20B PROBLEM SET 5: TAYLOR SERIES, APPLICATIONS
1. a) Show that if f and g are continuous on [a, b] and g(x) 0, x [a, b] then for some c (a, b)
Z b
Z b
f (x)g(x) dx = f (c)
g(x) dx
a
a
b) Assuming that f (n+1) (xo + th) is continuous on [0, 1], show
Z 1
f
20B PROBLEM SET 1: ORBITAL MECHANICS, NEWTON AND KEPLERS LAWS
1. For orbital mechanics we want to be able to recognize an ellipse in polar coordinates when we see
one. We already know that the standard ellipse can be
parametrized by ~
() = (a cos , b sin
20B PROBLEM SET 4: ISOCHRONE, LENGTH, SURFACE AREA, TAYLOR SERIES
1. In class we showed that the time for the mass
to travel from (0, 0) to (R, 2R) along the cys
R
cloid was tb =
. We now must show that
g
if the mass started at an arbitrary point p1 it w
20B PROBLEM SET 3: GAUSSIAN INTEGRAL, DIFFERENTIAL EQUATIONS, BUILDING
INTEGRALS
1. In class we derived the recursion formula for the integral
We used this to obtain
2n 1
1 3 5
and
J(2n) =
2 2 4 6
2n
R /2
0
sinn xdx = J(n) obtaining J(n) = ( n1
n )J(n 2
20B PROBLEM SET 2: INTEGRALS AND PHYSICS
Z
a
1. a) Without using methods of integration or the FTOC find the value of
p
a2 x 2 dx by recognizing geometrically what
0
it represents .
2
2
y
x
+ 2 = 1 should be b/a times the area bounded by the circle of rad
20B PROBLEM SET 6: SERIES, POWER SERIES
1. a) Show that if the sum
P
k=1 ak
converges then limk ak = 0.
X
b) Does it follow that if lim ak = 0 then
ak converges ? If so, prove it, if not, give a counter example.
k
c) Does
X
k=1
[1 + 1/k]k converge ?
k=1
X