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MATH 221: Calculus and Analytic Geometry
Prof. Ram, Fall 2006
HOMEWORK 1
DUE September 11, 2006
(6) What do 2 + 3, 2 +(5) What are the complex numbers and why do we care?(4) What are the real numbers and why do we care?(3) What are the rational numbers and why do we care?(2) What are the nonnegative integers and why do we care? (1) What are the positive integers and why do we care?Problem A. Numbers
75 (^) , 94
75 (^) , 2 + 1
4 and 2 +
2 mean?
(7) What do
x 2 , x 1 and
x mean?
(8) What do
a
b , a
cb (^) , ba
dc
mean?
(9) What do 2
3 , 2
(^75) (^) , ( 32 ) 7 5 , 2^
x , 2
1 . 4 and 2
√ 2 mean?
(10) What do
a b , a (^) cb ,^ ( ba^ ) d^ c , 2^
x , and
x 2 mean?
(11) What do
x x and
x √ x mean?
(1) Find a complex number^ Problem B. Computing with complex numbers
z such that
z (^) +
w
=
w
for all other complex numbers
w .
(2) Find a complex number
x such that
xw
w
for all other complex numbers
w .
(3) Compute (
i ) + (2 + 5
i ) and graph the result.
(4) Compute (
i ) −
(
−
5 i ) and graph the result.
(5) Compute (4 + 8
i )(
i ) and graph the result.
(6) Compute
i
i
and graph the result.
(7) Compute (
i ) 3 and graph the result.
(8) Compute
i and graph the result.
(9) Compute
a (^) +
(^) bi
and graph the result, where
a, b
R
(10) Compute (
i ) + (7 + 2
i ) and graph the result.
(11) Compute (
i ) (^) −
(^) (
(^) −
(^6) i ) and graph the result.
(12) Compute (
i )(3 + 2
i ) and graph the result.
(13) Compute
i
i and graph the result.
(14) Compute 1
1 / 4 and graph the result.
(15) Compute 16
1 / 4 and graph the result.
(16) Compute (
1 / 3 ) 4 and 27
(4+
/
graph the result.
(17) Compute 1 +
(18) Compute 1
5 and 1
(19) Compute 1 +
Problem C. Functions (1) What is
x 2 ?
(2) What is
e x ?
(3) What is sin
(^) x ?
(4) What is cos
(^) x ?
(5) What is tan
x ?
(6) What is cot
(^) x ?
(7) What is sec
(^) x ?
(8) What is csc
(^) x ?
(9) What is sinh
x ?
(10) What is cosh
(^) x ?
(11) What is tanh
(^) x ?
(12) What is coth
(^) x ?
(13) What is sech
(^) x ?
(14) What is csch
(^) x ?
(15) What is
x ?
(16) What is ln
(^) x ?
(17) What is sin
− 1 x^ ?
(18) What is cos
− 1 x ?
(19) What is tan
− 1 x ?
(20) What is cot
− 1 x ?
(21) What is sec
− 1 x ?
(22) What is csc
− 1 x ?
(23) What is sinh
− 1 x^ ?
(24) What is cosh
− 1 x ?
(25) What is tanh
− 1 x ?
(26) What is coth
− 1 x ?
(27) What is sech
− 1 x^ ?
(28) What is csch
− 1 x^ ?
Problem D. Function identities
(1) Explain why
x = 1 +
x
x 2
x 3
(^) · · ·
(2) Explain why
x n −
1
x −
1
= 1 +
x
x 2
(^) x 3
(^) · · ·
(^) x n − 1 .
(3) Find all possibilities for
c 0 , c 1 , c 2
...
so that
f (^) ( x ) =
c 0 (^) +
(^) c 1 x (^) + (^) c 2 x 2 +^ (^) c 3 x 3 +^ (^) · · ·
satisfies
f ( x
y ) =
f (^) ( x ) f (^) ( y ).
(4) Explain why
e x = 1 +
x
x 2
x 3
x 4
x 5
x 6
(5) Explain why ln
x is the inverse function to
e x .
(6) Verify the identity
e x
y
e x e y
.^
(7) Verify the identity
e − x =
e x (^).
(8) Verify the identity (
e x ) n =
e nx
.
(9) Verify the identity
e 0 = 1.
(10) Verify the identity ln(
xy
) = ln
(^) x
(^) y .
(11) Verify the identity
(^) ln
(^) x
= ln(
/x
).
(12) Verify the identity ln
x n =
n (^) ln
(^) x .
(14) Explain why cos(13) Verify the identity ln 1 = 0.
(^) x
= 1
x 2
x 4
x 6
(15) Explain why sin
(^) x
=
x (^) −
x 3
x 5
x 7
(16) Verify the identity
e ix
= cos
(^) x
(^) i (^) sin
x .
(17) Verify the identity cos
2 x^
2 x^ = 1
(18) Verify the identity sin(
x ) =
(^) sin
(^) x .
(19) Verify the identity cos(
x ) = cos
(^) x .
(20) Verify the identity sin(
x
y ) = sin
(^) x
cos
(^) y
(^) x
sin
(^) y .
