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 2
z
=
y
,
z
(0, 0) = 0,
x
y
a2 x 2 y 2
z
(0, 0) = 0
y
1
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 perpendicular to r(t). Solution. Notice rst that at the time t wher
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 that 2w 2w 2w = . uv x2 y 2 By the chain rule, w w x w
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 = cos sin , y = sin sin , z = cos , express the partial
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 computing site use E to take the exponent and there is no need
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 proof reads as follows: Proof. Suppose that > 0 is given.
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 tangent at t = 1, where is it at t = 2? Solution. First n
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 function such that g (1, 0) = 1, and g (1, 0) = i 3j. Calcul
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, y ) = xy = Q(x, y ) and the same disk for P = 2y, Q =
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 proofs other than the sort covered in a Euclidean geometry c
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 complex plane. Heres how we tell if c belongs to the Mand
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 = Q(x, y ) and the same disk for P = 2y, Q = x. . 2. (1
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 of the textbook; each section is a rectangle of length
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 rectangle D = [0, 1] [0, 1]. Find a transformation T su
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 ] and in [0, ]. Solution. Here, T = (3 sin sin , 2 cos
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, y) = 2x2 + 3y 2 = c, c = 6, 12
1
(i)
(ii)
(iv)
2
(iii
Solution 1, Ma 1c Prac 2014
April 7, 2014
1
Problem 2.1.3
1
x2 y 2 = 1
x2 y 2 = 1
2x2 + 3y 2 = 12
2
x2 y 2 = 0
2x2 + 3y 2 = 6
Problem 2.1.10
(a)f (x, y) = x2 + y 2 + 1
The level curves are intersections of the graph with the planes z = c. As c
goes from t
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 the textbook without worrying too much about the quality
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 the textbook without worrying too much about the quality
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 derivatives w/x and w/y: (d) w = x/y
3. 2.3.10
Compute the ma
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
sin(x + y)dA =
0
0
R
1
( cos(1 + y) + cos(y)dy = sin(2)
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 + y 2 )2 k and S is the surface of the
cylinder x2 + y 2
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
sin(x + y)dA =
0
0
R
1
( cos(1 + y) + cos(y)dy = sin(2)
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 calculated over.
cos(x)
/2
y sin(x) dydx
0
0
5.3.8
Let D b
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 the textbook without worrying too much about the quality
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).
3.3:
In the following three exercises, nd the critica
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 [0, ]. 2. (15 Points) Section 7.3, Exercise 15 (a) Find
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 = [0, 1] [0, 1]. Find a transformation T such that D i
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 textbook; each section is a rectangle of length 5 and widt
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) Set up a triple integral giving the integral of a functi