqrt (112)$
$c=2\pm \sqrt(3)$
$c=-2\pm \sqrt(5)$

Q6 Find the number $c$ guaranteed by the mean value theorem for derivatives for $f(x)=(x+1)^{3}, [-1, 1]$
$c=\frac{-\sqrt (3) \pm 2}{\sqrt(3)}$
$c=\frac{-\sqrt (2) \pm 1}{\sqrt(3)}$
$c=1\pm \sqrt(5)$
$c=\frac{-\sqrt (5) \pm 2}{\sqrt(5)}$

Q7 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 $f'( c) = 0$, $f(x)=\frac{x^{2}-2x-3}{x+2}, [-1, 3]$
$c=-2\pm\sqrt(5)$
$c=-1\pm\sqrt(5)$
$c=-2\pm 2\sqrt(5)$
$c=-2\pm\sqrt(5)$

Q8 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 $f(x)=x(x^{2}-x-2), [-1, 1]$
$c=\frac{-1}{2}$
$c=\frac{-1}{3}$
$c=\frac{-2}{3}$
$c=\frac{-2}{5}$

Q9 Find the two x-intercept of $f(x)=x^{2}-3x+2$
x=1, 3
x=1, 1
x=-2, 2
x= 1, 2

Q10 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)

Q1 Find the total differential of the function $f(x,y)=x^{2}+3xy$ wth respect to x, given that $y=sin^{-1} x$.
$2x+2sin^{-1} x+\frac{x}{(2-2x^{2}}^{\frac{1}{2}}$
$2x+3sin^{-1} x+\frac{3x}{(1-x^{2}}^{\frac{1}{2}}$
$x+sin^{-1} x+\frac{2x}{(1-x^{2}}^{\frac{1}{2}}$
$2x+sin^{-1} x+\frac{3x}{(1-x^{3}}^{\frac{1}{2}}$

Q2 Find the total differential of the function $f(x,y)=y e^{x+y}$
$d f=[y e^{x+y}]dx+[(1+y)e^{x+y}]dy$
$d f=[y e^{x+y}]dx-[(1+y)e^{x+y}]dy$
$d f=[y e^{x-y}]dx+[(1+y)e^{x-y}]dy$
$d f=[y e^{x-y}]dx-[(1+y)e^{x-y}]dy$

Q3 Evaluate the second partial derivative of the functon $f(x,y)=2x^{3}y^{2}+y^{3}$
$\frac{\partial^{2}f}{\partial x^{2}}=12xy, \frac{\partial^{2} f}{\partial y^{2}}=x^{3}+y, \frac{\partial^{2} f}{\partial x\partial y}=2x^{2}y$
$\frac{\partial^{2}f}{\partial x^{2}}=12x^{2}y^{2}, \frac{\partial^{2} f}{\partial y^{2}}=4x+6y, \frac{\partial^{2} f}{\partial x\partial y}=10x^{2}y$
$\frac{\partial^{2}f}{\partial x^{2}}=12xy^{2}, \frac{\partial^{2} f}{\partial y^{2}}=4x^{3}+6y, \frac{\partial^{2} f}{\partial x\partial y}=12x^{2}y$
$\frac{\partial^{2}f}{\partial x^{2}}=5x^{3}y^{2}, \frac{\partial^{2} f}{\partial y^{2}}=6x^{3}+6y, \frac{\partial^{2} f}{\partial x\partial y}=2x^{2}y^{2}$

Q4 Find the first partial derivative of the functon $f(x,y)=2x^{3}y^{2}+y^{3}$
$\frac{\partial f}{\partial x}=6x^{2}y^{2}, \frac{\partial f}{\partial y}=4x^{3}y+y^{2}$
$\frac{\partial f}{\partial x}=6x^{3}y^{3}, \frac{\partial f}{\partial y}=4x^{4}y+y^{2}$
$\frac{\partial f}{\partial x}=x^{2}y, \frac{\partial f}{\partial y}=2x^{3}y+y$
$\frac{\partial f}{\partial x}=x^{2}y^{2}, \frac{\partial f}{\partial y}=x^{3}y+y^{2}$

Q5 Evaluate the stationary points of the function $f(x,y)=xy\left(x^{2}+y^{2}-1\right)$
$c=3\pm \sqrt(3)$
$(0,0), (0,0), (0, 0), \pm \left(0, \frac{1}{2}\right), \pm \left(0, -\frac{1}{2}\right)$
$(0,0), (0,0), (\pm 1, 0), \pm \left(\frac{1}{2}, \frac{1}{2}\right), \pm \left(\frac{1}{2}, 0\right)$
$(0,0), (0,\pm 1), (\pm 1, 0), \pm \left(\frac{1}{2}, \frac{1}{2}\right), \pm \left(\frac{1}{2}, -\frac{1}{2}\right)$

