Calculus Honors. Gorman Learning Center (052344) Basic Course Information

Transcription

1 Calculus Honors Gorman Learning Center (052344) Basic Course Information Title: Calculus Honors Transcript abbreviations: HCalc A / HCalc B Length of course: Full Year Subject area: Mathematics ("c") / Calculus UC honors designation? Yes Prerequisites: Algebra 1 (Required) Algebra 2 (Required) Geometry (Required) Pre Calculus (Required) Co-requisites: None Integrated (Academics / CTE)? No Grade levels: 11th, 12th Course learning environment: Classroom Based Course Description Course overview: The Calculus Honors course incorporates Weil College Preparatory Curriculum s 4 major principles: Thinking Skills, Appropriate Assessments, Cross-Curricular Activities and Pertinent Technology Use. The fundamental areas of the mathematics curriculum include the following: modeling mathematical thinking, solving problems, developing analytic ability and logic, experiencing mathematics in depth, appreciating the beauty and fascination of mathematics, building confidence, communicating, and becoming fluent in mathematics. The course follows the principles put forth in the ICAS Statement on Competencies in Mathematics Expected of Entering College Students and the Common Core Standards for Mathematical Practice. The school s curriculum includes Honors level courses for 10th-, 11th- or 12th-grade students to encourage them to undertake more challenging studies in high school. Courses include

2 additional topics, deeper study of topics and higher standards of evaluation from the normal course work. Calculus Honors is an in-depth study of functions, graphs, limits, derivatives, definite integrals, antiderivatives, differential equations and real-world applications of differentiation and antidifferentiation. Concepts will be explored graphically, numerically, algebraically, and verbally. Subtopics include products, quotients, the calculus of logarithmic functions, growth and decay, plane and solid figures, algebraic calculus of motion. Honors students will study additional integration techniques including Integration by Parts, Trig integrals, Trig substitutions, Reduction and Improper Integrals. Technology tools such as graphing calculators and other graphing applications will be used regularly in this course. The course follows the AP curriculum is designed primarily for students who do not want to take the Advance Placement Exam. Course content: Limits and Their Properties This unit develops the concepts of limits and continuity that are the foundation of the differentiation and integration. The Tangent-Line Problem and the Area Problem are described as the major questions that lead to the initial development of calculus. Starting with graphical estimations for limits, a formal definition of the limit of a function is defined. Properties of limits are explored, unbounded and oscillating functions are considered, and one-sided limits are defined. Tests for continuity are established. Learning vocabulary, use of formal concepts and specialized symbols are stressed. The unit presents the Intermediate Value Theorem, Infinite limits and applications. Area & Tangent Activity: Students determine the slope of a line tangent to a curve using the limit of changing secant lines. Students will use limits of rectangular areas to estimate the area under a curve. In the Area Problem, students will use an increasing number of rectangles to approximate the area under a curve. Students will then reason what they would calculate if they used an infinite number of rectangles. Research, Reading and Writing: To support cross-curricular instruction and reinforce reading and writing skills, students are given a weekly assignment to read and/or research a problem or topic related to calculus. They will write at least a three-paragraph response which is graded using the English department rubric. Limit problems: As an oral quiz, each student is given a separate problem. Students must describe how to find a limit for the given function (which may be presented algebraically, graphically, or numerically) and explain what the limit means in the context of the problem. Problems that Inspired Calculus: Students will investigate the Tangent Line Problem and Area Problems for polynomials using estimating processes. After studying finite values, students will develop answers to infinite iterations.

