Advancing from Calculus I to Calculus II:
Software available to help you decide if you are ready for Calculus II
http://www.math.buffalo.edu/rur_index.html
You should be able to accomplish all of the following WITHOUT a calculator
Algebra:
Graph functions of the form: C œ 0 ÐB +Ñ , with right/left shifts of +units and up/down shifts of , units
http://www.youtube.com/watch?v=3Q5Sy034fok
Exponent rules
http://www.youtube.com/watch?v=Kr16rdBMX4o&feature=channel_page
Complete the square
http://www.youtube.com/watch?v=xGOQYTo9AKY&feature=channel_page
http://www.youtube.com/watch?v=zKV5ZqYIAMQ&feature=channel_page
Long Division (use this for rational functions if the degree of the top is the degree of the bottom)
http://www.purplemath.com/modules/polydiv2.htm
http://www.youtube.com/watch?v=l6_ghhd7kwQ
Rationalize the numerator or denominator of a rational function
http://xrl.us/rationalizenumerator
Trigonometry:
Know the exact values of all trig functions on the standard angles
http://www.youtube.com/watch?v=cIVpemcoAlY&feature=channel_page
Evaluate the exact values of inverse trig functions
http://tutorial.math.lamar.edu/Extras/AlgebraTrigReview/InverseTrig.aspx
Be able to use trig identities to simplify or expand trigonometric expressions
http://www.templejc.edu/dept/Math/CMartinez/math2412/notes7_2.pdf
Be able to use right triangles to evaluate compositions of trig functions and inverse trig functions
example: Evaluate sin(cos" BÑ
Let ) be the angle such that cos ) œ B (i.e. cos" B = )) and draw an appropriate triangle:
Use the Pythagorean Theorem to label the third side of the triangle
È" B#
Use this triangle to find sin(cos" BÑ œ sin ) =È" B#
Graph trigonometric functions with translations, change in period, change in amplitude
http://www.youtube.com/watch?v=80c_F0-7ZxE
Know these identities:
sin# B cos# B œ "
sin ( BÑ œ sin B
cos Ð BÑ œ cos B
Be able to use the above formulas to derive the following:
" tan# B œ sec# B
sin #B œ #sinBcosB
cos #B œ cos# B sin# B
Be able to use the above formulas to derive the following:
#B
#B
sin# B œ "cos
cos# B œ "cos
#
#
Calculus:
http://www.youtube.com/watch?v=EX_is9LzFSY
http://www.youtube.com/watch?v=Q9OkFTDG4fY&feature=related
http://justmathtutoring.com/ (lots of good videos)
Differentiation
Product Rule
C œ 0 ÐBÑ1ÐBÑ
Quotient Rule
Cœ
Chain Rule
C œ 0 Ð1ÐBÑÑ
C w œ 0 w ÐBÑ1ÐBÑ 0 ÐBÑ1w ÐBÑ
0ÐBÑ
1ÐBÑ
Cw œ
1ÐBÑ0 w ÐBÑ0 ÐBÑ1w ÐBÑ
[1ÐBÑ]#
C w œ 0 w Ð1ÐBÑÑ1w ÐBÑ
Formulas you should know:
Note that in the formulas below ? is a function of B and chain rule applies:
. ,
.B ?
.?
œ ,?," .B
.
.B ln
|?| œ
" .?
? .B
. ?
.B /
œ /?
.
.B sin ?
.?
.B
.?
œ cos ? .B
.
.B cos?
.?
œ sin? .B
.
.B tan
.?
? œ sec# ? .B
.
.B sec ?
.?
œ sec? tan? .B
.
.B cot
.?
? œ csc# ? .B
.
.B csc ?
.?
œ csc ? cot ? .B
.
1
.?
.B arcsin ?= È "?# .B
.
" .?
.B arctan ?= "?# .B
.
,
.B Ò?ÐBÑÓ
œ ,?ÒÐBÑÓ," .?ÐBÑ
.B
Integration :
Understand definite integral as signed area between a curve and the B-axis.
http://www.youtube.com/watch?v=LkdodHMcBuc
http://www.intmath.com/Applications-integration/2_Area-under-curve.php
Formulas you should know:
' ?8 .? œ
G
' /? .? œ /? G
G
' ?" .? œ lnl?l G
?8"
8"
' ln B .B Á
"
B
' sin? .? œ cos? G
' cos? .? œ sin ? G
' tan?.? œ lnksec ?k G
' sec# ? .? œ tan ? G
' csc# ? .? œ cot ? G
' sec? tan ? .? œ sec ? G
' csc? cot ? .? œ csc ? G
'
'
.?
