CALCULUS II, FINAL EXAM 1
MA 126 - CALCULUS II
Friday, December 10, 2010
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Section: . . . . . . . . . Instructor Name: . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
FINAL EXAM
Closed Book. No calculators are permitted.
PART I
Each question is worth 4 points.
Part I consists of 10 questions. Clearly write your answer (only) in the space
provided after each question. You need not show your work for this part of the
exam. No partial credit is awarded for this part of the exam! CHECK YOUR
ANSWERS!
Question 1
√
Find the angle between the vectors u =< 1, 1, 2 > and v =< 1, 1, 0 >. (Give the angle in
degrees or radians.)
Answer: . . . . . . . . . . . . . . . . . . . . .
Question 2
Find the area of the parallelogram generated by the vectors u =< 0, 0, 1 > and v =< 0, 2, 0 >.
(Your answer must be a real number!)
Answer: . . . . . . . . . . . . . . . . . . . . .
CALCULUS II, FINAL EXAM 2
Question 3
Find the parametric equations of the line that passes through the point P (2, 1, 3) and is
perpendicular to the plane 5x + 2y + 3z = 1.
Answer: . . . . . . . . . . . . . . . . . .
Question 4
Find an equation of the plane that passes through the point P (1, 1, 2) and is perpendicular
(normal) to the line with parametric equations x = 1 + 2t, y = 1 − t, z = 2 + 3t.
Answer: . . . . . . . . . . . . . . . . . .
Question 5
Use the Fundamental Theorem of Calculus to find the derivative of the function
Z x √
g(x) = ln ( t) dt.
3
Answer: . . . . . . . . . . . . . . . . . .
CALCULUS II, FINAL EXAM 3
Question 6
Determine whether the improper integral is convergent or divergent. Evaluate the integral if
it is convergent. Z ∞
1
√
3
dx
1 x4
Answer: . . . . . . . . . . . . . . . . . .
Question 7
√
Find the area of the region bounded by the curve y = x and the line y = x.
Answer: . . . . . . . . . . . . . . . . . .
Question 8
Z
Evaluate the indefinite integral sin2 (x) cos3 (x) dx.
Answer: . . . . . . . . . . . . . . . . . .
CALCULUS II, FINAL EXAM 4
Question 9
Z
x+2
Evaluate the indefinite integral dx.
x+1
Answer: . . . . . . . . . . . . . . . . . .
Question 10
∞
X 1
Determine whether the alternating series (−1)n is divergent, absolutely convergent,
n=1
n5/3
or conditionally convergent.
Answer: . . . . . . . . . . . . . . . . . .
CALCULUS II, FINAL EXAM 5
PART II
Each problem is worth 12 points.
Part II consists of 5 problems. You must show your work on this part of the
exam to get full credit. Displaying only the final answer (even if correct) without
the relevant steps will not get full credit. CIRCLE YOUR ANSWER!
Problem 1
Two planes are given by the equations x + y − z = 2 for the plane P1 and x − y + z = 4 for
the plane P2 .
(a) Find the coordinates of a point of intersection of the planes P1 and P2 .
(b) Find the normal vector (i.e., the vector perpendicular) to the plane P1 and the normal
vector to the plane P2 .
(c) Find the parametric equations of the line of intersection of the planes P1 and P2 .
CALCULUS II, FINAL EXAM 6
Problem 2
This problem has two separate questions. (Answer all the questions!)
(a) Find the length of the arc of the circular helix with vector equation
r(t) = h4 cos(t), 4 sin(t), 3ti when −1 ≤ t ≤ 1.
(b) Determine whether the (improper) integral
Z ∞
ln(x)
dx
1 x3
is convergent or divergent. Evaluate the integral if it is convergent.
CALCULUS II, FINAL EXAM 7
Problem 3
Evaluate the following integrals (clearly show the techniques of integration you use):
√
Z
1
(a) √ e x dx
2 x
Z
(b) x2 ln(x) dx
Z
7
(c) dx.
x2 − x − 12
CALCULUS II, FINAL EXAM 8
Problem 4
This problem has two separate questions. (Answer all the questions!)
(a) Find the area of the region enclosed by the parabola x = y 2 − 5y and the parabola
x = 3y − y 2 .
√
(b) The region enclosed by the curve y = x and the parabola y = x2 and is rotated about
the horizontal line y = −2. Find the volume of the solid obtained in this way.
CALCULUS II, FINAL EXAM 9
Problem 5
This problem has two separate questions. (Answer all the questions!)
(a) Find the radius and interval of convergence of the power series
∞
X (−1)n
√ (x − 4)n .
n=1
n
Be sure to check any endpoints that exist!
(b) Find the Maclaurin series for the function f (x) = ex , and use it to write out the
2
Maclaurin series for the function g(x) = e−x .
CALCULUS II, FINAL EXAM 10
DO NOT ENTER ANY PROBLEM SOLUTIONS OR WORK ON THIS
PAGE.
Summary of scores on problems - for grading purposes only.
Points
Part I
Questions 1 – 10
Part II
Problem 1
Problem 2
Problem 3
Problem 4
Problem 5
Total Test Score
CALCULUS II, FINAL EXAM 11
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CALCULUS II, FINAL EXAM 12
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CALCULUS II, FINAL EXAM 13
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