Download Final Exam Review

MATH 137 CALCULUS 2
SPRING 2012
FINAL EXAM REVIEW
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The examples I've listed aren't meant to be exhaustive, so don't just study these examples and
expect to have covered all your bases. Instead, each example should just give you an idea of the
type of problem to expect.
Integration by parts
EX
 6 x cos(2 x)dx
EX

ln( x )
dx
x2
EX
x e
EX
 arctan( 4 x)dx
EX

3
x
dx
Improper integration
EX


0
xe x dx (u-sub also)
2
1
0
2 x ln( x)dx (improper because
of the 0; have to use parts also)
EX

0

1
dx
2x  5
EX


2
4
dx (look like
x(ln( x)) 3
something we've done recently?)
Area between curves
EX Find the area bounded by f ( x)  cos( x) and g ( x)  sin( x) between x  0 and x   .
EX Find the area of the region in the first quadrant bounded by y  x 3 and y  2 x  x 2 .
Volumes of revolution
EX Find the volume of the solid obtained by rotating the region bounded by
y  40 x  5 x 2 in the first quadrant about the line x  10 . (have to use shells)
EX Find the volume of the solid obtained by rotating the region bounded by
y  40 x  5 x 2 and y   x 2  8 x about the line y  2 . (use discs)
EX Find the volume of the solid obtained by rotating the region bounded by y  x and
y
x about the line x  3 . Now try it about the line y  3 .
Separable first order differential equations
EX Find a general solution to the differential equation
dy
 xy cos(x) .
dx
EX Solve the initial value problem:
dy
2x
 2
, y (0)  3 .
dx y  x 2 y 2
EX Solve the initial value problem:
dy y cos( x)

, y (0)  0 .
dx
1 y2
EX
Find a general solution to the differential equation x cos( x)  (2 y  e 3 y ) y' .
Differential equation (other)
EX Sketch the slope field for y '  ( y  4)( y  x) and sketch on it solutions which start at
(-3, 2) and (-4, 6).
EX Given the differential equation y'  y 2 ( y  6)( y  5) , sketch a graph of possible
solutions by finding the equilibrium solutions and then sketching a representative
solution from each region they define. Finally, label each equilibrium solution as stable,
semi-stable, or unstable.
EX Find the values of r which would make y  e rx a solution of y ' ' '2 y ' '24 y '  0
Convergence of sequences
EX Determine whether the following sequences converge or diverge.
a) an 
n
ln( n)
b) an 
n 2  3n
n!
c) a n 
cos 2 (n)
2n
d) a n 
arctan( n)
1  1n
Convergence of series
EX Determine whether the following series converge or diverge.
a)
3n  7 n
 6n
c)

b)
n
3
 n(ln( n))
d)
n6  2
2
(1) n 2n 2
 n 1
EX* Explain the difference between “if” and “iff”. Use convergence tests as examples.
Power series
EX Find the interval of convergence for the following power series.
(1) n (2 x  1) n

n
EX Find the first five nonzero terms (not just the coefficients) of the Taylor expansion of
a) f ( x)  ln( x) about a 
1
2
b) f ( x)  5 sin( x) about a   .
EX Write the following function as a power series.
2
a) f ( x)  x arctan( x)
sin( x4 )
b) f ( x) 
x
c*) f ( x)  2 x 3 ln( 2 x  3)
d*) f ( x)  x 2 e  x
4
EX Evaluate  2 x cos(3x)dx using MacLaurin series.
Regions in R 3
EX Describe the region represented in R 3 by the following equations and inequalities.
a)
yx
z0
b)
x2  y 2  z 2  1
y 2  z 2  16
Vectors


EX Let a  2,3,1 and b   3,1,2 .


a) Find  a  3b
 
b) Find a  b


c) Find the angle between a and b
 
d) Find a  b
Lines and planes in R 3
EX Find the equation of the line through the points (2,5,1) and (1,1,4) .
EX Find an equation of the plane through the points (1,2,1) , (2,5,1) and (1,1,4) .
x  1 t
EX Find the equation of the plane containing the point (3,3,3) and the line y  t
z  1  2t
.