+# ?#
.?
È +# ?#
œ arcsin ?+ G
œ +" arctan ?+ G
Techniques you should be able to use:
Substitution
http://www.youtube.com/watch?v=qclrs-1rpKI
http://www.youtube.com/watch?v=qclrs-1rpKI
#
#
# B
B#
B
Break up numerator B %B œ B% B% ÐCautionx B%B
$ Á % B$ Ñ
Add and subtract the same number to regroup terms
"
' B#B%"
' B##B%
'
example: ' B##B$
# +%B& .B œ
+%B& .B œ
+%B& .B B# +%B& .B
For the first integral, use substitution; for the second, complete the square to obtain something for tan"
Common mistakes ÐDON'T MAKE THEM!):
"
B
B#
CB
"
C
Á
Á
B
C
#
BC
"
B
B#
B$
"
C
Á
Á
#
$
"
BC
"
B
"
C
B# "
B
"
BC
ÁB"
sin #B Á #sin B
sin Ð#BÑ
B
ÐB CÑ# Á B# C #
ÈB# C # Á B C
BÐC DÑ Á BC D
"
B
/+, Á /+ /,
' 68B .B Á
Á B
Á sin Ð#Ñ
sin ÐBÑ
sin (CÑ
Á
Á
B
C
"
B
-
Useful numbers to know (even though your calculator can find them):
ln (1) = 0
ln (0) does not exist
ln (/) = 1
/! œ "
sin (0) = 0
sin ( 1# Ñ œ "
sin (1) = 0
sin (21)=0
cos (0) = 1
cos ( 1# Ñ œ 0
cos (1) = "
cos (21Ñ œ "
tan (0) = 0
tan ( 1# ) does not exist
tan ( 14 ) = 1
Practice Problems:
Algebra:
1. Graph the following functions of the form: C œ 0 ÐB +Ñ , using standard graphs and shifts.
a) 0 ÐBÑ œ ÐB $Ñ# "
b) 0 ÐBÑ œ B# #B $ (complete the square first)
c) 0 ÐBÑ œ ÐB #Ñ$
Trigonometry:
2Þ Give the exact values:
a) cos
1
$
e) cos" "
b)sin
1
%
f) tan" "
c) tan 1'
d) cot
1
#
g) csc" #
h) sin" "#
3Þ Given the following triangle, find the 6 trigonometric functions of )
4. Write in terms of cos +B only note this skills will help you put trig functions in integrable form:
a). cos% B
b) sin# Bcos# B
sin# B
c) "
cosB
5. Verify the Identities
a) cos# #B sin# #B œ cos 4B
c)
sec B
tan# B
œ Ðsin BÑ# ÐcosBÑ
b) sin$ B cos% B œ cos% B sinB cos' BsinB
6. Given C œ arcsin B" ß find cot C .
7. Graph the following pairs of functions.
a)
0 ÐBÑ œ sin B
1ÐBÑ œ #sin B
c)
0 ÐBÑ œ cos B
1ÐBÑ œ " cos B
b)
0 ÐBÑ œ tan B
1ÐBÑ œ tan #B
Calculus:
8. Differentiate:
a) C œ
B%
)
"
%B#
d) C œ sin (3B# Ñ
c) C œ
/B
e) 0 ÐBÑ œ ln "/
B
f) 1Ð:Ñ œ :# tan :#
#
#
g) C œ sin B#cos B
9. Write the equation of the line tangent to the curve C œ $B# #B at the point (1, 5).
"0Þ Find the area between the curve C œ sec# B and the B-axis on [0, 1% Ó
11Þ Use the picture below to approximate the integral
a) Using 4 intervals and the right endpoint
b) Using 2 intervals and the midpoint rule
"
B
b) C œ ln ÐB# #Ñ
'"& 0 ÐBÑ .B
12. Find the anti-derivative:
a)
'
d)
' B ÈB BE .B
%.B
b)
' Ð )>$ "&>& ,>Ñ.>
c)
' $/2B .B
e)
' ÈB
f)
' BÈB.B
#
#
#
'
g)
Š B"
"
#
j)
"
B$ ‹.B
B"
'!" B B"
.B
#
$
h) '
1
%
k)
'
1
%
)B#
.B
sin#B .B
arctanB
"B# .B
1
cos B
'
i)
.B
! "sinB
l)
'
"
$B .B