AM105b, Spring 2005 FIRST AND SECOND ORDER ODEsESSENTIALS General solution to linear first order ode : y + a ( x ) y = h( x ) y ( x ) = cy1 ( x) + y P ( x ) with homogeneous solution y1 ( x ) = e

x0 a ( ) d
x
x
h( ) d x1 y ( ) 1 where c , x0 an
Calculus I Fri 1PM Quiz 5
ID:
Name:
Signature:
1 Find the interval of convergence of the following series
10 Points
X
(1)n+1 (x + 2)n
.
n3n
n=1
2 Consider the following series
4+6
Points
X
2n xn
.
n
n=1
(a) Find the series radius of convergence R.
(b) On
Calculus I Fri 10AM Quiz 5
ID:
Name:
Signature:
1 Find the interval of convergence of the following series
10 Points
X
nxn
.
4n (n2 + 1)
n=1
2 Consider the following series
5+5
Points
n
2
X
x +1
n=0
3
.
(a) Find the series radius of convergence R.
(b) On
Calculus I Fri 1PM Solution Quiz 4
ID:
Name:
Signature:
1 Determine whether the following sequences converge or diverge, and find the limit of
3+7 each convergent sequence.
Points
sin n
(a)
n
Solution. Since n1
sin n
n
1
n,
by the Sandwich theorem for se
Calculus I Fri. 1PM Quiz 2
ID:
Name:
Signature:
1 Indicate whether the statement is true(T) or false(F). (2 points for correct answer, 1
10 Points point for incorrect answer, and 0 for no answer.)
(1) f = O(g) implies f = o(g). (
(2) sin1
1
x
grows faste
Calculus I Fri 1PM Quiz 6
ID:
Name:
Signature:
1 Use Taylors formula with a = 0 and n = 3 to find the cubic approximation of
1
at x = 0.
(1 x)
Also, give an upper bound for the magnitude of the error in the approximation when
10 Points f (x) =
x 0.1, by
MAS101 2014 FRI 1PM Quiz 1 ID:
Name:
Signature:
1 Find derivative of y with respect to x.
5+5+5
Points
(a) y = ln (ln (ln (x)
(b) tan (y) = ex + ln x
1
(c) y = 2 + 2x + x2 + x x
2 Evaluate the integrals.
5+5
Points
(a)
Z
(b)
Z
e r
dr
r
/2
7cos (t) sin (t
Calculus I Fri 10AM Quiz 6
ID:
Name:
Signature:
1 Find the first three nonzero terms in the Maclaurin series for the function f (x).
10 Points
f (x) = cos(tan1 x)
2 Let a = (2, 0) and b = (1, 1).
10 Points Describe the set of vectors cfw_x = sa + tb  0 s
Calculus I Fri 10AM Quiz 4
ID:
Name:
Signature:
1 Determine whether the following sequence converge or diverge, and nd the limit of
3+7 each convergent sequence.
Points
cos n
(a)
2n
(b) (21/n + 31/n )n
2 Determine how many terms should be used to estimate
Calculus I Fri. 10AM Quiz 2
ID:
Name:
Signature:
1 Indicate whether the statement is true(T) or false(F). (2 points for correct answer, 1
10 Points point for incorrect answer, and 0 for no answer.)
(1) f = o(g) implies f = O(g). (
(2) tan1
1
x
grows fast
Calculus I Fri 1PM Quiz 4
ID:
Name:
Signature:
1 Determine whether the following sequence converge or diverge, and find the limit of
3+7 each convergent sequence.
Points
sin n
(a)
n
(b)
3n + 1
3n 1
n
2 Determine how many terms should be used to estimate t
MAS101 2014 FRI 10AM Quiz 1 ID:
Name:
Signature:
1 Find derivative of y with respect to x.
5+5+5
Points
(a) y = ln (sec (x) + tan (x)
(b) xy = y x
(c) y =
x x2
x e
2 Evaluate the integrals.
5+5
Points
(a)
Z
(b)
Z
0
Mar. 14, 2014
tan(x) ln(cos(x)dx
/4
ta
Calculus I Fri 1PM Quiz 5
ID:
Name:
Signature:
1 Find the interval of convergence of the following series
10 Points
X
(1)n+1 (x + 2)n
.
n3n
n=1
Solution: We apply the Ratio test.
