Math 53 - Hutchings
GSI: Nicolas Bray
Final, Fall 2003, Solutions
1. The chain rule tell us that
d
dt f (r(t)|t=0
=
f (r(0) r (0) = 2e2 , e2 3, 4 = 10e2 .
2. (a) A useful fact about spheres is that if the point x, y, z is on a sphere S then
the vector x,
Math 53 - Review for Midterm 1
GSI: Santiago Canez http:/math.berkeley.edu/scanez/courses/math53/fall05/
1. Consider the curve given by the parametric equations x = cos t, y = sin 2t. (a) Find the points on the curve at which there is either a horizontal
Math 53 Midterm 2
Monday, Nov 21, 2011 1:10 - 2:00
Directions: Do all the work on these pages; use reverse side if needed. Answers without
accompanying reasoning may only receive partial credit. No books, notes, calculators, or
electronic devices. Please
Math 53 Midterm 1
Monday, Oct 10, 2011 1:10 - 2:00
Directions: Do all the work on these pages; use reverse side if needed. Answers without
accompanying reasoning may only receive partial credit. No books, notes, calculators, or
electronic devices. Please
Math 53 Midterm #2, 4/10/07, 3:40 PM 5:00 PM (please do not leave the exam between 4:45 and 5:00) No calculators or notes are permitted. Each of the 6 questions is worth 10 points. Please write your solution to each of the 6 questions on a separate sheet
Math 53 - Midterm 1 Review
GSI: Santiago Canez
1. Eliminate the parameter to nd a cartesian equation of the curve given by the parametric equations x = 2 sinh t and y = 3 cosh t. 2. Find parametric equations for the curve of intersection of the surfaces y
MATH 53 PRACTICE MIDTERM #1
Problem #1 (a). Compute (b). Compute
f x
and
f y
for f (x, y) = (x2 + y)2 .
g for g(x, y, z) = sin(xy 2 z 3 ).
Problem #2. Show that the limit lim
(x,y)(0,0)
x8 + y 5 x8 + 7y 4
does not exist. Problem #3. Find the length of the
Name: Math 53, Section 002 Midterm 1 July 8, 2003
Justify all answers. 1. Find the area of the region enclosed by the curve r = 2 + 2cos().
1
2. Calculate the tangent of the angle between the vectors a = (2, 3, 6) and b = (2, 3, 6).
2
3. Let P be the poin
Math 53 Midterm #1, 10/2/03, 8:10 AM 9:30 AM No calculators or notes are permitted. Each of the 6 questions is worth 10 points. Please write your solution to each of the 6 questions on a separate sheet of paper with your name on it. Please put a box aroun
Math 53 Final, 5/18/07, 12:30 PM 3:30 PM No calculators or notes. Each question is worth 10 points. Please write your solution to each of the 10 questions on a separate sheet with your name, SID#, and GSI on it. (If you are removing an incomplete for prof
Math 53 - Hutchings
GSI: Nicolas Bray
Final, Spring 2007, Solutions
1. Letting g (x, y, z ) = x2 + y 2 + z 2 , we want to nd solutions to g = 1 and
f = g which
works out to the four equations:
2(x 2) = 2x (1 )x = 2 x =
2
1
2
1
1
2(z 1) = 2z (1 )z = 1 z =