For Exercises 1-16, identify which functions shown here ( f , g , h , and so on) have the given characteristics f x = sin π 2 x + 3 g x = − 3 cos 1 2 x − π 4 h x = 3 sin − 1 2 x − π 5 k x = − 3 sec 2 x + π m x = 2 csc 2 x − π 2 − 3 n x = 3 tan x − π 2 p x = − 2 cot 1 2 x + π t x = − 3 + 2 cos x Has a vertical shift downward from the parent graph
For Exercises 1-16, identify which functions shown here ( f , g , h , and so on) have the given characteristics f x = sin π 2 x + 3 g x = − 3 cos 1 2 x − π 4 h x = 3 sin − 1 2 x − π 5 k x = − 3 sec 2 x + π m x = 2 csc 2 x − π 2 − 3 n x = 3 tan x − π 2 p x = − 2 cot 1 2 x + π t x = − 3 + 2 cos x Has a vertical shift downward from the parent graph
Solution Summary: The author explains that the given functions have a vertical shift downward from the parent graph.
For Exercises 1-16, identify which functions shown here (
f
,
g
,
h
,
and so on) have the given characteristics
f
x
=
sin
π
2
x
+
3
g
x
=
−
3
cos
1
2
x
−
π
4
h
x
=
3
sin
−
1
2
x
−
π
5
k
x
=
−
3
sec
2
x
+
π
m
x
=
2
csc
2
x
−
π
2
−
3
n
x
=
3
tan
x
−
π
2
p
x
=
−
2
cot
1
2
x
+
π
t
x
=
−
3
+
2
cos
x
Has a vertical shift downward from the parent graph
EXAMPLE 3
Find
S
X
√√2-2x2
dx.
SOLUTION Let u = 2 - 2x². Then du =
Χ
dx =
2- 2x²
=
信
du
dx, so x dx =
du and
u-1/2 du
(2√u) + C
+ C (in terms of x).
Let g(z) =
z-i
z+i'
(a) Evaluate g(i) and g(1).
(b) Evaluate the limits
lim g(z), and lim g(z).
2-12
(c) Find the image of the real axis under g.
(d) Find the image of the upper half plane {z: Iz > 0} under the function g.
k
(i) Evaluate
k=7
k=0
[Hint: geometric series + De Moivre]
(ii) Find an upper bound for the expression
1
+2x+2
where z lies on the circle || z|| = R with R > 10. [Hint: Use Cauchy-Schwarz]
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