Math 115, Fall 2010
Nguyen, Scedrov
Final Exam
December 17, 2010
Name:
Penn ID #:
Show all your work. A correct answer without supporting work receives little or no
credit!
Problem 1
Problem 2
Problem 3
Problem 4
Problem 5
Problem 6
Problem 7
Problem 8
Pr
Math 115, Fall 2012
Scedrov
Final Exam
December 18, 2012
Name:
Penn ID #:
Show all your work. A correct answer without supporting work receives little or no
credit!
Problem 1
Problem 2
Problem 3
Problem 4
Problem 5
Problem 6
Problem 7
Problem 8
Problem 9
MATH 115 - FINAL EXAM
December 14, 2011
Name
Student no. (from ID)
Rec. Day & time
This is a Multiple choice, closed book, no calculator exam. You may use a 5 8 card.
Show all your work.
PUT YOUR ANSWERS ON THE ANSWER SHEET (page 19). Next to the
number f
Math 115, Spring 2010
A. Scedrov, K. Zhu
Final Exam
May 4, 2010
Name:
Penn ID #:
Show all your work. A correct answer without supporting work receives little or no
credit!
Problem 1
Problem 2
Problem 3
Problem 4
Problem 5
Problem 6
Problem 7
Problem 8
Pro
Math 115
Final Exam Spring 2009 May 7, 2009
1. Evaluate
31
0 x 3
e y dydx.
e 1 2
3
(A) 1 (E) e 1
(B) e (F)
1 e 1 2
(C) 1 e (G) e
(D)
(H) None of these
2. Find the z coordinate of the point on the plane x + 2 y + 3 z = 13 closest to the point (1,1,1) .
(A)
Math 115, Fall 2010
Nguyen, Scedrov
Final Exam
December 17, 2010
Name:
Penn ID #:
Show all your work. A correct answer without supporting work receives little or no
credit!
Problem 1
Problem 2
Problem 3
Problem 4
Problem 5
Problem 6
Problem 7
Problem 8
Pr
Math 115, Spring 2012
Scedrov
Final Exam
May 4, 2012
Name:
Penn ID #:
Show all your work. A correct answer without supporting work receives little or no
credit!
Problem 1
Problem 2
Problem 3
Problem 4
Problem 5
Problem 6
Problem 7
Problem 8
Problem 9
Prob
Math 115 Final Exam Fall 2006 1. Consider the surface z = f (x, y) = 2x2 + y 2 . Find the tangent plane to the surface at the point (x, y, z) = (1, 1, 3) and nd where this plane intersects the z-axis. Plane intersects the z-axis at z = A. 3 B. 2 C. 0 D. 3
Math 115 Final Exam December 2002 1. Consider the surface z = f (x, y) = 2x2 + y 2 . Find the tangent plane to the surface at the point (x, y, z) = (1, 1, 3) and nd where this plane intersects the z-azis. Plane intersects the z-axis at z = A.3 B.2 C.0 D.
Math 115
Final Exam
Answers at the end
Fall 2004
1. The tangent plane to the ellipsoid x2 /4 + y 2 + z 2 /9 = 3 at the point (2, 1, 3) intersects the xaxis at the point:
A. (4, 0, 0) E. (6, 0, 0)
B. (3, 0, 0) F. (6, 0, 0)
C. (3, 0, 0) G. (1, 0, 0)
D. (2,
Math 115
Final Exam
Answers at the end
Fall 2005
1. X is a continuous random variable on the interval [0,1] whose density function is of the form kx2 for some constant k. What is Var(X)? A. 1/80 E. 5/80 B. 1/40 F. 3/40 C. 3/80 G.
1 80
D. 1/20 H.
3 80
2. A
Exercise 4.3. Using index notation, prove the identity
( f v) = f v+( f )v.
Now repeat this exercise, but without using index notation (i.e. writing out the left-hand sides of the identity
in full, and showing that it can be rearranged to give the right-
Math 115
Final Exam Spring 2009 May 7, 2009
1. Evaluate
31
0 x 3
e y dydx.
e 1 2
3
(A) 1 (E) e 1
(B) e (F)
1 e 1 2
(C) 1 e (G) e
(D)
(H) None of these
2. Find the z coordinate of the point on the plane x + 2 y + 3 z = 13 closest to the point (1,1,1) .
(A)
Math 115, Spring 2010
A. Scedrov, K. Zhu
Final Exam
May 4, 2010
Name:
Penn ID #:
Show all your work. A correct answer without supporting work receives little or no
credit!
Problem 1
Problem 2
Problem 3
Problem 4
Problem 5
Problem 6
Problem 7
Problem 8
Pro
Math 115 Exam 2 Answers 1. A fair coin is tossed 7 times. What is the probability that there are exactly 3 heads.
7! 7 6 5 4 3 2 1 1 35 1 1 C ( 7,3) = = 2 = 128 2 2 3! 4! 3 2 1 4 3 2 1
3 4 7
2. A person draws two socks at random out of a drawer containin
Math 115 Exam 2 Extra Problems Bivariate Distributions 1. Suppose the joint p.d.f. of a pair of random variables on the rectangle 0 < x < 2, 0 < y < 1 is given by f(x,y) = 1/2. Compute Prob(X > Y). 2. Suppose the joint p.d.f. of a pair of random variables
PARTIAL DERIVATIVES
TANGENT PLANES
Suppose a surface S has equation z = f(x, y),
15.4
where f has continuous first partial derivatives.
Tangent Planes and
Linear Approximations
Let P(x0, y0, z0) be a point on S.
In this section, we will learn how to:
Appr
Some applications of these polar coordinates
Using polar (or cylindrical) coordinates the area within a circle of radius R,
RR
0
R 2
d dr, comes out immediately
as R2. Using spherical polar coordinates the volume of a sphere of radius R,
0r
RR
0
R
0
R 2
0
General Orthogonal Curvilinear Coordinates
The two sets of polar coordinates above have a feature in common: the three sets of coordinate lines are
orthogonal to one another at all points. The corresponding unit vectors are also orthogonal.
General orthog
Spherical Polar Coordinates
These are coordinates (r, , ), where r measures distance from the origin, measures angle from some
chosen axis, called the polar axis, and measures angle around that axis (see Fig 5.2.) To relate them to
Cartesian coordinates w