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 ………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 ………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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