3 Trig Squeezes: Students will use the Squeeze Theorem to prove the Special Trig Limits. Students will use the same reasoning when they are given other functions whose limits require the Squeeze theorem to solve. Differentiation The unit introduces the process of differentiation. The process and definition of differentiation are developed using limits as applied to the secant line transformed to the slope of a tangent line. With this concept, differentiation is defined and formulas for a variety of functions including linear, polynomial, rational, radical, logarithmic and trigonometric. Students are presented with basic vocabulary and symbols used throughout calculus. Manipulative differentiation formulas are proven including sum, product, quotient and chain rules. Students study the concept and application of implicit differentiation and related rates in Physics and the real world. In to the Water: Students will model and analyze the motion of an Olympic diver that jumps off a 10-meter platform using the basic physics kinematic formulas. Students will calculate the average speed that the diver experiences and the instantaneous velocity upon impact with the water. At the end of the activity, students will understand that the relationship between these kinematic equations is based in basic calculus. Derivative Representations: Students are each given one of the two definitions of derivative. From this limit they must orally explain what the limit represents. That is, they must state that it is the derivative, tell the function that is being differentiated, and give the point (if applicable) at which it is being differentiated. Differentiation Rules: The product and quotient rules are proven in a traditional manner after following an activity that asks students to use their calculators to consider a function, h(x), that is approximated by the product of the local linear approximations of two other functions, f and g, in an attempt to discover the formula for the product rule before it s analytic proof. When they determine the product of these approximations and consider its derivative at a point x=a, they have essentially derived the product rule. Students use linear approximations as a way of comprehending nonlinear functions. Applications of Differentiation Using the concepts of differentiation, students will analyze functions by finding local extrema and using limits to identify any potential asymptotes. They will gain an understanding of the properties and uses of critical numbers for function analysis. Students will be introduced to Rolle s Theorem and the Mean Value Theorem and understand that these theorems are foundational for future study. Students will apply first and second derivative tests, use basic definitions to describe the properties of functions, and sketch graphs of functions. This includes max/min points, horizontal and vertical asymptotes, concavity and limits at infinity. Finally,

4 students will learn techniques in optimization and apply them to physics and other real-world problems. Graph Sketching: Students will be provided with multiple graphs of higher degree polynomials and will determine critical points-max/min, increasing/decreasing functions, concavity, points of inflection, etc. Motion: Working in pairs, students answer velocity/acceleration problems both given graphs and given a verbal problem. Students must incorporate their knowledge and skill using the first and second derivatives to solve these problems. All answers are required to be given in complete sentences. They then trade with a partner who grades their problems and discusses why their explanations and mathematical support may or may not be clear. Rainbows: Rainbows are formed when light strikes rain drops and is reflected and refracted. The Laws of Refraction involves sine functions. Students will be tasked with finding the minimum angle of deflection. Optimization: Working in groups, students are given a lab to complete. They must find the maxima and minima of various functions using the derivative. They are required to determine the various functions given a real-world problem (e.g., cost, revenue, or volume) and support their claims with clear derivatives and full sentences. Discussion among members of their respective groups is required. Integration This unit turns to the process of working with anti-derivatives. Students will be presented with notation and how to find antiderivatives and indefinite integrals of functions. They will observe how there is a loss of information when completing a derivative on a function which requires the addition of C (the constant of integration) to create the general integral. Students will then use initial conditions to determine a specific value of C to determine the particular solution. Students will notice that derivatives and integrals/antiderivatives are inverse operations and will apply basic integration rules to a wide-variety of functions. Students will review sequences and series to prepare for various types of Riemann Sums (including left-hand, right-hand, center, and trapezoidal) to find the area under a curve through approximation. Generalization of this will define definite integrals and its properties and, using The Fundamental Theorem of Calculus, find the area under a curve with defined boundaries. Students will explore the Mean Value

