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AP Calculus AB

Course Syllabus Math Department

E-Mail: Emily.lucas@pikeville.kyschools.us

Contact Times: 1:15 P.M. – 2:05 P.M. School Phone: 606-432-0185

Grade Level: 11-12 Credit: 1 hour Fees: None Prerequisite: Algebra I, Algebra II, Geometry, Pre-Calculus

Course Description: This course teaches all topics associated with Functions, Graphs, and Limits; Derivatives; and Integrals as delineated in the Calculus AB Topic Outline in the AP Calculus Course Description. Course Design and Philosophy: Students  do  best  when  they  have  an  understanding  of  the  conceptual  underpinnings  of  calculus.  Rather  than  making  the  course  a  long  laundry  list  of  skills  that  students  have  to  memorize,  the  “why’  behind  the  major  ideas  is  stressed.  If  students  can  grasp  the  reasons  for  an  idea  or  theorem,  they  can  usually  figure  out  how  to  apply  it  to  the  problem  at  hand.    Teaching Strategies:  During  the  first  few  weeks,  extra  time  is  spent  familiarizing  students  with  graphing  calculators.  Students  are  taught  the  rule  of  three:  Ideas  can  be  investigated  analytically,  graphically,  and  numerically.  Students  are  expected  to  relate  the  various  representations  to  each  other.  The  graphing  calculator  is  used  to  help  students  develop  an  intuitive  feel  for  concepts  before  they  are  approached  through  typical  algebraic  techniques.    Finding  a  root,  sketching  a  function  in  a  specified  window,  approximating  the  derivative  at  a  point  using  numerical  methods,  approximating  the  value  of  a  definite  integral  using  numerical  methods,  and  other  calculator  functions  will  ultimately  be  emphasized.    It  is  important  for  them  to  understand  that  graphs  and  tables  are  not  sufficient  to  prove  an  idea.    Verification  always  requires  an  analytic  argument.      It  is  important  to  maintain  a  high  level  of  student  expectation.  Students  will  rise  to  the  level  that  is  expected  of  them.  A  teacher  needs  to  have  more  confidence  in  the  students  than  they  have  in  themselves.      Communication  is  stressed  as  a  major  goal  of  the  course.    Students  are  expected  to  explain  problems  using  proper  vocabulary  and  terms.  Many  times  students  take  the  role  of  teacher  in  their  small  groups  or  upon  occasion  to  the  whole  class.    Communication  between  classmates  is  encouraged  and  facilitated.    

jbelcher� 7/27/12 4:29 PMComment [1]: C3 – The course provides students with the opportunity to work with functions represented in a variety of ways – graphically, numerically, analytically, and verbally – and emphasizes the connections among these representations. jbelcher� 7/27/12 4:29 PMComment [2]: C5 – The course teaches students how to use graphing calculators to help solve problems, experiment, interpret results, and support conclusions.

jbelcher� 7/27/12 4:29 PMComment [3]: C4 – The course teaches students how to communicate mathematics and explain solutions to problems both verbally and in written sentences.

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Resource Materials: Primary Textbook Title: Calculus, 7th edition Author: Larson/Hostetler/Edwards Publisher: Houghton Mifflin Company Copyright: 2002 ISBN: 0-618-14918-X Resource Textbook Title: Calculus, 5th edition Author: James Stewart Publisher: Brooks/Cole Copyright: 2003 ISBN: 0-534-39366-7 Resource Title: Multiple-Choice & Free-Response Questions in Preparation for the AP Calculus (AB) Examination, 7th edition Title: Accompanying Student Solutions Manual, 7th edition Author: David Lederman & Lin McMullin Publisher: D&S Marketing Systems, Inc. Copyright: 1999 Resource Title: Fast Track to a 5: Preparing for the AP Calculus AB and BC Examinations Author: Sharon Cade, Rhea Caldwell, & Jeff Lucia Publisher: McDougal Little Copyright: 2006 ISBN: 0-618-14944-9 Technology Resources: Geometer’s Sketchpad, Derive, TI-84 Plus Graphing Calculator Online Help: http://archives.math.utk.edu/topics/calculus.html http://www.math.com/homeworkhelp/Calculus.html http://hotmath.com/ Grading Policy: Nine Weeks Grade will comprise:

