Calculus Derivative Definition Notes
The derivative is the function for which at x = k, the parent functions slope at x = k can
be determined.
It finds the slope of the tangent line at x = k. For example, the derivative of the function,
F(x) = x2, is re
Math 215 HW #11 Solutions
1. Problem 5.5.6. Find the lengths and the inner product of
x=
2 4i
4i
and y
2 + 4i
.
4i
Answer: First,
x
2
= xH x = [2 + 4i 4i]
2 4i
= (4 + 16) + 16 = 36,
4i
so x = 6. Likewise,
y
2
2 + 4i
= (4 + 16) + 16,
4i
= y H y = [2 4i 4i]
Math 215 HW #11
Due 5:00 PM Thursday, April 29
Reading: Sections 5.55.6 from Strangs Linear Algebra and its Applications, 4th edition.
Problems: Please follow the guidelines for collaboration detailed in the course syllabus.
1. Problem 5.5.6.
2. Problem 5
Math 215 HW #10 Solutions
1. Problem 5.2.14. Suppose the eigenvector matrix S has S T = S 1 . Show that A = S S 1 is
symmetric and has orthogonal eigenvectors.
the eigenvectors of A. Then, since S T = S 1 ,
Proof. Suppose S = [v1 . . . vn ], where vi are
Math 215 HW #10
Due 5:00 PM Thursday, April 15
Reading: Sections 5.25.4 from Strangs Linear Algebra and its Applications, 4th edition.
Problems: Please follow the guidelines for collaboration detailed in the course syllabus.
1. Problem 5.2.14.
2. Problem
Math 215 HW #9 Solutions
1. Problem 4.4.12. If A is a 5 by 5 matrix with all |aij | 1, then det A
big formula or pivots should give some upper bound on the determinant.
. Volumes or the
Answer: Let vi be the ith column of A. Then
|vi | =
a2i + a2i + a2i
Math 215 HW #9
Due 5:00 PM Thursday, April 8
Reading: Sections 4.4, 5.15.2 from Strangs Linear Algebra and its Applications, 4th edition.
Problems: Please follow the guidelines for collaboration detailed in the course syllabus.
1. Problem 4.4.12.
2. Probl
Math 215 HW #8 Solutions
1. Problem 4.2.4. By applying row operations to produce an upper triangular U , compute
2 1 0
0
1
2 2 0
1 2 1 0
2
3 4 1
and
det
det
0 1 2 1 .
1 2 0 2
0
0 1 2
0
2
53
Answer: Focusing on the rst matrix, we can subtract twice
Math 215 HW #8
Due 5:00 PM Thursday, April 1
Reading: Sections 4.14.3 from Strangs Linear Algebra and its Applications, 4th edition.
Problems: Please follow the guidelines for collaboration detailed in the course syllabus.
1. Problem 4.2.4.
2. Problem 4.2
Math 215 HW #7 Solutions
1. Problem 3.3.8. If P is the projection matrix onto a k -dimensional subspace S of the whole
space Rn , what is the column space of P and what is its rank?
Answer: The column space of P is S. To see this, notice that, if x Rn , t
Math 215 HW #7
Due 5:00 PM Thursday, March 25
Reading: Sections 3.33.4 from Strangs Linear Algebra and its Applications, 4th edition.
Problems: Please follow the guidelines for collaboration detailed in the course syllabus.
1. Problem 3.3.8.
2. Problem 3.
Math 215 HW #6 Solutions
1. Problem 3.1.14. Show that x y is orthogonal to x + y if and only if x = y .
Proof. First, suppose x y is orthogonal to x + y. Then
0 = x y, x + y
= (x y)T (x + y)
= xT x + xT y yT x yT x
= x, x + x, y y, x y, y
= x, x y, y
sinc
Math 215 HW #6
Due 5:00 PM Thursday, March 18
Reading: Sections 3.13.2 from Strangs Linear Algebra and its Applications, 4th edition.
Problems: Please follow the guidelines for collaboration detailed in the course syllabus.
1. Problem 3.1.14.
2. Problem 3
Math 215 HW #4 Solutions
1. Problem 2.3.6. Choose three independent columns of U . Then make two other choice. Do the same for A. You have found bases for which spaces? 2341 2341 0 6 7 0 0 6 7 0 U = 0 0 0 9 and A = 0 0 0 9 . 4682 0000 Solution: The most o
Math 215 HW #5
Due 5:00 PM Thursday, February 25
Reading: Sections 2.32.4, 2.6 from Strangs Linear Algebra and its Applications, 4th edition.
