Introductory Physics Volume Two

C Power Series Expansions 191
The Fundamental Theorem of Calculus
Recall that the distance traveled between time t 1 and time t 2 is
equal to the area under the v versus t graph between these times.
v(t)
Area = ∫ t 2
t 1
v(t)dt
t
t 1 t 2
But the distance traveled is also x(t 2 ) − x(t 1 ). So we find that
∫ t2
v(t)dt = x(t 2 ) − x(t 1 )
t 1
This works for any pair of functions where one is the derivative of the
other. So in general
IF
f(t) = dg
dt
THEN
∫ t2
t 1
f(t)dt = g(t 2 ) − g(t 1 )
§ C.3 Power Series Expansions
(1 + x) N =
∞
1
1 − x = ∑
x n = 1 + x + x 2 + x 3 + x 4 . . .
sin(x) =
n=0
∞∑
n=0
cos(x) =
e x =
N∑
n=0
∞∑
n=0
(−1) n
(2n + 1)! x2n+1 = x − 1 6 x3 + . . .
∞∑
n=0
(−1) n
(2n)! x2n = 1 − 1 2 x2 + . . .
1
n! xn = 1 + x + 1 2 x2 + 1 6 x3 + . . .
N!
N!
n!(N − n)! xn = 1 + Nx +
2!(N − 2)! x2 + · · ·