> )+( -+bjbj99 @>[{_[{_-#vvD
5555HJJJJJJ$nn5555HHo&40
nn
v
: AP CALCULUS (AB)
Text: Larson, Hostetler, Edwards Calculus of a Single Variable 8th edition Houghton Mifflin Complany.
Supplementary Materials: Included are free response questions from past AP exams which are used for in class assignments and discussion.
Unit 1: Laying the groundwork
Limits and Continuity
Estimating limits from tables and graphs
(Use of TI-83 to explore the graphs of functions)
Left hand limits, right hand limits, and limits
Limits and the graphs of piece-wise defined functions
Limits by algebraic methods
Limits at infinity and the graphs of functions
Infinity as a limit and the graphs of functions
Continuity and limits
Intermediate value theorem
Formal definition of limits and continuity
The Limit Theorems
Introduction to the Derivative
This unit opens with the development of the difference quotient as an average rate of change of a function with a discussion about the nature of the difference quotient as delta x becomes small. Students then break into groups and graph the difference quotient for the function x3 using a small value for delta x. Students are then instructed to find a simple formula for the curve that they see (which they quickly recognize as a parabola). The various groups then report their results and the lessons wraps up with the calculation of the derivative function using limits.
The derivative as an instantaneous rate of change
(including the relationship to the average rate of change)
(velocity and acceleration are included as examples of rates of change)
Estimating the derivative at a point from a table of function values
Using the definition of the derivative to find the derivative function.
Unit 2: Properties of the Derivative
The power, product, quotient, and chain rule
The product rule is introduced by means of a calculator experiment.
Students are asked to use the calculator to determine if the derivative of a product
is the product of the derivative for xsin(x). After the product rule has been
correctly derived (using the definition of the derivative) students are taught to
graph the derivative using their calculator by entering y=nDeriv(Asin(A),A,X)
and using this graph and the dy/dx function on their calculators to confirm the
product rule.
Also investigated by TI-83 is the nature of the derivatives of odd and even
functions as well as their sum, product and compositions. After the students have
formed their conjectures they are asked to prove at least two of their conjectures
in groups and present to the class.
Finding the equation of the tangent line
Applications: Position, Velocity, and Acceleration along a line
Differentiability
(includes the examination of points where a function fails to have a
derivative and why this happens)
This topic begins with the students zooming in on a particular point of a
differentiable function until it appears to be linear. Then the students are
instructed to use the trace function of the calculator to find a second point on this
line which are used to find the slope. Next the students find the slope at the
zoomed in point by means of paper and pencil. Students quickly draw the
conclusion that the line they see in their calculator is the tangent line to the
curve (which leads into the linear approximation of functions), Next, the vertex
of an absolute value function is zoomed in on and the students are asked to
discuss the meaning of the result of this experiment. Finally the graph of the cube
root of x is explored and students discuss the derivative of this function at zero.
Differentials definition and use in writing linear approximations for functions
Mean Value Theorem
Unit 3: Trigonometric functions
The Derivatives of trigonometric functions
This unit opens with students graphing the sine function on their calculators.
Next, they construct a table of values for d(sin x)/dx using the dy/dx function in
their calculators. From this they are asked to speculate on the nature of the derivative function for sine. This is followed up with a challenge to the students to prove their conjecture and a discussion ensues as to how this can be done. In the course of writing the proof, students are permitted to use their calculators to give evidence for the value of relevant limits.
The power, product, quotient, and chain rule.
Applications: velocity, acceleration, and position
Unit 4: Implicit Differentiation
Finding derivatives implicitly
Finding the equation of the tangent line
Finding inverse trigonometric functions by implicit differentiation
Applications: related rate problems
Unit 5: Analyzing the properties of functions using derivatives
Curve sketching using derivatives
This includes the locating of relative extrema, the locating of inflection points,
Determining the intervals on which a given function is increasing and decreasing,
and determining the concavity of the function on various intervals.
Understanding the relationship between the graphs of a function and the graphs of its
derivatives
Students are given the graph of the derivative of a function and will be asked questions about the function and about the second derivative of the function. Much of this work is done in groups. There is an emphasis on writing explanations and justifications using standard mathematical language and notation.
Absolute maximums and minimums
Finding absolute extrema on various intervals
The extreme value theorem
Optimization problems
Unit 6: Indefinite Integrals
Evaluating definite integrals (including the change of variable method)
Solving differential equations
Slope fields (the relationship of slope fields to the solution of differential equations)
Applications include position, velocity, and acceleration along a line.