Test and Quizzes – 70% of grade Homework and related assignments – 30% of grade

Grading Scale: 93-100 A 83-92 B 73-82 C 63-72 D 62-Below F

Performance Standards and Expectations:

• Students will use graphing calculators to understand the relationships between the analytic and graphical characteristics of functions, such as predicting and explaining the observed local and global behavior of a function.

jbelcher� 7/27/12 4:29 PMComment [4]: C2 – The course teaches all topics associated with Functions, Graphs, and Limits; Derivatives; and Integrals as delineated in the Calculus AB Topic Outline in the AP Calculus Course Description.

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Performance Standards and Expectations: (Continued) • Students will find limits algebraically, estimate limits graphically and numerically, and use limit

notation correctly. • Students will know the definition and properties of a continuous function as they relate to limits

and graphical interpretations; explain continuity at a point, and be able to identify functions which are continuous.

• Students will understand asymptotes in terms of graphical behavior and in terms of limits involving infinity.

• Students will find horizontal and vertical asymptotes using limits. • Students will be able to compare relative magnitudes of functions and their rates of change (for

example, contrasting exponential growth, polynomial growth, and logarithmic growth). • Students will be able to explain and derive the limit definition of the derivative with the

understanding of derivative as an instantaneous rate of change, and explain the relationship between differentiability and continuity.

• Students will be able to estimate derivatives of functions graphically, numerically, and using a graphing calculator.

• Students will be able to explain the difference between average and instantaneous rate of change, approximate rates of change from graphs and tables, and sketch a graph of f´(x) given the graph of f(x).

• Students will be able to use implicit differentiation when applicable, such as the derivative of an inverse function.

• Students will be able to explain how to find critical points and extreme values, state and apply the Mean Value Theorem.

• Students will be able to sketch a graph of f(x), given characteristics of f, f´, f´´ (increasing and decreasing behavior of f and the sign of f´; relationship between the concavity of f and the sign of f´´; points of inflection as places where concavity changes). In addition, curve sketching will draw upon the student’s knowledge of domain and range, vertical and horizontal asymptotes, symmetry, continuity and differentiability.

• Students will be able to use Calculus to solve application problems such as related rates, optimization (absolute and relative extrema), velocity, speed, acceleration, and position.

• Students will be able to differentiate and integrate basic functions (power, exponential, logarithmic, trigonometric, and inverse trigonometric functions) using rules for the derivative, such as sums, products, quotients, chain, and implicit.

• Students will understand that rules for differentiation are derived from the limit definition of the derivative.

• Students will be able to define a definite integral as a limit of Riemann sum using left, right, and midpoint summation. In addition, students will use trapezoidal sums to approximate definite integrals of functions represented algebraically, graphically, and by tables of values.

• Students will be able to solve separable differential equations and use them in modeling (in particular, studying the equation y´ = ky and exponential growth).

• Students will be able to interpret differential equations using slope fields and understand the relationship between slope fields and solution curves to differential equations.

• Students will be able to use the Fundamental Theorem of Calculus to solve Definite Integrals and understand its relationship to rate of change of a quantity over an interval. In addition, students will use the Fundamental Theorem to represent a particular anti-derivative, and the analytical and graphical analysis of functions so defined.

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Performance Standards and Expectations: (Continued) • Students will solve anti-derivatives using substitution of variables (including change of limits

for definite integrals) and recognize anti-derivatives that follow directly from derivatives of basic functions.

• Students will be able to find specific anti-derivatives using initial conditions, including applications to motion along a line.

• Students will be able to use integrals to find accumulated change, the area of a region, volume of a solid of revolution (Disk, Washer, and Shell method), volume of a solid by known cross section, average value of a function, and the distance of a particle along a line.

