Calculus AB
Syllabus
Course Title: Calculus
Prerequisites: Algebra I, Algebra II, Geometry, Pre-Calculus
Calculus Course Goals
Students will be able to understand derivatives and antiderivatives through graph analysis.
Students will be able to find derivatives of functions and understand their mathematical applications.
Students will develop a strong foundation of calculus concepts, techniques, and applications to prepare students for more advanced work.
Length of Course: 2 Tri-mesters
Course Overview
Calculus is intended to give students the opportunity to analyze and apply their mathematical knowledge in real world applications. Students will use their knowledge of equations and graphs to better understand derivatives as a measurement of slope. Students will analyze various rates of change using derivatives. Students will see how rates of change affect original measurements of population, position, temperature, and other various quantities. This class will comer everything in the Calculus AP topic outline as it appears in the AP Calculus Course Description, including integration by parts.
Textbook
The primary textbook is Calculus 7th edition by Larson, Hostetler, and Edwards. Additionally, Calculus from a Graphical, Numerical, and Symbolic Points of View by Ostebee and Zorn is used for supplemental material. Other supplemental material includes Calculus - Graphing Calculator Labs for Students written by Benita Albert and Phyllis Hillis.
Technology
Students will use the “Advanced Placement AP CD CalculusAB” to explore and practice the various topics outlined on this CD. They will be able to do this in the computer lab. All students have access to a TI-83 plus graphing calculator. Calculator Based Rangers and Calculator Based Labs (2) will be utilized by the students to investigate various topics covered in this course.
Course Planner
Pre-Calculus Review
Students complete a summer assignment to review pre-calculus topics. This is given in the form of multiple choice and open ended questions. This is scored as a test. Students must complete the summer assignment and have it finalized the first day of school to be eligible to complete the course.
Student Activity 1 - (Slinkies as a trigonometric Function)
Student Activity 2 - (Ball Bounce as a Piecewise Function)
Limits and Their Properties
Finding Limits Graphically and Numerically
Using Local Linearity to find Limits
Evaluating Limits Analytically
Continuity and One-sided Limits
Infinite Limits
Limits at Infinity
Calculator Based Lab-Important Limits and their Extensions
Derivatives - A graphical Approach
Amount functions and rate functions: The Idea of the Derivative
Estimating Derivatives using slope of a graph
The geometry of derivatives
The geometry of Higher-Order Derivatives
Average and Instantaneous Rates
Student Activity 3 (Function Derivative Cards - A Matching Game)
Calculator Based Lab- Numerical Derivatives
Differentiation
The Derivative and the Tangent Line problem
Basic Differentiations Rules and Rates of Change (The Power Rule)
Trigonometric Functions and Derivatives
The Product and Quotient Rules and Higher-Order Derivatives
The Chain Rule
Implicit Differentiation
Related Rates
L’Hopital’s Rule
Application of Differentiation
Extrema on an Interval
Rolle’s Theorem and Mean Value Theorem
Increasing and Decreasing Function and the First Derivative Test
Concavity and the Second Derivative Test
Continuity and Differentiability
Optimization Problems
Newton’s Method
AP - Released Open Response Questions
Calculator Based Lab - Mattie’s Mean Value Adventure
Integration
Antiderivatives and Indefinite Integration
Slope Fields
Area under the Curve
Riemann Sums
Definite Integrals
Fundamental Theorem of Calculus
Integration by Substitution
Numerical Integration
Student Activity 4 (Riemann Sum)
Logarithmic, Exponential, and other Transcendental Functions
The natural Logarithmic Function: Differentiation
The natural Logarithmic Function: Integration
Derivatives of Inverse Functions
Exponential Functions: Differentiation and Integration
Bases other than e and Applications
Differential Equations: Growth and Decay
Student Activity 5 (Newton’s Law of Cooling)
Student Activity 6 (Finding the Area of Pizza)
Applications of Integration
Area of a region between two Curves
Volume: The Disk Method
Volume: The Shell Method
Integration Techniques
Basic Integration Rules
Integration by Parts
Trigonometric Integrals
Student Activities
Student Activity 1 - Slinky
1.Using a CBR and Calculator, create a sine or cosine function by oscillating a slinky over the CBR.
