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
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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
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Name: Solmliom Km& Student Id: (>0
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Problem1234567
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of 10 10 10 10 10 10 10
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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.
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Discussion Section: (Circle 4pm, 5pm, 6pm (Jamie), 6pm (Colin), 7pm, 8pm)
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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