(1)n+2 (x + 2)n+1 (1)n+1 (x + 2)n
n
x + 2
x + 2
/
.
=
n+1
n
(n + 1)3
n
Applied Mathematics 105b
February 2008 reprint/revision of February 1997 text
J. R. Rice
Notes on solutions of (linearized) ode systems near critical points Let {x} denote a column of n functions x1(t), x2(t), ., xn(t) such that {x} = [x1(t), x2(t
Applied Mathematics 105b
Feb 2008 (revision of Feb 1997 version)
J. R. Rice
Notes on linear ode's with constant coefficients (see also Greenberg, 1998, Sect. 3.4 & 3.7) Linear operator L with constant coefficients: dny d n 1y dy Ly + a1 n 1 + . +
Exact Differentials and Integrating Factors J. R. Rice, AM 105b, 6 February 2008 Consider a first order ode which is given in the form
dy = dx
M (x, y) . N(x, y)
To solve such an ode is equivalent, after multiplying by dx, and rearranging, to find
Jett Roberts
ENG 1320
October 2nd 2016
Essay #2
Student athletes should not be paid for their sport of choice
because college athletes already receive scholarships for their sport, and
the schools top priority is to educate, not to hire entertainers.
Stud
Calculus I Fri 10AM Quiz 3
ID:
Name:
Signature:
1 Evaluate the following integrals.
5+5
Z
Points (1)
1
0
x+1
dx
x2 + 2x
Solution. Let u = x2 + 2x, then du = 2x + 2.
Z
0
1
x+1
dx =
x2 + 2x
Z
0
3
du
= lim
2 u b0+
Z
b
3
du
= lim [ u]3b = 3
2 u b0+
Each sim
Calculus I Fri 1PM Quiz 3
ID:
Name:
Signature:
1 Evaluate the following integrals.
5+5
Z
Points (1)
2
x2
2
dx
1
Solution.
Z
2
2
dx =
2
x 1
Z
1
1
dx (+2 points)
x1 x+1
2
x 1 b
= lim ln
b
x + 1 2
b 1
ln 2 1
= lim ln
b
b + 1
2+1
= ln 3
(+3 points)
Z
(2)
Calculus I Fri 10AM Quiz 5
ID:
Name:
Signature:
1 Find the interval of convergence of the following series
10 Points
X
nxn
.
4n (n2 + 1)
n=1
Solution: We apply the Ratio test.
n+1
(n + 1)x(n+1)
nxn
n2 + 1
x
x
/
=
.
n
2
2
(n+1)
2
n
(n + 1) + 1 4
4
4
(
Calculus I Fri 1PM Quiz 6 Sol.
ID:
Name:
Signature:
1 Use Taylors formula with a = 0 and n = 3 to find the cubic approximation
1
at x = 0.
(1 x)
Also, give an upper bound for the magnitude of the error in the approxi
10 Points of f (x) =
mation when x
Calculus I Fri 10AM Solution Quiz 4
ID:
Name:
Signature:
1 Determine whether the following sequences converge or diverge, and find the limit of
3+7 each convergent sequence.
Points
cos n
(a)
2n
Solution. Since 21n
cos n
2n
1
2n ,
by the Sandwich theorem
Calculus I Fri 10AM Quiz 6 Sol.
ID:
Name:
Signature:
1 Find the first three nonzero terms in the Maclaurin series for the function
10 Points f (x).
f (x) = cos(tan1 x)
Solution:
cos x = 1
x2
x4
x6
+
+
2!
4!
6!
tan1 x = x
(+2 points)
x3
x5
x7
+
+
3
5
7
i
I Feb 4, 2008
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