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MA 126 - CALCULUS II December 07, 2007
Name (Print last name first):......................................................
Signature:............................................................
Student ID Number (last four digits):........................
Section:......... Instructor Name:.............................................
FINAL EXAM
Closed Book. No calculators are permitted.
PART I - Basic Skills
Each question is worth 5 points.
Part I consists of 8 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 center and the radius of the sphere x^2 + y^2 + z^2 + 2x โ 8 y + 6z + 1 = 0.
Answer:.....................
Question 2
Find a vector that is orthogonal to both of the vectors < 1 , โ 1 , 2 > and < โ 1 , 0 , 3 >.
Answer:.....................
Question 3
Find the parametric equations of the line through the point (3, 2 , โ5) and parallel to the vector < โ 1 , 1 , 6 >.
Answer:..................
Question 4
Evaluate the indefinite integral (^) โซ 1 x cos(ln x) dx.
Answer:..................
Question 5
Find the derivative f โฒ(x) if
f (x) =
โซ (^) x
3
et 2 dt.
Answer:..................
PART II - Problem Solving skills
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
The velocity function (in miles per second) of an object moving along a line is given by
v(t) = 2t โ 4 , 0 โค t โค 4.
(a) Find the displacement (in miles) of the object during the given time interval.
(b) Find the distance traveled (in miles) by the object during the given time interval.
Problem 2
This problem has two separate questions. (Answer all questions!)
(a) Find the parametric equations for the line passing through the points (2, 0 , 1) and (โ 1 , 3 , 2), and then determine the coordinates of points where the line intersects the coordinate planes, i.e., the xy-plane, the xz-plane, and the yz-plane.
(b) Consider two planes given by the equations x + y + z = 1 and x โ y = 1, respectively.
(b 1 ) Find the symmetric equations for the line of intersection of the two planes.
(b 2 ) Find an equation of the plane containing the point (0, 1 , 1) and the line of intersection obtained in (b 1 ) above. [Hint: Some info from your calculations in (b 1 ) might prove useful here, or al- ternatively, you might use two points on the line obtained in (b 1 ) and the point given here to proceed!]
Problem 4
(a) Find the length of the arc traced by the curve
y =
x^3 3
4 x when 1 โค x โค 2. (Simplify your answer!)
(b) Find the volume of the solid generated by rotating the region bounded by the parabo- las y = x^2 โ 1 and y = 1 โ x^2 about the x-axis.
Problem 5
This problem has two separate questions. (Answer all the questions!)
(a) Test the following series for convergence. In order to receive credit, you must state the test that you are using and give the reasoning for your conclusion.
โ^ โ
n=
ln n n
(b) Consider the series โโ
n=
(x + 1)nโ^1 enโ^1
(b 1 ) Find the values of x for which the series converges. (Write your answer in interval notation!)
(b 2 ) Find the sum of the series for those values of x. (You must simplify your answer!)
SCRATCH PAPER
SCRATCH PAPER