5 Theorem for Integrals, used to calculate the average value of a function on an interval. Students will practice the Second Fundamental Theorem of Calculus. Students will then do detailed practice in calculating more complex integrals and using substitution methods and changing of variables. Slope Fields: After being introduced to the idea of a slope field, students are given a series of differential equations, asked to draw the slope field, and then to describe its solution curve. They then analytically solve the differential equation to reinforce their assertions. To emphasize the usefulness of slope fields, students are then given a differential equation they cannot solve and are asked about the solution curves. They quickly sketch the slope field to answer probing questions and understand their purpose. Fundamental Theorem of Calculus: Students are given questions in which they must use the Fundamental Theorem of Calculus. Within these problems, they are often required to calculate a definite integral with their calculators. In addition, they must answer questions about extrema and inflection points of g using calculus if given a function and a graph of f. Logarithmic, Exponential and other Transcendental Functions This unit uses the previously presented knowledge of differentiation and basic integration concepts and applies it to more complex functions including logarithmic, exponential, trigonometric, inverse, and inverse trigonometric functions. Students review inverse functions and their graphs and the relationships with the original functions. Using this knowledge, students differentiate and integrate Logarithmic and Exponential Functions as well as the general Inverse of a function. With that information, the differentiation and integration of inverse Trigonometric Functions are derived. Kinematics of objects: Using piecewise functions that representing velocity, students will develop a model as a method of calculating the final position of an object. Since at times the velocity will have been negative, students will also need to determine the total distance traveled by the object. With these two results, students must discuss the difference in what each amount represents. Inverse Trig Proofs: Each student will be given an inverse trigonometric function. They will develop the proof of its integral and make an oral presentation to the class. St. Louis Arch: Students will learn how the St. Louis Arch was constructed using the hyperbolic cosine function. The equation will be provided with lower and upper bounds. Students will determine the height above the ground of the center of the highest triangle, the height of the arch, and the width at ground level.

6 Differential Equations and Applications of integrations Students explore the concept of simple differential equations, a more extensive use of slope fields and Euler s Method of approximation. Additionally, students study finding general and particular solutions of differential equations and separation of variables techniques. Applications of these functions are explored including compounding interest, radioactive half-life modeling, bacterial growth, and logistic growth functions. An extension of integration applications is explored, including calculating area between two curves, calculating volume using the disk and shell methods, and determining arc length and surface of revolution. Students apply these principles to applications including Moments, centers of mass, centroids, fluid pressure and fluid force. Students study the additional processes of integration (Parts, Trig integrals, Trig substitutions, Reduction and Improper Integrals) and apply to specific problems. The Washer and Shell Game: Students will be provided with a variety of simple curves that are revolved around the x-axis and/or y-axis. After determining the form, students will determine surface areas and volumes of the objects. Students will also describe why they picked the particular method used to determine the volume. The Shape of Saturn: Students will research the shape of the planet Saturn. They will then use integration techniques to determine its volume and the differences with a spherical model. Their results will be presented to the class and discussed. Tidal Pools: Using real-world tidal models, students will describe water flow using integration techniques. Extending those models, students will develop equations that approximate the force and energy of those flows and discuss possibilities of its use in generating power. Honors Final Exam Details: Honors Calculus includes a comprehensive final exam. This exam covers all topics throughout the course and is similar in nature to the current AP Calculus-AB exam. Students will display an understanding of material taught in the course through the exam. Students will work individually or in pairs to develop and complete a cumulative project that will utilize the skills learned in the course. A rubric will be provided to students containing the expectations of types and quantities of suggested key concepts demonstrated as well as expectations for development and presentation of the work. Suggested topic may include: Hurricane Modeling, Lissajous Curves, building a Harmonograph, a river navigation problem, contour maps and gradients, Centroids and Center of Mass, Investigation of Ideal Gas Laws using 3D plotting techniques, and researching and presenting Maxwell s Equations.

7 Final Exam Final Exam Honors Calculus Gorman Learning Center Instructions: Work all of the following problems on your own paper. Give thorough explanations to receive full credit for 1. (5 points) Determine the limit of the trigonometric function (if it exists) 2. (5 points) Determine the limit of the trigonometric function (if it exists) 3. (20 points) Use the precise definition of limit to prove that 4. (15 points) Find the derivative of the function by using the definition of the derivative. 5. (10 points) Find the derivative of the function 6. (20 points) The position function of a particle moving along the -axis is 1. a) Find the velocity of the particle 2. b) Find the open interval(s) in which the particle is moving to the left 3. c) Find the position of the particle when the velocity is 4. d) Find the speed of the particle when the position is 5. (20 points) Analyze and sketch a graph of the function. Label any intercepts, relative extrema, points of inflection, and asymptotes. 8. (10 points) Solve the differential equation See next page 9. (15 points) Suppose a curve is given by the equation 10. Find by implicit differentiation. 11. Find the slope of the curve at the point 1, 2). 12. (20 points) An air traffic controller spots two planes at the same altitude converging on a point as they fly at right angles to each other (see figure below). One plane is 150 miles