Problems: Please follow the guidelines for collaboration detailed in the course syllabus.
1. Problem 2.3.6.
2. Pr
Math 215 HW #4 Solutions
1. Problem 2.1.6. Let P be the plane in 3-space with equation x + 2y + z = 6. What is the
equation of the plane P0 through the origin parallel to P? Are P and P0 subspaces of R3 ?
Answer: For any real number r, the plane x + y + =
Math 215 HW #4
Due 5:00 PM Thursday, February 18
Reading: Sections 2.12.2 from Strangs Linear Algebra and its Applications, 4th edition.
Problems: Please follow the guidelines for collaboration detailed in the course syllabus.
1. Problem 2.1.6.
2. Problem
Math 215 HW #3
Due 5:00 PM Thursday, February 11
Reading: Sections 1.61.7 from Strangs Linear Algebra and its Applications, 4th edition.
Problems: Please follow the guidelines for collaboration detailed in the course syllabus.
1. Problem 1.6.6.
2. Problem
Math 215 HW #2 Solutions
1. Problem 1.4.6. Write down the 2 by 2 matrices A and B that have entries aij = i + j and
bij = (1)i+j . Multiply them to nd AB and BA.
Solution: Since aij indicates the entry in A which is in the ith row and in the j th column,
Math 215 HW #2
Due 5:00 PM Thursday, February 4
Reading: Sections 1.41.5 from Strangs Linear Algebra and its Applications, 4th edition.
Problems: Please follow the guidelines for collaboration detailed in the course syllabus.
1. Problem 1.4.6
2. Problem 1
Math 215 HW #1 Solutions
1. Problem 1.2.3. Describe the intersection of the three planes u + v + w + z = 6 and u + w + z = 4
and u + w = 2 (all in four-dimensional space). Is it a line or a point or an empty set? What
is the intersection if the fourth pla
Math 215 HW #1
Due 5:00 PM Thursday, January 28
Reading: Sections 1.11.3 from Strangs Linear Algebra and its Applications, 4th edition. Reading
and understanding the material from the textbook is an important part of the course, so please do
not skip this
Analysis II (Math 318)
Spring 2011
1
Technicalities
Instructor: Clay Shonkwiler (cshonkwi@haverford.edu)
Oce: KINSC H210 (phone: 795.3367)
Course web page: http:/www.haverford.edu/math/cshonkwi/teaching/m318s11
Texts: Understanding Analysis, by Stephen Ab
Math 318 HW #10
Due 3:00 PM Friday, April 29
Reading:
Wilcox & Myers 3237.
Problems:
1. Exercise 34.36.
Comment: The unit ball not being compact is a common feature of innite-dimensional
normed linear spaces which can be a serious nuisance. In fact, one c
Math 318 HW #9 Solutions
1. Let A be a bounded, measurable set and let (fn ) be a sequence of measurable functions on A
converging to f . Suppose L(A) and that |fn (x)| (x) for all x A and all n = 1, 2, . . .
Show that
f g dm
fn g dm =
lim
n A
A
if g is m
Math 318 HW #9
Due 5:00 PM Thursday, April 14
Reading:
Wilcox & Myers 2830.
Problems:
1. Let A be a bounded, measurable set and let (fn ) be a sequence of measurable functions on A
converging to f . Suppose L(A) and that |fn (x)| (x) for all x A and all n
Math 318 HW #8 Solutions
1. (a) Prove Chebyshevs inequality, which says that if f is nonnegative and measurable on the
bounded, measurable set A, then
mcfw_x A : f (x) c
1
c
f dm.
A
Proof. Let c R and dene
Ac := cfw_x A : f (x) c,
which is measurable sin
Math 318 HW #8
Due 5:00 PM Thursday, April 7
Reading:
Wilcox & Myers 2127.
Problems:
1. (a) Prove Chebyshevs inequality, which says that if f is nonnegative and measurable on the
bounded, measurable set A, then
mcfw_x A : f (x) c
(b) Show that if
A |f |d
Math 318 HW #7 Solutions
1. Exercise 20.12. Let C be the Cantor set. Let D [0, 1] be a nowhere dense measurable set
with m(D) > 0. Then there is a non-measurable set B D. At each stage of the construction
of C and of D, a certain nite number of open inter