Advancing from Calculus II to Calculus III
See also: Advancing from Calculus I to Calculus II
Software available to help you decide if you are ready for Calculus III:
http://www.math.buffalo.edu/rur_index.html
Other useful websites:
http://justmathtutoring.com/ (lots of good instructional videos)
http://tutorial.math.lamar.edu/Classes/CalcII/CalcII.aspx (Calculus II lecture notes)
Topics:
(you should be able to accomplish all of the following WITHOUT a calculator)
Algebra:
Complete the square
http://www.youtube.com/watch?v=xGOQYTo9AKY&feature=channel_page
http://www.youtube.com/watch?v=zKV5ZqYIAMQ&feature=channel_page
Solve a System of Equations (including nonlinear equations)
http://www.purplemath.com/modules/syseqgen3.htm
Recognize Conic Sections
http://www.stewartcalculus.com/data/ESSENTIAL%20CALCULUS%20Early%20Transcendentals/upfiles/ess-reviewofconics.pdf
Work with Parametric Curves
http://tutorial.math.lamar.edu/Classes/CalcII/ParametricEqn.aspx
Trigonometry:
Be able to compute quickly 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
Use the Pythagorean Theorem to label the third side of a right triangle
Calculus I and II:
Methods of finding limits
L'Hopital's Rule for limits
http://www.math.hmc.edu/calculus/tutorials/lhopital/
.C
Find .B
for a curve given in rectangular form, parametric form, or polar form
http://www.youtube.com/watch?v=pe5HtTwYte8
http://tutorial.math.lamar.edu/Classes/CalcII/PolarTangents.aspx
Integration techniques from Calculus I and II
The Substitution Rule
Integration by Parts
http://www.youtube.com/watch?v=dqaDSlYdRcs
Trigonometric Integrals and Trigonometric Substitution
http://www.math.northwestern.edu/~mlerma/courses/math214-2-03f/notes/c2-trigint.pdf
Integration of Rational Functions by Partial Fractions
http://www.math.ucdavis.edu/~kouba/CalcTwoDIRECTORY/partialfracdirectory/PartialFrac.html
Improper Integrals
http://archives.math.utk.edu/visual.calculus/4/improper.2/index.html
Applications:
Arc Length of a Curve
http://archives.math.utk.edu/visual.calculus/5/arclength.1/index.html
Area Between Curves
Area Enclosed by a Polar Curve
http://tutorial.math.lamar.edu/Classes/CalcII/PolarArea.aspx
Center of Mass
http://tutorial.math.lamar.edu/Classes/CalcII/CenterOfMass.aspx
Linear approximation of a function near a specific point
http://www.sosmath.com/calculus/diff/der06/der06.html
Find the maximum or minimum of a continuous function on a closed, bounded interval
http://www.math.umn.edu/~maxwell/docs/OptimizationSlides.pdf
Formulas you should know:
Differentiation:
Product Rule
C œ 0 ÐBÑ1ÐBÑ
Quotient Rule
Cœ
Chain Rule
C œ 0 Ð1ÐBÑÑ
. ,
.B ?
.?
œ ,?," .B
.
.B ln
|?| œ
" .?
? .B
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
. ?
.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 arcsec ?= ?È ?# " .B
Integration:
' ?8 .? œ
?8"
8"
' ln B .B Á
"
B
G
G
.
" .?
.B arctan ?= "?# .B
' /? .? œ /? G
' ?" .? œ lnl?l G
' sin? .? œ cos? G
' cos? .? œ sin ? G
' sec# ? .? œ tan ? G
' csc# ? .? œ cot ? G
'
'
' tan?.? œ lnksec ?k G
' sec? tan ? .? œ sec ? G
'
.?
È +# ?#
.?
+# ?#
œ arcsin ?+ G
œ +" arctan ?+ G
' sec?.? œ lnksec? tan?k G
' csc? cot ? .? œ csc ? G
.?
?È ?# +#
œ arcsec k?+k G
Practice problems:
"Þ Integrate the followingÞ
a) ' B /#B .B
d) '
b) ' > lnÐ> "Ñ .>
$B
$B# # .B
g) ' sin# B .B
e) '
h) '
c) ' /#B sin B .B
f) ' sin# B cos$ B.B
'B"$
B# &B' .B
B$
È *B# .B
#Þ Find the limits.
B# B"#
#
BÄ$ B B'
a) lim
c) lim /
BÄ!
#B "
B
B#
B
/
BÄ∞
b) lim
d) lim /B ÈB
BÄ∞
3. Identify the following conic sections:
a) B# #B $C # œ '
b) B# #B $C œ '
c) B# #B $C # "#C œ "#
d) #B $C # "#C œ '
4. Find all solutions to the following systems (here Bß Cß Dß and - denote real numbers)
%B œ #Ba)  )C œ #C B# C # œ *
c)

#BC # D # œ #B#B# CD # œ #C #B# C # D œ #D B# C # D # œ "
C/BC œ $B# b)  B/BC œ $C # B$ C $ œ "'
d)

CD œ #BBD œ %C BC œ 'D #
B #C # $D # œ '
5. Assuming +ß ,ß and - are constants, differentiate the following with respect to BÞ
B# ,#
B# +#
a) C œ +B arctan(,BÑ
b) 0 ÐBÑ œ
c) C œ sin (B# +$ Ñ B/+,B
d) 1ÐBÑ œ /#
,ln (B-Ñ
.C
6. Find .B
for the following parametric curves. Then find all points where the curve has a vertical tangent
and all points where the curve has a horizontal tangent.
a) B œ > sin>
b) B œ >$ $>
c) B œ " ln >
C œ > #cos>
C œ ># "!>
C œ arctan>
! Ÿ > Ÿ #1
"! Ÿ > Ÿ (
!&>&∞
.C
7. Find .B
for the following curves given in polar form. Then find all points where the curve has a vertical
tangent and all points where the curve has a horizontal tangent.
a) < œ # cos )
b) < œ # sin )
8. Describe/sketch the parametric curve given below.
! Ÿ > Ÿ #1
a) B œ cos > C œ sin >
b) B œ cos > C œ sin >
1 Ÿ > Ÿ #1
c) B œ # cos > C œ # sin > ! Ÿ > Ÿ #1
d) B œ sin > C œ cos >
! Ÿ > Ÿ #1
9.
a)
b)
c)
Parameterize the curves described below.
A circle of radius 5 centered at ( 1,4) , oriented counterclockwise.
The line going through the point (3, #Ñ having a slope of 6.
The piece-wise smooth curves shown below.
10. Find the area enclosed by the curves C œ B$ B# )B " and C œ #B$ B# &B "
11. Find the area enclosed by the polar curve < œ % % cos ).
12. Let 0 ÐBÑ œ ln B. Find 0+ ÐBÑ, the linear approximation of 0 ÐBÑ for values of B near B œ +. Use your
result to compute an approximate value of ln (1.1) without using a calculator. Then use a calculator to
compute a more accurate value of ln (1.1). How big is the error associated with using the linear
approximation?
13. Find the maximum and the minimum of the function
a) 0 ÐBÑ œ B# on the interval Ò "ß $Ó
b) 0 Ð>Ñ œ #>$ $># "#> % on the interval Ò %ß #Ó