Q6 Use Leibnitz theorem to evaluate the fourth derivative of $\left(2x^{3}+3x^{2}+x+2\right)e^{2x}$
$16\left(2x^{3}+15x^{2}+31x+19\right)e^{2x}$
$8\left(x^{2}+5x^{2}+3x+14\right)e^{2x}$
$10\left(3x^{2}+10x^{2}+3x+15\right)e^{2x}$
$16\left(3x^{2}+5x^{2}+2x+3\right)e^{2x}$

Q7 Compute the third derivative of $\sin x In x$ using Leibnitz theorem
$(2x^{-2}-3x^{-2})\cos x-(3x^{-3}+In 2x) \sin x$
$(x^{-3}-x^{-2})\cos x-(x^{-2}+In x) \cos x$
$(2x^{-3}-3x^{-1})\sin x-(3x^{-2}+In x) \cos x$
$(3x^{-3}-4x^{-1})\sin x-(3x^{-2}+In x) \sin x$

Q8 Use Leibnitz theorem to find the second derivative of $\cos x \sin 2x$
$2 \sin x (2-9\cos^{2} x)$
$2 \sin x (1-5\cos^{3} x)$
$3 \sin x (2-9\sin^{2} x)$
$2 \cos x (3-5\cos^{2} x)$

Q9 Compute the n-th differential coefficient of $y=x\log_{e}x$
$(-1)^{n-2}\frac{(n+2)!}{x^{n+1}}\left(n^{3}+2\right)$
$(-1)^{n-2}\frac{(n-2)!}{x^{n-1}}\left(n^{3}-2\right)$
$(-1)^{n-1}\frac{(n-1)!}{x^{n-2}}\left(n^{2}-2\right)$
$(-1)^{n+1}\frac{(n+1)!}{x^{n+2}}\left(n^{2}+2\right)$

Q10 Obtain the n-th differential coefficient of $y=(x^{2}+1)e^{2x}$
$2^{n-3}e^{4x}(x^{2x}+nx+n^{3}-n+4)$
$2^{n-2}e^{2x}(4x^{3x}+5nx+n^{3}-n+4)$
$2^{n-2}e^{2x}(4x^{2x}+4nx+n^{2}-n+4)$
$2^{n}e^{x}(4x^{2x}+4nx-n+4)$

Q1 If a and b are non-collinear vectors and $A=(x+y)a+(2x+y+1)b$
x=1,y=1
x=2,y=4
x=2,y=1
x=4,y2

Q2 The following forces act on a particle P:$F_{1}=2i+3j-5k$, $F_{2}=-5i+j+3k$,$F_{3}=i-2j+4k$,$F_{4}=4i-3j-2k$, Find the magnitude of the resultant
$2i-j$
$2i-j+k$
$2i-j-2k$
$i-j-k$

Q3 Given the scalar defined by $\phi(x,y,z)=3x^{2}z-xy^{2}+5$,find $\phi$ at the points (-1,-2,-3)
12
5
19
19

Q4 Find a unit vector parallel to the resultant vector $A_{1}=2i+4j-5k$,$A_{2}=1+2j+3k$
$\frac{3}{7}i+\frac{6}{7}j-\frac{2}{7}k$
$\frac{1}{7}i+\frac{63}{7}j-\frac{4}{7}$
$\frac{2}{7}i-\frac{3}{7}j-\frac{5}{7}$
$\frac{3}{5}i+\frac{6}{5}j-\frac{2}{5}$

Q5 If $A_{1}=3i-j-4k$, $A_{2}=-2i+4j-3k$,$A_{3}=i+2j-k$, find $\left|3A_{1}-2A_{3}+4A_{3}\right|$
$\sqrt (398)$
$\sqrt (112)$
$\sqrt (214)$
$\sqrt (81)$

Q6 A car travels 3km due north, then 5km northeast. Determine the resultant displacement
7.43
5.61
9.51
4.53

Q7 Let a and b be vectors, then $a \times b= ab\sin \theta$ is the ââ‚¬¦ââ‚¬¦ââ‚¬¦product
product
scalar
vector
none of the above

Q8 Given that $A_{1}=2i-j+k$,$A_{2}=i+3j-2k$,$A_{3}=3i+2j+5k$ and $A_{4}=3i+2j+5k$,Find scalars a, b, c such that $A_{4}=a A_{1} +b A_{2}+c A_{3}$
a=1,b=-1,c=1
a=-2,b=1,c=-3
a=2,b=3,c=-1
a=-2,b=-1,c=2

Q9 Given that $A_{1}=3i-2j+k$,$A_{2}=2i-4j-3k$,$A_{3}=-i+2j+2k$, find the magnitudes of $2A_{1}-3 A_{2}-5 A_{3}$
5
$\sqrt 5$
$\sqrt 30$
$\sqrt 15$

Q10 Find the magnitude of vector $A=3i-2j+2k$
3
2
1
5
Q1 If a and b are non-collinear vectors and $A=(x+y)a+(2x+y+1)b$
x=1,y=1
x=2,y=4
x=2,y=1
x=4,y2

