(1) Find the Taylor series centered at 0 for the
function 111(1 + 1:).
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(2) Find the Taylor series centered at 1 for the
function (new.
cm a: WDQ
The Final Exam 21C, Spring 2011
There are 20 problems. Each problem is worth 10 points.
Justify your answers by citing relevant theorems and/or showing calculations.
Name:
ID:
(1)
(2)
(3)
(4)
(5)
(6)
(7)
(8)
(9)
(10)
Total
(11)
(12)
(13)
(14)
(15)
(16)
1. (10 pt)
Complete the following text to valid deﬁnitions. ( W § (Li’- ' )
a) A point (3:0, yo) in the r—y—plane is an interior point of a set R of the x—y—plane if
0e disk 81; qgcsi'k'vt 1110(in wewd (Xo,jvv)
Liv) 117M151 (g
b) A point (Io, yo) in the .
1. (10 pt) (a) Complete the following text to a valid deﬁnition.
The sequence {an} converges to the number L if for every positive
number 6 there is
such that for all
MOLWWA/QR‘PS m, >N
wehave \QM—“Ll < E,
’ (b) Using this deﬁnition, prove that the sequen
MAT21C (Koppe, Spring 2013):
Practice Midterm 1
Please ﬁll out the following.
Name: Solmliom Km& Student Id: (>0
We ﬁll out the following when we grade this exam.
Problem1234567
Points
of 10 10 10 10 10 10 10
Total:
Please write your answers in the pr
1
Problem 1 (10 pts)
1
as a geometric series and write it using the summation notation.
1 + x2
What is the interval of convergence of this geometric series?
(a) (4 pts) Express
(b) (4 pts) Find the Maclaurin series for tan1 x, (which is also denoted by ar
MAT 21C Practice Final Exam
The actual exam will take place on Monday, June 9, 2014 from 8am to 10am at GIEDT 1001.
Last Name:
First Name:
Student ID #:
Discussion Section: (Circle 4pm, 5pm, 6pm (Jamie), 6pm (Colin), 7pm, 8pm)
Name of Left Neighbor:
Name
Notes on Gradient Vectors, Tangent Planes, and Normal Lines
I think some of you wondered why the gradient vector
F (x 0 , y 0 , z 0 ) = F x (x 0 , y 0 , z 0 ), F y (x 0 , y 0 , z 0 ), F z (x 0 , y 0 , z 0 )
(1)
is perpendicular to the level surface F (x,
An Example Revealing f x y (a, b) = f y x (a, b)
For those who are curious, I would like to give you an example where the
order of the partial derivatives taken at a particular point matters quite a bit.
Let us consider the following function (see Figure
MAT 21C, Spring 2014
Quiz 8 05.27.14
Name:
Student ID:
Show all work clearly and in order, and circle your nal answers. Justify your answers algebraically whenever possible. You have 15 minutes to take this 10 point quiz.
1.
(3 points) Find the partial de
MAT 21C, Spring 2014
Quiz 7 05.20.14
Name:
Student ID:
Show all work clearly and in order, and circle your nal answers. Justify your answers algebraically whenever possible. You have 15 minutes to take this 10 point quiz.
1.
(3 points) Find the distance f