Download Calculus II Final Exam: December 2012 and more Exams Calculus in PDF only on Docsity!
MA 126 - CALCULUS II
Saturday, December 08, 2012
Name (Print last name first):......................................................
Signature:........................................................................
Section:......... Instructor Name:.............................................
FINAL EXAM
Closed Book - No calculators!
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
Use vectors to decide whether the triangle with vertices P (1, 0 , 1), Q(1, 0 , 0), and R(2, 3 , 0) is right-angled.
Answer:.....................
Question 2
Find the area of the parallelogram generated by the vectors u =< 1 , 3 , โ 1 > and v =< 1 , 0 , โ 1 >.
Answer:.....................
Question 3
Find the parametric equations of the line that passes through the point P (1, 2 , 1) and is parallel to the vector v =< 2 , โ 5 , 3 >.
Answer:..................
Question 4
Find an equation of the plane that passes through the point P (1, โ 1 , 0) and is perpendicular
(normal) to the line with symmetric equations
x โ 1 2
y โ 1 โ 1
z โ 2 3
Answer:..................
Question 5
Use the Fundamental Theorem of Calculus to find the derivative of the function
g(x) =
โซ (^) x
0
arctan (t^2 ) dt.
Answer:..................
Question 9
Evaluate the indefinite integral
x^2 1 + x^2
dx.
Answer:..................
Question 10
Determine whether the alternating series
โ^ โ
n=
(โ1)n^
n^2 + 6 n^4 + 2
is divergent, absolutely conver-
gent, or conditionally convergent.
Answer:..................
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 - no credit for unsubstantiated answers. CIRCLE YOUR ANSWER!
Problem 1
Two planes are given by the equations x + y + z = 1 for the plane P 1 and x โ 2 y + 3z = 1 for the plane P 2.
(a) Find the coordinates of a point of intersection of the planes P 1 and P 2.
(b) Find the normal vector (i.e., the vector perpendicular) to the plane P 1 and the normal vector to the plane P 2.
(c) Find the symmetric equations of the line of intersection of the planes P 1 and P 2.
Problem 3
Evaluate the following integrals (clearly show the techniques of integration you use):
(a)
x
ln(x)
dx
(b)
x cos(x) dx
(c)
โx^2 + x + 2 (x โ 1)(x^2 + 1)
dx.
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 โ y and the parabola x = y โ y^2.
(b) The region enclosed by the curve y =
x and the line y = x and is rotated about the horizontal line y = โ1. Find the volume of the solid obtained in this way.
(b) Use the Maclaurin series of the function cos(x) =
โ^ โ
n=
(โ1)n^
x^2 n (2n)!
to write out the
Maclaurin series for the function g(x) = cos(x^2 ) and then write out the Maclaurin
series expansion of
0
cos(x^2 ) dx. (Do not compute and add up the terms of your series!)
Using the above information, find the minimum number of terms needed to approximateโซ 1
0
cos(x^2 ) dx with an error less than 0.01?
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 โ 10
Part II
Problem 1
Problem 2
Problem 3
Problem 4
Problem 5
Total Test Score
SCRATCH PAPER
(Scratch paper will not be graded)
SCRATCH PAPER
(Scratch paper will not be graded)
