HW 2 SOLUTIONS, MA 1C PRAC 2013
1. 2.3.2
Evaluate the partial derivatives z/x and z/y for the given function at the indicated points:
(a) z = a2 x2 y 2 ; (0, 0), (a/2, a/2)
Solution.
z
=
x
x
a2 x 2 y
1
Mathematics 1c: Solutions, Homework Set 4
Due: Monday, April 26 at 10am. 1. (10 Points) Section 4.1, Exercise 14 Show that, at a local maximum or minimum of the quantity r(t) , r (t) is perpendicula
1
Mathematics 1c: Solutions, Homework Set 3
Due: Monday, April 19th by 10am. 1. (10 Points) Section 3.1, Exercise 16 Let w = f (x, y ) be a function of two variables, and let x = u + v, y = u v. Show
1
Mathematics 1c: Homework Set 2
Due: Monday, April 12th by 10am. 1. (10 Points) Section 2.5, Exercise 8 Suppose that a function is given in terms of rectangular coordinates by u = f (x, y, z ). If x
1
Mathematics 1c: Solutions to homework Set 1
1. (10 Points) Using the computing site or otherwise, draw the graphs of the following functions: (a) f (x, y ) = 3(x2 + 2y 2 )ex y ; Tip: On the computin
Finding N : a vague outline Math 8 2009, Chris Lyons Lets suppose you have a sequence cfw_an of real numbers, and youre trying to prove that
n
lim an = L
by using the denition of the limit. Your pro
Mathematics 1c: Solutions, Midterm Examination Due: Monday, May 3, at 10am 1. Do each of the following calculations. (a) If a particle follows the curve c(t) = et1 i (t 1)j + sin(t)k and ies o on a ta
1
Mathematics 1c: Solutions, Final Examination Due: Wednesday, June 9, at 10am 1. (a) [7 points] Let f : R3 R2 be dened by f (x, y, z ) = e2xy , x2 z 2 4x + sin(x + y + z ) and let g : R2 R be a funct
1
Mathematics 1c: Solutions, Homework Set 8
Due: Tuesday, June 1 at 10am. 1. (10 Points) Section 8.1, Exercises 3c and 3d. Verify Greens theorem for the disk D with center (0, 0) and radius R and P (x
Calculus of One and Several Variables and Linear Algebra
MA 1C

Spring 2009
Some remarks about math and proofs
Math 8 2009, Chris Lyons While I cant speak about international standards, students at the average American high school receive little exposure to mathematical proof
Calculus of One and Several Variables and Linear Algebra
MA 1C

Spring 2009
How to dene the Mandelbrot set Math 8 2009, Chris Lyons The famous Mandelbrot set is a subset of the complex plane:
The black points signify the members of the Mandelbrot set. Fix a number c in the co
Calculus of One and Several Variables and Linear Algebra
MA 1C

Spring 2009
1
Mathematics 1c: Homework Set 8
Due: Tuesday, June 1 at 10am. 1. (10 Points) Section 8.1, Exercises 3c and 3d. Verify Greens theorem for the disk D with center (0, 0) and radius R and P (x, y ) = xy
1
Mathematics 1c: Solutions, Homework Set 5
Due: Monday, May 10th at 10am. 1. (10 Points) Section 5.1, Exercise 4 Using Cavalieris principle, compute the volume of the structure shown in Figure 5.1.11
1
Mathematics 1c: Solutions, Homework Set 6
Due: Monday, May 17 at 10am. 1. (10 Points) Section 6.1, Exercise 6 Let D be the parallelogram with vertices (1, 3), (0, 0), (2, 1) and (1, 2)
and D be the
1
Mathematics 1c: Solutions, Homework Set 7
Due: Monday, May 24 at 10am. 1. (10 Points) Section 7.3, Exercise 6. Find an expression for a unit vector normal to the surface x = 3 cos sin , for in [0, 2
Ma 1c Prac Assignment 1
Due 2pm Monday, April 7, 2014.
1
Problem 2.1.3
Match the level curves of the following functions with their visual descriptions.
(a) f (x, y) = x2 y 2 = c, c = 0, 1, 1
(b) f (x
Solution to HW 3, Ma 1c Prac 2014
Remark : every function appearing in this homework set is suciently nice
at least C 3 following the jargon from the textbookwe can apply all kinds of
theorems from th
Solution to HW 4, Ma 1c Prac 2014
Remark : every function appearing in this homework set is suciently nice
at least C 3 following the jargon from the textbookwe can apply all kinds of
theorems from th
HW 2 , MA 1C PRAC 2014
1. 2.3.2
Evaluate the partial derivatives z/x and z/y for the given function at the indicated points:
(a) z = a2 x2 y 2 ; (0, 0), (a/2, a/2)
2. 2.3.3
Find the two partial deriva
HOMEWORK 6 SOLUTIONS
5.2.2(c)
Evaluate the integral R sin(x + y)dA on the region R = [0, 1] [0, 1]
Solution
Using Fubinis theorem we can write this as an iterated integral to get
1
1
sin(x + y)dx dy
s
HOMEWORK 7
7.6.4
Let F(x, y, z) = 2xi 2yj + z 2 k. Evaluate
F dS,
S
where S is the cylinder x2 + y 2 = 4 with z [0, 1].
7.6.14
Evaluate the surface integral S F ndA, where F(x, y, z) = i + j + z(x2 +
HOMEWORK 5 SOLUTIONS
5.2.2(c)
Evaluate the integral R sin(x + y)dA on the region R = [0, 1] [0, 1]
Solution
Using Fubinis theorem we can write this as an iterated integral to get
1
1
sin(x + y)dx dy
s
HOMEWORK 6
5.2.2(c)
Evaluate the integral
R
sin(x + y)dA on the region R = [0, 1] [0, 1]
5.3.4(d)
Evaluate the following integral. Additionally, sketch the region of R2 that this
integral is being cal
Solution to HW 4, Ma 1c Prac 2014
Remark : every function appearing in this homework set is suciently nice
at least C 3 following the jargon from the textbookwe can apply all kinds of
theorems from th
3.2:
4. Determine the secondorder Taylor formula for f (x, y) = 1/(x2 + y 2 + 1)
about (0, 0).
10. Let f (x, y) = xcos(y) ysin(x). Find the secondorder taylor approximation for f at the point (1, 2)
Calculus of One and Several Variables and Linear Algebra
MA 1C

Spring 2009
1
Mathematics 1c: Homework Set 7
Due: Monday, May 24 at 10am. 1. (10 Points) Section 7.3, Exercise 6. Find an expression for a unit vector normal to the surface x = 3 cos sin , for in [0, 2 ] and in [
Calculus of One and Several Variables and Linear Algebra
MA 1C

Spring 2009
1
Mathematics 1c: Homework Set 6
Due: Monday, May 17 at 10am. 1. (10 Points) Section 6.1, Exercise 6 Let D be the parallelogram with vertices (1, 3), (0, 0), (2, 1) and (1, 2)
and D be the rectangle D
Calculus of One and Several Variables and Linear Algebra
MA 1C

Spring 2009
1
Mathematics 1c: Homework Set 5
Due: Monday, May 10th at 10am. 1. (10 Points) Section 5.1, Exercise 4 Using Cavalieris principle, compute the volume of the structure shown in Figure 5.1.11 of the tex
Calculus of One and Several Variables and Linear Algebra
MA 1C

Spring 2009
1
Review Example 1, Chapter 8. Let W be the region in the octant x 0, y 0, z 0, bounded by the three planes y = 0, z = 0, x = y , and by the sphere x2 + y 2 + z 2 = 1. (a) Find the volume of W . (b) S