for Review beginning of Math 242
This document covers some topics that you should have learned in Math 241
or whatever your equivalent course was.
If x = 3, evaluate x2 , x2 , 2x2 , and 2x2 .
If x = 3, evaluate x2 , x2 , 2x2 , and 2x2 .
Expand and simplify:
(a + b)(x y + z ) =
(a + b)(a b) =
(a + b)(a2 ab + b2 ) =
(a + b)(a b) =
2 3 2 3
=
2+ 3 2 3
x2 + x + x
x2 + x x
=
x2 + x + x
Be very aware that
a+b
(a + b)2
sin(a + b)
n(a + b)
is
is
is
is
VERY
VERY
VERY
VERY
DIFFERENT
DIFFERENT
DIFFERENT
DIFFERENT
from a + b
from a2 + b2
from sin a + sin b
from n a + n b
Play around with this to convince yourself! Try some nice numbers, like
a = 1 and b = 1. (For the sine function, examples of nice numbers might
be a = /2 and b = /2.)
1
Solve the inequalities. In other words, for each inequality, nd all values of
x that make the inequality true.
x2 < 9
x2 > 9
|x 3| < 2
|x 3| > 2
Sketch the graphs of each of the following.
y = x2
x2 + y 2 = 4
x2 + y 2 = 2
y = x2 /3
x = y 2 /3
Where does y = 1 + x intersect y = 1 + x2 ?
Where does x = y 2 intersect x2 + y 2 = 2?
Where does y = x4 /2 y intersect = 4x?
Sketch the graphs of each of the following. Label any obvious special
points involving nice numbers. Are there asymptotes? What happens
when x approaches or ?
y = sin x
y = cos x
y = ex
y = nx
y = tan x
2
Fill in the following table:
sin
cos
tan
0
6
4
3
2
Make a similar table for angles in the second, third, and fourth quadrants.
Be aware of the rules for exponents and logarithms:
am an = am+n
am
= amn
n
a
(am )n = amn
n(ab) = n a + n b
n
a
= na nb
b
n(ab ) = b n a
n(a b) = THERE IS NO RULE FOR THIS.
, csc
, and cot
? What about other angles?
6
6
6
1
1
3
, arcsin
, and arcsin
?
What are arcsin(1), arcsin(0), arcsin
2
2
2
What are sec
3
Evaluate the following derivatives.
d
nx = ?
dx
d
sin x = ?
dx
d
tan x = ?
dx
d
xn = ?
dx
dx
e =?
dx
d
cos x = ?
dx
d
sec x = ?
dx
d sin x
=?
dx x
d
x2 tan x = ?
dx
d
x2 tan x2 = ?
dx
d
n(sec x) = ?
dx
d
arctan x = ?
dx
If f (x) = tan x, nd f (x), f (x), f (x), and f (x).
Evaluate the following limits.
x2 9
=?
x3 x 3
x2 + 1
=?
x 3x2 1
sin x
=?
x 0 x
5
=?
x ex
lim
lim
lim
lim
lim
x
17
=?
nx
lim
+
x0
4
17
=?
nx