AP Calculus BC
Name: ___________________
1.
2.
3.
Power Series and Euler’s Identity
4-13-16
Using e ix = cosx + i sin x , verify.
a.
1
cosx sin y = [cos(x + y) + cos(x − y)]
2
b.
d
cosx = −sin x
dx
c.
sin 2x = 2sin x cosx
d.
sin(−x) = −sin x
e.
sin 2 x + cos2 x = 1
Evaluate.
a.
ln(3 − 4i)
b.
ln(−2 + 2i)
c.
ln(4i)
d.
ln(−π)
Derive a power series for each function below. Write at least three terms for each series and then write the series
in sigma notation, if possible.
a.
f (x) = x 4 sin 2x
b.
f (x) = cos(x −1)3
c.
f (x) = e x
d.
f (x) = ∫ t 2 tan−1 t 3dt
e.
f (x) =
1
x −1
f.
f (x) =
3
x
g.
f (x) =
3x
x + 2x − 2
h.
f (x) =
6
x +2
i.
Expand f (x) = cosx as a Taylor Series about x =
3
x
0
4
2
π
.
2