In#13,sketchtheparametriccurves(onyourcalculator)andidentifythosewhichdefineyasafunction
ofx.Ineachcase,eliminatetheparametertofindanequationthatrelatesxandydirectly.
1)
x = 2t + 3 and y = 4t 3 fortintheinterval [ 0,3]
2)
x = tan t and y = sec t fort
In#13:
(a) Setupanintegralforthelengthofthecurve
(b) Useyourcalculatortofindthelengthofthecurve.
1)
x = sin y
3)
y=
e x + e x
2
0 y
2)
x = 1 y2
1
1
y
2
2
3 x 3
In#47,findtheexactlengthofthecurveanalyticallybyantidifferentiation.Youwillneedtosimplif
Testing Convergence HW
In#15,usetheIntegralTesttodetermineconvergenceordivergenceoftheseries.
1)
1
3n
n =1
2)
nen
3
2
n =1
4)
n
5)
n =1
3)
1
n +1
n =1
1
4n + 1
n =1
In#69,usetheAlternatingSeriesTesttodetermineconvergenceordivergenceoftheseries.
(1) n
Volumes: Cross Sections of Known Solids
Homework
1)
Find the volume of the solid whose base is bounded by the circle x 2 + y 2 = 4 , with
cross sections taken perpendicular to the x-axis that are:
(a)
(b)
(c)
(d)
2)
squares
equilateral triangles
semicircl
WHY?
While we know how to represent an object that moves along a straight line, it
becomes harder to do when that object is moving on a path in a plane with
magnitude and an infinite number of directions to pursue.
Vectors are designed to show us the magn
TESTING TESTING
TESTING FOR CONVERGENCE
INCLUDING AT ENDPOINTS
WHAT COULD HAPPEN?
1)The series could diverge
2)The series could converge absolutely
3)The series could converge conditionally
The n-th Term Test
PROS Quick test, should be your first try
CONS
TAYLORS THEOREM
GOING FORWARD
If we approximate a function represented by a power series by its Taylor
polynomial, it is important to know how to determine the ERROR in the
approximation. The error in question comes from cutting the series off after a
cer
Taylor Theorem HW
In#13,findtheTaylorpolynomialoforderfourforthefunctionat x = 0 ,anduseittoapproximate
thefunctionat x = 0.2 .
1)
e 2 x
2)
5sin( x)
3)
(1 x) 2
In#45,findtheMaclaurinseriesforthefunction.
4)
xe x
5)
sin 2 x
Hint: sin 2 x = 1 cos 2 x
TAYLOR SERIES
CHALLENGE!
Construct a polynomial P( x) = a0 + a1 x + a2 x 2 + a3 x3 + a4 x 4
with the following behavior at x = 0:
P(0) = 1
P(0) = 2
P(0) = 3
P(0) = 4
P (4) (0) = 5
Sounds hard right? But luckily, the predictability of the differentiation o
Taylor Series HW
1)
ConstructthefourthorderTaylorpolynomialat x = 0 forthefunction.
(a) f ( x) = 1 + x 2
(b)
f ( x) = e 2 x
2)
ConstructthefifthorderTaylorpolynomialandtheTaylorseriesforthefunctionat x = 0
(a) f ( x) =
1
x+2
(b)
f ( x) = e1 x
For#34,u
RADIUS
OF
CONVERGENCE
The Convergence Theorem
for Power Series
Note: R is the radius of convergence, and the set of all values of
x for which the series converges is the interval of convergence.
The radius of convergence completely determines the interval
Radius of Convergence HW
For#18,findtheradiusofconvergenceofthepowerseries.
1)
xn
2)
n=0
(1)
n
(4 x + 1) n
n=0
( x 2)n
10n
n =0
3)
4)
xn
n n 3n
n =1
n( x + 3)n
5n
n =0
5)
6)
n !( x 4)
n
n =0
7)
(2)
n
(n + 1)( x 1)
n
( x + )n
n
n =1
8)
n=0
Fo
Polar Functions
Why Polar?
Polarequationsenableustodefinesomeinterestingandimportantcurves
thatwouldbedifficultorevenimpossibletodefineintheformy=f(x).
y = sin 3x
y = cos 5 x
x = sin 3t
y = cos 5t
Polar Coordinates
(r , )
r givesthedirecteddistancefromOto
If x and y are given as functions
x = f (t ),
y = g (t )
over an interval of t-values, then the set of points ( x, y ) = ( f (t ), g (t )
defined by these equations is a parametric curve. The equations are parametric
equations for the curve.
The variable