Unit 7: The Definite Integral
The definition of the definite derivative as the limit of a Riemann sum.
Properties of the definite integral
Approximating definite integrals using Riemann sums and
Trapezoidal sums when a function or graph of a function is given.
Evaluating definite integrals using the TI-83 calculator
The Fundamental Theorem of Calculus
Students are presented with a partial proof of this theorem.
Students use the fundamental theorem to evaluate definite integrals
Change of variable formula for the definite integral
Functions defined by Integrals
Finding local extrema of functions defined by integrals
Problems involving functions defined by an integral whose graph is depicted.
Using a function defined by an integral to represent a particular anti-derivative
A calculator exercise students use their calculators to graph xsin(x). They are then asked to ketch a graph of what they think the graph of EMBED Equation.3 would look like (included locating its critical and inflection points) from -2( to 2(. Students then graph this function by entering y=fnInt((Asin(A), A, 0, x) to check or confirm their sketches.
Applications
Distance and displacement from velocity
Area of a plane region (includes using the graphing calculator to find the
intersection points of plane curves)
Average value of a function
Approximating definite integrals from tables of data by Riemann sums and
Trapezoidal sums
Other applications emphasize the integral of a rate as accumulated change
and setting up an approximating Riemann Sum for a definite integral.
(example: the rates at which water enters and leaves a tank are given as functions of time. Students are then asked to find the maximum volume of water in the tank). There is a strong emphasis on using the graphing calculator to solve equations and calculate definite integrals.
Students will work these problems in groups and a strong emphasis is
placed on students making a written justification for their answers in
mathematically correct notation.
Unit 8: Volume
Calculating the volume of a solid with known cross-sections
Calculating the volumes of solids of revolution by means of disc and washer cross-
sections
Strong emphasis in this unit on using the graphing calculator
Unit 9: Logarithmic and Exponential Functions
This unit opens with a proof that the integral of 1/x is a logarithm and the number e is
defined on this basis. The derivative rules for the natural log and for the exponential function then quickly follow.
LHopitals Rule (including an investigation of the rates of growth of various functions)
Applications involving exponential growth and decay (with a focus on solving
differential equations where the rate of growth is proportional to the quantity
present.)
This unit serves as a review of all of the concepts, skills, and applications contained in
this course through the lens of exponential and logarithmic functions.
34<SU^wxy45##'#(#)#*#y#z###-+Ƿub[[ jphs$jhhsEHUmHnHu!jC
hsCJUV_HaJjhsUhG hsH*hshO*hs5'hGhG6B*H*OJQJ_Hph333hG6B*OJQJ_Hph333$hhs6B*OJQJ_Hph333!hhsB*OJQJ_Hph333hGB*OJQJ_Hph333hs5\y & , C m
&
V
` ^` gds
`P^`Pgds
`gdGgdG`^``gds!
&0`
P@`^``gds@`@
`
`PN
`P^`Pgds
`gdG
`v^`gds
` ^` gdsT()S orow
`P^`Pgds
`P^`P
`P^`Pgds:W2w
`P^`P^gds^`gds`gds
bUb()*+,-^gds`gds
`P^`P-KL\]^_`, n !R!!!!3"^gdsgds
`P^`P3""###$*$v$$$%%c%%&'R'v'w'y'z'{'
`gdG
`P^`Pp^pgds^`gds^gds^`gds{''''(*(j(k(l(m(n(((()y))#*w****+,+-+
`P^`Pgds^`gds
`P^`P> 00P:ps/ =!"#*$L%*LDpDd
B
SA?BHɍt0i̳k$D- Tɍt0i̳k\?6xtxTMo@;-m *
8R!8 **T@שUgy4UT$ˍw8ٵJkϼyf-c }q>R/~J
4-
&e K^B03n+5sq!u{sߨ0A2gĿ'L=,nF=&"5
31*"p@0 !'Ь" [Huv
^*?2TXD~۬+b40M`92j;v-qQ]UR6;Sߨ<'r,L
xhJ8_ŖߒHdma!Oho%-%CF4r8x){k>ƌw1]^҈wIu~6I?h5Y:b(E ֢Y"<ȓfJ
~YTHP!
