Name: TA Name: Math 20C. Midterm Exam 1 April 28, 2006
Section Number: Section Time:
No calculators or any other devices are allowed on this exam. Write your solutions clearly and legibly; no credit will be given for illegible solutions. Read each questio
Name: Math 20C. Midterm Exam 2 July 23, 2004
Student Number:
Read each question carefully, and answer each question completely. Show all of your work. No credit will be given for unsupported answers. Write your solutions clearly and legibly. No credit wil
Name: Math 20C. Midterm Exam 2 July 23, 2004
Student Number:
Read each question carefully, and answer each question completely. Show all of your work. No credit will be given for unsupported answers. Write your solutions clearly and legibly. No credit wil
UCSD 20052006 Spring
Math 20C Calculus & Analytic Geometry For Science & Engineering
Formulae for the rst mid-term examination
For a = a1 i + a2 j + a3 k and b = b1 i + b2 j + b3 k, ab= a2 a3 aa aa i+ 3 1 j+ 1 2 k= b2 b3 b3 b1 b1 b2 ijk a1 a2 a3 b1 b2 b3
Quiz 1 Solution for MATH 20C, 2009
1. For R = (1, 4, 3), nd the point P such that v = P R has components < 3, 2, 3 > and sketch v. Ans. Let P = (a, b, c). Then v = P R =< 1 a, 4 b, 3 c >=< 3, 2, 3 > So, 1 a = 3, a = 2, 4 b = 2, b = 6, 3c=3
c=0
P = (2, 6,
Quiz 2 Solution for MATH 20C, 2009
1. Consider three points P = (1, 1, 5), Q = (3, 4, 3), R = (1, 5, 7) in three space. (a) Use the cross product to nd the area of the triangle with vertices P , Q and R. Ans. P Q =< 2, 3, 2 >, P R =< 0, 4, 2 >. ij k 2 3 2
Solution of Quiz 3 for MATH 20C, 2009
1. Describe the domain and the range of the function f (x, y, z ) = Ans. Domain of f = cfw_(x, y, z )|9 x2 y 2 z 2 0 = cfw_(x, y, z )|x2 + y 2 + z 2 9 which is the interior and the boundary of a sphere centered at the
Quiz 4 Solution for MATH 20C, 2009
1. Evaluate the limit or determine that the limit does not exist. y2 (x,y )(0,0) x2 + y 2 lim Ans. We choose two dierent path approaching (0, 0) and nd that there exist two limit values along dierent paths. First, along
Quiz 5 Solution for MATH 20C, 2009
1. Let f (x, y ) = xex
2 y
and P = (1, 1). fP . , xex
2 y
(a) Find the rate of change of f in the direction of Ans. f =< ex and at P = (1, 1), f (P ) =< 3, 1 >. D
f (P ) f (P )
2 y
+ 2x2 ex
2 y
>
= | f (P )| =
32 + 1 =
Quiz 6 Solution for MATH 20C, 2009
1. Let f (x, y ) = ex xey . (a) Find the critical points of the function f . Ans. fx = ex ey = 0 ex = ey , or x = y, x=0 fy = xey = 0
By solving two equations x = y and x = 0, we get x = 0, y = 0. So, the critical point
Quiz 7 Solution for MATH 20C, 2009
1. For the following double integral, (a) sketch the region and (b) change the order of integration and evaluate.
1 0 y 1
sin x dxdy x
Ans. (b) The domain of integration is D = cfw_(x, y )|0 x 1, 0 y x.
1 0 0 x
sin x dyd
Review Solution for Midterm II, MATH 20C, 2009
1
Chapter 13
1. A parking ramp follows the curve in space traced by the function r(t) =< cos t, sin t, t > What is the curvature function (t) of the ramp? Ans. Note that the curvature formula is (t) = |r (t)
Review with Solution for Midterm I, MATH 20C, 2009
1
Chapter 12.
1. [12.2] Find an equation of a sphere if one of its diameters has end-points (2, 1, 4) and (4, 3, 10). Ans. The center of the sphere is the mid-point of the given two end points of one diam
Extra Homework on 14.7 and 14.8
Section 14.7 1. Find the points on the plane x y + z = 4 that is closest to the point (1, 2, 3). 2. Find three positive numbers whose sum is 100 and whose product is a maximum. 3. Find the volume of the largest rectangular
Print Name: TA Name: Math 20C. Midterm Exam 2 May 26, 2006
Section Number: Section Time:
No calculators or any other devices are allowed on this exam. Write your solutions clearly and legibly; no credit will be given for illegible solutions. Read each que
Print Name: TA Name: Math 20C. Midterm Exam 2 May 26, 2006
Section Number: Section Time:
No calculators or any other devices are allowed on this exam. Write your solutions clearly and legibly; no credit will be given for illegible solutions. Read each que
Name: TA: Math 20C. Midterm Exam 1 April 25, 2007 Sec. No:
PID: Sec. Time:
Turn o and put away your cell phone. You may use any type of handheld calculator; no other devices are allowed on this exam. You may use one page of notes, but no books or other as
Name: TA: Math 20C. Midterm Exam 2 May 23, 2007 Sec. No:
PID: Sec. Time:
Turn o and put away your cell phone. You may use any type of handheld calculator; no other devices are allowed on this exam. You may use one page of notes, but no books or other assi
Mathematics 20C Spring 2009 Name: ID:
Test 1 Section:
Be sure to show all work. Answers without full justication are worth no credit. Remember that you can often check your answers. You may use a single page of notes, but no other aids are allowed. This t
Mathematics 20C Spring 2009 Name: ID:
Test 2 Section:
Be sure to show all work. Answers without full justication are worth no credit. Remember that you can often check your answers. You may use a single page of notes, but no other aids are allowed. This t
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Print Name: Math 20C. Midterm Exam 1 October 17, 2005
Student Number:
Read each question carefully, and answer each question completely. Show all of your work. No credit will be given for unsupported answers. Write your solutions clearly and legibly. No c
Print Name: Math 20C. Midterm Exam 1 October 17, 2005
Student Number:
Read each question carefully, and answer each question completely. Show all of your work. No credit will be given for unsupported answers. Write your solutions clearly and legibly. No c
Name: TA Name: Math 20C. Midterm Exam 1 April 28, 2006
Section Number: Section Time:
No calculators or any other devices are allowed on this exam. Write your solutions clearly and legibly; no credit will be given for illegible solutions. Read each questio
Name: Math 20C. Midterm Exam 1 July 9, 2004
Student Number:
Read each question carefully, and answer each question completely. Show all of your work. No credit will be given for unsupported answers. Write your solutions clearly and legibly. No credit will
Name: Math 20C. Midterm Exam 1 July 9, 2004
Student Number:
Read each question carefully, and answer each question completely. Show all of your work. No credit will be given for unsupported answers. Write your solutions clearly and legibly. No credit will
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Print Name:
Student Number: Section Time:
Math 20C. Midterm Exam 2 November 21, 2005
Read each question carefully, and answer each question completely. Show all of your work. No credit will be given for unsupported answers. Write your solutions clearly an