MATH 1B Mathematics | UC Berkeley加州大学 | final exam数学代考

MATH 1B FINAL (LEC 002)
PROFESSOR PAULIN
DO NOT TURN OVER UNTIL INSTRUCTED TO DO SO.
CALCULATORS ARE NOT PERMITTED
THIS EXAM WILL BE ELECTRONICALLY SCANNED. MAKE
SURE YOU WRITE ALL SOLUTIONS IN THE SPACES
PROVIDED. YOU MAY WRITE SOLUTIONS ON THE BLANK
PAGE AT THE BACK BUT BE SURE TO CLEARLY LABEL
THEM
Z
tan(x) dx = ln |sec(x)| + C
Z
sec(x) dx = ln |sec(x) + tan(x)| + C
cos2
(x) = 1 + cos(2x)
2
sin2
(x) = 1 − cos(2x)
2
e
x = 1 + x +
x
2
2 +
x
3
6 + · · · =
P∞
n=0
x
n
n!
sin x = x −
x
3
3! +
x
5
5! −
x
7
7! +
x
9
9! − · · · =
P∞
n=0(−1)n x
2n+1
(2n+1)!
cos x = 1 −
x
2
2! +
x
4
4! −
x
6
6! +
x
8
8! − · · · =
P∞
n=0(−1)n x
2n
(2n)!
arctan x = x −
x
3
3 +
x
5
5 −
x
7
7 +
x
9
9 − · · · =
P∞
n=0(−1)n x
2n+1
2n+1
ln(1 + x) = x −
x
2
2 +
x
3
3 −
x
4
4 +
x
5
5 − · · · =
P∞
n=1(−1)n−1 x
n
n
(1 + x)
k = 1 + kx +
k(k−1)
2! x
2 +
k(k−1)(k−2)
3! x
3 + · · · =
P∞
n=0 
k
n

x
n
limn→∞(
n+1
n
)
n = e
Name:
Student ID:
GSI’s name:
Math 1B Final (LEC 002)
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Math 1B Final (LEC 002), Page 2 of 15
This exam consists of 10 questions. Answer the questions in the
spaces provided.

1. (30 points) Compute the following integrals:
   (a)
   Z
   arctan(x)dx
   Solution:
   PLEASE TURN OVER
   Math 1B Final (LEC 002), Page 3 of 15
   (b)
   Z
   (
   √
   x
   2 − 1)3
   x
   8
   dx
   You should express your final answer in terms of x using an appropriate right
   triangle.
   Solution:
   PLEASE TURN OVER
   Math 1B Final (LEC 002), Page 4 of 15
2. (30 points) Determine if the following series are absolutely convergent, conditionally
   convergent or divergent. You do not need to show your working.
   (a)
   X∞
   n=2
   log2
   (n + 1) − log2
   (n − 1)
   Solution:
   (b)
   X∞
   n=1
   √
   n4 − 1
   n3 + 3
   Solution:
   (c)
   X∞
   n=1
   sin(πn)
   n
   Solution:
   (d)
   X∞
   n=1
   2
   n + 4n + 5n
   6
   n − 4
   n
   Solution:
   (e)
   X∞
   n=2
   (−1)nn tan( 1
   n2
   )
   Solution:
   PLEASE TURN OVER
   Math 1B Final (LEC 002), Page 5 of 15
3. (30 points) Determine the interval of convergence of the following power series
   X∞
   n=1
   (2x + 1)3n
   8
   n
   √3 n + 1
   .
   Carefully justify your answer.
   Solution:
   PLEASE TURN OVER
   Math 1B Final (LEC 002), Page 6 of 15
4. (30 points) Calculate the Maclaurin Series of the following function.
   f(x) = (x − 1)2
   sin(x).
   Express your final answer in sigma-notation.
   What is the value of f
   (100)(0)?
   Solution:
   PLEASE TURN OVER
   Math 1B Final (LEC 002), Page 7 of 15
5. (30 points) Find a general solution to the following differential equation
   x
   2
   y
   0 = e
   x − 4xy, x > 0.
   Solution:
   PLEASE TURN OVER
   Math 1B Final (LEC 002), Page 8 of 15
6. (30 points) Find an equation for the orthogonal trajectory to the family of curves
   y = 2 + √3
   x
   2 + k (k any constant)
   which contains the point (3, 2).
   Solution:
   PLEASE TURN OVER
   Math 1B Final (LEC 002), Page 9 of 15
7. (30 points) Give an example of a differential equation y
   0 = F(y) with the property that
   limx→∞ y(x) = 1 for any initial condition y(0) = y0. Give a precise formula and draw
   the graph of y
   0 versus y. Justify your answer with the method of direction fields.
   Solution:
   PLEASE TURN OVER
   Math 1B Final (LEC 002), Page 10 of 15
8. (30 points) Are there any solutions to
   y
   00 − 6y
   0 + 10y = 0
   satisfying the conditions y(0) = 0 and y(π) = 1? Carefully justify your answer.
   Solution:
   PLEASE TURN OVER
   Math 1B Final (LEC 002), Page 11 of 15
9. (30 points) Find a general solution to the following differential equation.
   y
   00 − y = 2(3x
   2 + 3x + 1)e
   x
   Solution:
   PLEASE TURN OVER
   Math 1B Final (LEC 002), Page 12 of 15
10. (30 points) Find a power series (centered at 0) which is a solution to
    y
    00 = 2xy0 + 2, y(0) = 0, y0
    (0) = 0.
    Solution:
    PLEASE TURN OVER
    Math 1B Final (LEC 002), Page 13 of 15
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    Math 1B Final (LEC 002), Page 14 of 15
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    PLEASE TURN OVER
    Math 1B Final (LEC 002), Page 15 of 15
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    END OF EXAM

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