8 from the point moving at 450 miles per hour. The other plane is 200 miles from the point moving at 600 miles per hour. 13. a) At what rate is the distance between the planes decreasing? 14. b) How much time does the air traffic controller have to get one of the planes on a different path? 11. (15 points) Find the equation of the tangent line T to the graph of f at the given point. Use this linear approximation to complete the table. 12. (20 points) Evaluate the definite integral by the limit definition. 13. (15 points) Find the Integral See next page, 14. (15 points) Find the volume of the solid generated by revolving the region bounded by the graph of the equation about the x-axis 15. (15 points) Find the time required for an object to cool from to by evaluating where is time in minutes 16. (20 points) Gasoline is increasing in price according to the equation where is the dollar price per gallon and is the time in years, with representing An automobile is driven miles a year and gets miles per gallon. The annual fuel cost is Estimate the annual fuel cost in a) and b) 17. (10 points) Evaluate the definite integral 18. (20 points) A conical tank (with vertex down) is 10 feet across the top and 12 feet deep. If the water is flowing into the tank at a rate of 10 cubic feet per minute, find the rate of change of the depth of the water when the water is 8 feet deep.

9 19. (10 points) Use the graph to determine the limit, and discuss the continuity of the function. 20. a) b) c) See next page. 20. (15 points) Let be twice differentiable and one-to-one on an open interval. Show that its inverse function satisfies If is increasing and concave downward, what is the concavity of -1? 21. (25 points) The function is differentiable on the interval [. The table shows the values of for selected values of. Sketch the graph of, approximate the critical numbers, and identify the relative extrema. 22. (10 points) Find the indefinite integral. 23. (25 points) A trough is feet long and feet across the top (see figure). Its ends are isosceles triangles with altitudes of 3 feet. 24. a) If water is being pumped into the trough at 2 cubic feet per minute, how fast is the water level rising when is 1 foot deep? 25. b) If the water is rising at a rate of inch per minute when, determine the rate at which water is being pumped into the trough. See next page 24. (10 points) Write the limit as a definite integral on the interval [a, b, where c i is any point in the ith subinterval. Limit Interval 25. (25 points) a) Use differentiation to verify that 26. b) Use the result of part a) to find the volume of the solid generated by revolving each plane about the -axis.

10 Course Materials Textbooks Title Calculus Calculus: Multivaria ble Multivaria ble Calculus Author Publish er Larson, Hought Hostetler on & Mufflin Edwards William G. McCallu m, Wiley Deborah Hughes- Hallet, et al. Ron Larson, Bruce Edwards Literary Texts Cengag e Learnin g Editio n Website Prima ry 2005 Yes 6th Ed, th Ed, bcsid=8042 No No Title Author Publisher Edition Website Calculus: Advanced Calculus of Several Variables How to Ace the Rest of Calculus: The Streetwise Guide Scholarly Articles C. H. Edwards, Jr. Colin Adams, Abigail Thompson, Joel Hass Dover Books on Mathematics Times Books 2nd Ed., st Ed., 2001 Read in entirety Article title Journal Authors Volume/Issue/Date Website "Student Research Projects in Calculus". The Mathematical Association of America. M. Cohen, E. Gaughan, A. Knoebel, D. Kurtz and D. Pengelley 1991 No No

11 Article title Journal Authors Volume/Issue/Date Website "Writing Projects for Mathematics Courses: Crushed Clowns, Cars and Coffee to Go". Websites The Mathematical Association of America A. Crannell, G. LaRose, T. Ratliff and E. Rykken Title various PHET Simulations Author(s)/Editor(s)/Compiler(s) Affiliated Institution or Organization University of Colorado URL Multimedia Title Author Director Variety of titles from Connections Series The Man who Knew Infinity Selected parts of Episodes of Numb3ers Other James Burke Kenneth Chisholm Name of video series Connections series Date Website [ empty Matthew 2015 Brown Numb3ers CBS TV series Medium of Publication Video Movie on digital Digital copies Title Authors Date Course material type Website Various Publisher and Teacher various Power points from publisher, teacher and previous students