SOLUTIONS 1. Put a = P Q = (1, 1, 1), b = R Q = (2, 2, 1). (a) n = a b = (3, 3, 0). Plane 3(x + 3) + 3(y 1) = 0 (b) A = |a b|/2 = 3 2/2 (c) c = T Q = (1, 0, 2), Volume V = (a b) c = 3 2. (a) r (t) = (1, 2t, 3t2 ), T (1) = r (1)/|r (1)| = (1/ 14, 2/ 14, 3/
MATH 0240 Final Sample Exam 7
Problem 1. Find the minimum distance between the lines
(x, y, z ) = t(1, 1, 1), t R, and (x, y, z ) = (1, 2, 1) + s(1, 0, 1), s R.
Problem 2. Determine all points at which function f (x, y ) = xy attains
its absolute minima a
MATH 0240 Midterm Examination I Sample 1
This test consists of 11 problems. All work must be shown in order to
get credit. Please write legibly and explain your logic by words whenever
appropriate. If more space is needed, write on the back of the pages a
MATH 0240 Midterm Examination I Sample 2
This test consists of 11 problems. All work must be shown in order to
get credit. Please write legibly and explain your logic by words whenever
appropriate. If more space is needed, write on the back of the pages a
MATH 0240 Midterm Examination I Sample 3
This test consists of 11 problems. All work must be shown in order to
get credit. Please write legibly and explain your logic by words whenever
appropriate. If more space is needed, write on the back of the pages a
MATH 0240 Midterm Examination II Sample 1
This test consists of 8 problems. All work must be shown in order to
get credit. Please write legibly and explain your logic by words whenever
appropriate. If more space is needed, write on the back of the pages a
MATH 0240 Midterm Examination II Sample 2
This test consists of 10 problems, each worth 10 points. All work must
be shown in order to get credit. Please write legibly and explain your logic
by words whenever appropriate. If more space is needed, write on
MATH 0240 Midterm Examination II Sample 3
This test consists of 9 problems. All work must be shown in order to
get credit. Please write legibly and explain your logic by words whenever
appropriate. If more space is needed, write on the back of the pages a
Quiz 4
Name
1. Determine the equation of the line having a slope of 3 and passing through the
point (18, 13).
y 13 = 3(x 18)
y = 3x + 67
2. If f (x) = 2x + 1 and g (x) = x2 3x + 5, determine: (simplify)
(a) g (f (2) = 15
(b) f (g (x) = 2x2 6x + 11
3. Dete
MATH 0240 Final Sample Exam 6
Problem 1. Find the points on the cone z 2 = x2 + y 2 closest to the point
(1, 3, 4).
Problem 2. Express the volume of the tetrahedron with vertices at
points (0, 0, 0), (1, 0, 0), (0, 2, 0), (0, 0, 3) as a triple integral. W
College Algebra F inal Review
EXAM ONE QUESTIONS
YOU MAY USE
1.
x=-bb2-4ac2a
x+b22=c+b22
Use the square root to solve:
2.
Complete the square to solve:
x2+ 10x+18 =0
3.
Find the imaginary solution:
4m-72= -27
4.
Use the quadratic formula to solve:
m2- 8m
1. Determine whether the following vector elds F have a potential , i.e., whether
F = . Explain your answers.
(a) F = x + 2xy, x2 + 2z, 2x2 + z .
(b) F = 2xy + 2yz, x2 + z 2 + 2xz, 2yz + 2xy
2. Let D be a plane region which is bounded by a positively orie
1. (a) Find an equation of the plane determined by the three points P (4, 0, 2),
Q(3, 1, 1) and R(5, 1, 0).
(b) Find the area of the triangle whose vertices are P , Q, R.
(c) Find the volume of the parallelepiped through P , Q, R, and T (2, 1, 3).
2. The
1. Consider the points A(1, 0, 0), B (1, 2, 0), and C (1, 1, 3).
(a) Find (to the nearest degree) the angle ABC .
(b) Find the area of the parallelogram formed by A, B , and C .
(c) Determine whether the point D (1, 0, 42) is in the plane determined by A,
Math 240 Final Sample Exam 8
Problem 1. Find parametric equations of the line parallel to the planes
x + y + z = 1 and x y + 2z = 5, and passing through the point (1, 2, 3).
Problem 2. The curve is given parametrically as
r(t) =< t + 1, t2 + 1, t3 + t2 >
q q 8 p%` 8 f 8 %Y b G
i` w Y
f 8 f@w p` 8 f 8 Y b e s QjiQj@x 8 nQj s h k w m` 0Y b
i hf
e f i w
h d h d wf h
5 e q ' 8 f q 8 w i 7R2!7PGQECU'!PcWq45 e 8 f 8 w
A A A " H D A V A S
i e ` 8 |cf 8 U hR7I X w E2yR'7E!%GE7IX 2E!P4D
A X ( ) A A " D A V