#?iJТMV2"3U;Dd=ۇ]&%QL/x)R{!i+%_AN͎uu<˼81lUcU
r'~ gŹlA.Ħ`H$?@Root Entry
F,@Data
WordDocument @>ObjectPool`_1125647809F`p!Ole
CompObjNObjInfo
FMicrosoft EquationDNQEEquation.37
AsinAdA0x
Oh+'0 $Equation Native S1Table0
SummaryInformation(DocumentSummaryInformation8$0
P\h
t'AB CALCULUSMarcy L KeithNormal.dotmMicrosoft Office User2Microsoft Office Word@
@xf@@@
՜.+,0hp|
' ?#AB CALCULUSTitle
F Microsoft Word 97-2003 Document
MSWordDocWord.Document.89qw666666666vvvvvvvvv666666>6666666666666666666666666666666666666666666666666hH66666666666666666666666666666666666666666666666666666666666666666p62&6FVfv2(&6FVfv&6FVfv&6FVfv&6FVfv&6FVfv&6FVfv8XV~ 0@ 0@ 0@ 0@ 0@ 0@ 0@ 0@ 0@ 0@ 0@ 0@ 0@ 0@6666_HmH nH sH tH L`LNormal1$7$8$H$CJ_HaJmH sH tH DA DDefault Paragraph FontRiRTable Normal4
l4a(k (No List<& <Footnote ReferencePK![Content_Types].xmlN0EH-J@%ǎǢ|ș$زULTB l,3;rØJB+$G]7O٭VGRU1a$N% ʣꂣKЛjVkUDRKQj/dR*SxMPsʧJ5$4vq^WCʽD{>̳`3REB=꽻UtQy@\.X7<:+&
0h@>nƭBVqu ѡ{5kP?O&CנAw0kPo۵(h[5($=CVs]mY2zw`nKDC]j%KXK'P@$I=Y%C%gx'$!V(ekڤք'Qt!x7xbJ7 oW_y|nʒ;Fido/_1z/L?>o_;9:33`=S,FĔ觑@)R8elmEv|!ո/,Ә%qh|'1:`ij.̳u'kCZ^WcK0'E8S߱sˮdΙ`K}A"NșM1I/AeހQתGF@A~eh-QR9C 5
~d"9 0exp<^!~J7䒜t L䈝c\)Ic8E&]Sf~@Aw?'r3Ȱ&2@7k}̬naWJ}N1XGVh`L%Z`=`VKb*X=z%"sI<&n|.qc:?7/N<Z*`]u-]e|aѸ¾|mH{m3CԚ.ÕnAr)[;-ݑ$$`:Ʊ>NVl%kv:Ns_OuCX=mO4m's߸d|0n;pt2e}:zOrgI(
'B='8\L`"Ǚ
4F+8JI$rՑVLvVxNN";fVYx-,JfV<+k>hP!aLfh:HHX WQXt,:JU{,Z BpB)sֻڙӇiE4(=U\.O.+x"aMB[F7x"ytѫиK-zz>F>75eo5C9Z%c7ܼ%6M2ˊ9B"N
"1(IzZ~>Yr]H+9pd\4n(Kg\V$=]B,lוDA=eX)Ly5ote㈮bW3gp:j$/g*QjZTa!e9#i5*j5öfE`514g{7vnO(^ ,j~V9;kvv"adV݊oTAn7jah+y^@ARhW.GMuO
"/e5[s`Z'WfPt~f}kA'0z|>ܙ|Uw{@tAm'`4T֠2j
ۣhvWwA9ZNU+Awvhv36V`^PK!
ѐ'theme/theme/_rels/themeManager.xml.relsM
0wooӺ&݈Э5
6?$Q
,.aic21h:qm@RN;d`o7gK(M&$R(.1r'JЊT8V"AȻHu}|$b{P8g/]QAsم(#L[PK-![Content_Types].xmlPK-!֧60_rels/.relsPK-!kytheme/theme/themeManager.xmlPK-!g theme/theme/theme1.xmlPK-!
ѐ'
theme/theme/_rels/themeManager.xml.relsPK]
-#.>-+
-3"{'-+')-#:8@0(
B
S ?ow/#3<wxx%%JJ\\]]yy ,#,#/#3<wxx%%JJ\\]]yy ,#,#/#3<wxx%%JJ\\]]yy ,#,#/#Gs-#/#@,#,#,#,#-#p@UnknownG*Ax Times New Roman5^Symbol3.*Cx ArialC(MArialMTArialQ(MArial-ItalicMTArialC.,*{$ Calibri Light7.*{$ CalibriA$BCambria Math"A hy@?@? xx##P3 P?8&2!xxؕ}AB CALCULUS
Marcy L KeithMicrosoft Office UserCompObjr