Topics to be Covered:

Limits and Their Properties (≈4 weeks)

1.) Finding Limits Graphically and Numerically 2.) Evaluating Limits Analytically 3.) Continuity and One-Sided Limits

(Intermediate Value Theorem) 4.) Infinite Limits 5.) Limits at Infinity(Horizontal Asymptotes)

Differentiation (≈6 weeks)

1.) Derivative and the Tangent Line Problem 2.) Basic Differentiation Rules and Rates of Change 3.) Product and Quotient Rules and Higher-Order Derivatives 4.) Chain Rule 5.) Implicit Differentiation

Differentiation and Applications (≈6 weeks)

1.) Related Rates 2.) Extrema on an Interval

(Extreme Value Theorem) 3.) Rolle’s Theorem and the Mean Value Theorem 4.) Increasing and Decreasing Functions,1st Derivative Test,

Concavity and the Second Derivative Test. 5.) Curve Sketching 6.) Optimization

` Integration (≈5 weeks)

1.) Antiderivatives/Indefinite Integration 6.) Riemann Sum and Trapezoidal Rule 7.) Definite Integral 8.) Fundamental Theorem of Calculus 9.) Integration by Substitution

Logarithmic, Exponential, and other Transcendental Functions (≈6 weeks)

1.) Natural Logarithm (Differentiation and Integration of Inverse Functions

2.) Exponential (Differentiation and Integration) Logarithmic, Exponential, and other Transcendental Functions (cont.)

jbelcher� 7/27/12 4:29 PMComment [5]: C2 – The course teaches all topics associated with Functions, Graphs, and Limits; Derivatives; and Integrals as delineated in the Calculus AB Topic Outline in the AP Calculus Course Description.

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3.) Bases other than e 4.) Inverse Trig. Functions (Differentiation and Integration) 5.) Differential Equations (Growth and Decay) 6.) Slope Fields 7.) Differential Equations (Separation of Variables)

Applications of Definite Integrals (≈5 weeks)

1.) Areas of a region between two curves 2.) Volumes of Solids of Revolution 3.) Overview and Test Preparation 4.) ArcLength and Surfaces of Revolution 5.) Indeterminate Forms and L’Hopital’s Rule

Activities:

The following sample activities demonstrate ways to help students gain an increased understanding of calculus. Example  1    The  “table”  feature  of  the  TI  graphing  calculator  can  be  used  to  zoom  in  on  a    limit  numerically.    For  example,  to  find    

⎟⎠

⎞⎜⎝

⎛−

−→ 4

2lim 22 xx

x  

 we  view  the  values  of  the  function  from  x-­‐values  of  1.5  to  2.5  with  an  increment  step  of  0.1.  At  x  =  2  the  table  records  “error”  or  “not  defined.”  Students  should  see  that  the  y-­‐values  seem  to  follow  a  pattern.  Redo  the  process  beginning  at  1.9  with  a  step  size  of  0.01,  and  observe  that  the  y-­‐values  are  converging  to  0.25.  The  process  can  be  repeated  with  smaller  and  smaller  steps.    The  limit  can  also  be  shown  visually  by  graphing  the  function  in  a  window  that  has  a  pixel  step  of  0.1.  Trace  the  function  beginning  at  x  =  1.  Each  step  shows  the  corresponding  x-­‐  and  y-­‐coordinates,  but  at  x  =  2,  the  y-­‐coordinate  disappears.  It  “reappears”  when  the  tracing  continues  at  x  =  2.1.  Students  can  see  graphically  that  the  y-­‐coordinates  cluster  at  about  0.25  as  x  is  near  2.    For  comparison,  do  the  same  exploration  with    