2.Determine the equation for the function. Use graph link to print out all appropriate points needed to explain your work. Print out the data from the slinky with your equation graphed over the collected data. It should be a very close overlay.
3.Present your work on at least a quarter sized poster board. Be sure to explain every calculation and define every variable used. BE DESCRIPTIVE.
Student Activity 2 - Piecewise Function
1.Using a CBR and Graphing Calculator, collect data of a ball bouncing.
2.Printout your graph and label all major points of interest. You may want to make several printouts of these points using the graph link program.
3.Using your knowledge of parent functions, vertical and horizontal translations, and reflections determine a piecewise function for the first three or four - (¾) sections of the graph. Show all work for each equation. A parabola equation can only be used once. A line equation can only be used twice.
4.A printout of each equation and graph of the equation needs to be on your final poster board presentation.
5.A printout is needed of your collected data and the piecewise function overlaid on the data to show how closely the two match.
Student Activity 3 - Function Derivative Cards - A Matching Game
Students are given approximately 30 cards. The cards have either an equation, a graph of f(x), a graph of the first derivative, a graph of the second derivative, or a written explanation of the graph detailing maximums, minimums, and zeros of the original function. There are polynomial, trigonometric, logarithmic and exponential functions displayed on the cards. The students are to match an equation to its first and second derivative graph, the original graph, and the written explanation. This is completed with a partner.
Student Activity - 4 - (Riemann Sum)
The students are to create a velocity graph of a toy car using a graphing calculator and a calculator based ranger (CBR). The students are to find the area under the curve using the Riemann sums thus finding the distance the car travels.
1.Set up the calculator/CBR to record the velocity a car travels away from the CBR for 5 seconds. (Use feet/sec as your unit of measurement).
2.Using your recording beginning point and ending point, determine the area under the curve using Riemann sums with 10 subintervals. Show all work and needed points using TI-connect.
3.State what this area represents and how this is represented.
4.Display all of the above on the poster board.
Student Activity 5 (Newton’s Law of Cooling)
Students will use a Calculator Based Lab (CBL2) with a temperature probe and a graphing calculator to record the temperature of a cooling cup of coffee over time.
1.Set up the calculator to record the temperature of the cooling coffee every 5 seconds for 30 minutes.
2.Use Newton’s law of cooling and determine an equation to describe the collected data. Be sure to justify all numbers and show all calculations.
3.Using your equation, determine the time the coffee will be 90 degrees. Test your answer with your collected data.
4.Using TI-Connect to print the appropriate points to justify your answer. Prepare your work on a quarter size poster board.
Student Activity 6 (Finding the Area of Pizza)
Problem solving with circles Pizza
Interest in this investigation will be peak by bringing in a large pizza that they will eventually get to eat. The idea of finding the area of the pizza is developed through first exploring the graphs of various circles. Students will then investigate the inverse in function, its graph, and its derivative. From there, the students will consider cutting a pizza in slices with the same area using concentric circles and parallel lines. Ultimately the area of the pizza will be found using integration and applying the above explorations.
Student Assessment
Homework
Generally it will take up to 2 days to thoroughly discuss each section Comprehensive homework/class work will be assigned following instruction. Points will be awarded when students complete a homework quiz over an assigned section. Homework quizzes will be worth around 10 points each. Periodically, problems will be chosen from a homework set and graded.
Activities
Other points that students earn will come from activities/projects that they accurately complete. Projects will be worth anywhere from 30 points to 50 points.
Tests
Tests will be given at the end of each chapter or halfway through a large chapter. They will be worth 100 points. Memory and pop quizzes will be given throughout each tri-mester.