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10 Calculus Problems

Please see the attached file for the fully formatted problems.

Problems involve:
parametric equation of line segment,
volume of a parallelipiped,
sketching a plane gven the equation,
finding rectangular equations,
center and radius of a sphere using the equation of a sphere,
force vector problems.

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Solution Preview

1.)
The important points are: intersection of plane with 3 axes x, y, z.
x+2y+4z = 8
Interception on x axis: y=0,z=0, x = 8
Interception with y axis: x=0,z=0, y = 4
Interception with zaxis: x=0,y=0, z = 2
therefore,
three well known points are: (8,0,0), (0,4,0), (0,0,2)
See the attached file.

2.)
P(2,-1,4), Q(5,3,3)
x1-x2 = 2 - 5 = -3
y1-y2 = -1 - 3 = -4
z1-z2 = 4 - 3 = 1
therefore direction ratios of the line given as:
a:b:c = -3:-4:1
hence the parametric equation of straight line is given as:
x = x1 + a*r
y = y1 + b*r
z = z1 + c*r
=> (x = 2 - 3r, y = -1 - 4r, z = 4 + r) --Answer

3.)
postion vectors of P,Q,R:
P = (1,-1,2)=> vector OP = v(OP) = i - j + 2k
Q = (4,1,3) => v(OQ) = 4i + j + 3k
R = (-1,1,-1) => v(OR) = -i + j - k
v(PQ) = v(OQ) - v(OP) = 3i + 2j + k
v(RQ) = v(OQ) - v(OR) = 5i + 0.j + 4k
direction cosines of the normal to the plane:
normal vector to the plane = v(n)
v(n) = v(PQ) X v(RQ) = 8i - 7j - 10k
therefore, equation of the plane:
{v(r) - v(OP)}.v(n) = 0
because, r vector - OP vector will be in the plane and dot product ...

Solution Summary

Ten calculus problems are solved in detail.

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