⎟⎟⎠

⎞⎜⎜⎝

⎛

−

+→ 2

4lim2

2 xx

x  

 This  function  is  also  undefined  at  x  =  2,  but  the  y-­‐values  do  not  converge  as  x  approaches  2.  Instead,  the  values  “explode”,  giving  students  a  numerical  look  at  asymptotic  behavior.      Example  2      The  “Round  Barn”  assignment  that  follows  is  assigned  at  the  beginning  of  the  year.    Students  are  not  aware  of  ways  in  which  calculus  could  be  applied  to  the  problem.    Students  are  

jbelcher� 7/27/12 4:29 PMComment [6]: C5 – The course teaches students how to use graphing calculators to help solve problems, experiment, interpret results, and support conclusions. jbelcher� 7/27/12 4:29 PMComment [7]: C3 – The course provides students with the opportunity to work with functions represented in a variety of ways – graphically, numerically, analytically, and verbally – and emphasizes the connections among these representations.

jbelcher� 7/27/12 4:29 PMComment [8]: C4 – The course teaches students how to communicate mathematics and explain solutions to problems both verbally and in written sentences. jbelcher� 7/27/12 4:29 PMComment [9]: C3 – The course provides students with the opportunity to work with functions represented in a variety of ways – graphically, numerically, analytically, and verbally – and emphasizes the connections among these representations.

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reassigned  the  problem  at  the  end  of  the  school  year,  but  with  the  charge  that  they  must  use  calculus  in  significant  ways  to  justify  their  arguments.    Their  final  task  is  to  present  their  justifications  to  the  class  and  the  following  scoring  criteria  sheet  is  used  to  assess.    Classmates  and  the  teacher  fill  out  the  scoring  criteria  form,  complete  with  comments.    Comments  are  to  be  constructive  criticism.    Presenters  are  given  these  critiques  later  and  asked  to  read  them  looking  for  themes  representing  areas  of  improvement.          

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Agriculture has always played a large part in the history of West Virginia. Round barns, though never quite as common as the standard square or rectangular shaped barns, were seen as a time and money saver. The selling point for the round barn design was the large loft space, which allowed farmers to store hay in the barn, eliminating the cost of building and maintaining outbuildings for hay storage. This also reduced the farmer’s work since they did not need to haul the hay to the barn to feed the livestock. The Hamilton Round Barn, built in 1912 by Amos C. Hamilton as a dairy barn, also incorporates a technology more commonly associated with the Pennsylvania bank barns. It has two main entrances, one on the lower level for livestock and another, accessible by ramp, to the second level where the farmer would store equipment and silage. The above picture and description was taken from the Frontiers to

Mountaineers Heritage Tourism brochure on Mountaineer Architecture. There are two statements I would like for you to justify. Statement 1: “Round barns, though never quite as common as the standard square or rectangular

shaped barns, were seen as a time and money saver.” Statement 2: “The selling point for the round barn design was the large loft space, which allowed

farmers to store hay in the barn.”

Task 1: Mathematically support or contradict the claim made by statements 1 and 2 in the brochure. (Note: Hay could have been stored in square and rectangular barn lofts as well. So, statement 2 implies that the loft space of the round barn is more efficient.)

Follow-up question: Would the way hay is stored today, as compared to 1912, make the round barn more attractive or less attractive? Justify your answer mathematically.

Task 2: What other shapes might barns be built in? Compare at least two other shapes to a square, rectangle, and circle supporting claims of comparison mathematically.

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The Round Barn Scoring Criteria

1=strongly agree 2=agree 3=average performance 4=disagree 5=strongly disagree 1.) Addressed all questions. 1 2 3 4 5 Comments _________________________________________________________ _________________________________________________________ _________________________________________________________ 2.) Neat and well organized. 1 2 3 4 5 Comments _________________________________________________________ _________________________________________________________ _________________________________________________________ 3.) All propositions are 1 2 3 4 5 mathematically justified. Comments _________________________________________________________ _________________________________________________________ _________________________________________________________ 4.) Significant use of Calculus 1 2 3 4 5 Comments _________________________________________________________ _________________________________________________________ _________________________________________________________ Additional comments or suggestions for improvement _________________________________________________________ _________________________________________________________ _________________________________________________________ _________________________________________________________ _________________________________________________________ _________________________________________________________ _________________________________________________________