Q2 The following forces act on a particle P:$F_{1}=2i+3j-5k$, $F_{2}=-5i+j+3k$,$F_{3}=i-2j+4k$,$F_{4}=4i-3j-2k$, Find the magnitude of the resultant
$2i-j$
$2i-j+k$
$2i-j-2k$
$i-j-k$

Q3 Given the scalar defined by $\phi(x,y,z)=3x^{2}z-xy^{2}+5$,find $\phi$ at the points (-1,-2,-3)
12
5
19
19

Q4 Find a unit vector parallel to the resultant vector $A_{1}=2i+4j-5k$,$A_{2}=1+2j+3k$
$\frac{3}{7}i+\frac{6}{7}j-\frac{2}{7}k$
$\frac{1}{7}i+\frac{63}{7}j-\frac{4}{7}$
$\frac{2}{7}i-\frac{3}{7}j-\frac{5}{7}$
$\frac{3}{5}i+\frac{6}{5}j-\frac{2}{5}$

Q5 If $A_{1}=3i-j-4k$, $A_{2}=-2i+4j-3k$,$A_{3}=i+2j-k$, find $\left|3A_{1}-2A_{3}+4A_{3}\right|$
$\sqrt (398)$
$\sqrt (112)$
$\sqrt (214)$
$\sqrt (81)$

Q6 A car travels 3km due north, then 5km northeast. Determine the resultant displacement
7.43
5.61
9.51
4.53

Q7 Let a and b be vectors, then $a \times b= ab\sin \theta$ is the __________
product
scalar
vector
none of the above

Q8 Given that $A_{1}=2i-j+k$,$A_{2}=i+3j-2k$,$A_{3}=3i+2j+5k$ and $A_{4}=3i+2j+5k$,Find scalars a, b, c such that $A_{4}=a A_{1} +b A_{2}+c A_{3}$
a=1,b=-1,c=1
a=-2,b=1,c=-3
a=2,b=3,c=-1
a=-2,b=-1,c=2

Q9 Given that $A_{1}=3i-2j+k$,$A_{2}=2i-4j-3k$,$A_{3}=-i+2j+2k$, find the magnitudes of $2A_{1}-3 A_{2}-5 A_{3}$
5
$\sqrt 5$
$\sqrt 30$
$\sqrt 15$

Q10 Find the magnitude of vector $A=3i-2j+2k$
3
2
1
5

Q11 If a and b are non-collinear vectors and $A=(x+y)a+(2x+y+1)b$
x=1,y=1
x=2,y=4
x=2,y=1
x=4,y2

Q12 The following forces act on a particle P:$F_{1}=2i+3j-5k$, $F_{2}=-5i+j+3k$,$F_{3}=i-2j+4k$,$F_{4}=4i-3j-2k$, Find the magnitude of the resultant
$2i-j$
$2i-j+k$
$2i-j-2k$
$i-j-k$

Q13 Given the scalar defined by $\phi(x,y,z)=3x^{2}z-xy^{2}+5$,find $\phi$ at the points (-1,-2,-3)
12
5
19
19

Q14 Find a unit vector parallel to the resultant vector $A_{1}=2i+4j-5k$,$A_{2}=1+2j+3k$
$\frac{3}{7}i+\frac{6}{7}j-\frac{2}{7}k$
$\frac{1}{7}i+\frac{63}{7}j-\frac{4}{7}$
$\frac{2}{7}i-\frac{3}{7}j-\frac{5}{7}$
$\frac{3}{5}i+\frac{6}{5}j-\frac{2}{5}$

Q15 If $A_{1}=3i-j-4k$, $A_{2}=-2i+4j-3k$,$A_{3}=i+2j-k$, find $\left|3A_{1}-2A_{3}+4A_{3}\right|$
$\sqrt (398)$
$\sqrt (112)$
$\sqrt (214)$
$\sqrt (81)$

Q16 A car travels 3km due north, then 5km northeast. Determine the resultant displacement
7.43
5.61
9.51
4.53

Q17 Let a and b be vectors, then $a \times b= ab\sin \theta$ is the ââ‚¬¦ââ‚¬¦ââ‚¬¦product
product
scalar
vector
none of the above

Q18 Given that $A_{1}=2i-j+k$,$A_{2}=i+3j-2k$,$A_{3}=3i+2j+5k$ and $A_{4}=3i+2j+5k$,Find scalars a, b, c such that $A_{4}=a A_{1} +b A_{2}+c A_{3}$
a=1,b=-1,c=1
a=-2,b=1,c=-3
a=2,b=3,c=-1
a=-2,b=-1,c=2

Q19 Given that $A_{1}=3i-2j+k$,$A_{2}=2i-4j-3k$,$A_{3}=-i+2j+2k$, find the magnitudes of $2A_{1}-3 A_{2}-5 A_{3}$
5
$\sqrt 5$
$\sqrt 30$
$\sqrt 15$

Q20 Find the magnitude of vector $A=3i-2j+2k$
3
2
1
5

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## 5 Replies to “MTH281 TMA”

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