Curriculum Design Template. Course Title: Calculus Grade Level: 12. Topics in Differentiation Marking Period 2

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1 Curriculum Design Template Content Area: Mathematics Course Title: Calculus Grade Level: 12 Limits and Continuity Derivatives and Differentiation Marking Period 1 Topics in Differentiation Marking Period 2 Application of the Derivative Integration Marking Period 3 Applications of the Definite Integral Marking Period 4 Date Created: May 2012 Board Approved on: August 27, 2012

2 Calculus Topic Outline Limits Intuitive Approach Limits Computational Techniques Limits Involving Horizontal and Vertical Asymptotes Continuity & Differentiability Tangent Lines &Rates of Change Techniques of Differentiation Derivatives of Trigonometric Functions Chain Rule Local Linearity & Differentials Logarithmic and Exponential Functions Implicit Differentiation Derivatives of Logarithms and Exponentials Derivatives of Inverse Trigonometric Functions Related Rates L Hopital s Rule Increase, Decrease, Concavity and the Derivative Tests Relative and Absolute Extrema Applications of Absolute Maxima and Minima - Intermediate Value Theorem and Extreme Value Theorem Rectilinear Motion Rolle s Theorem and Mean-Value Theorem Indefinite Integrals Integration using U-Substitution Riemann Sum Left, Right, Midpoint, and Trapezoidal Approximations Definite Integral Fundamental Theorem of Calculus Average Value Definite Integrals using U-Substitution Integrals with Logarithms and Exponentials Area Between Two Curves Volumes by Disks Volumes by Washers Volumes by Shells Suggested Projects: Wikis, blogs, Google Docs group discussion and projects. ipad projects Interview a professional and determine how he/she uses math in the workplace. Posters, Power Points, songs, videos

3 Technology and 21 st -Century Life and Careers Standards A. Basic Computer Skills and Tools 3. Construct a spreadsheet, enter data, use mathematical or logical functions to manipulate and process data, generate charts and graphs, and interpret the results A. Basic Computer Skills and Tools 5. Produce a multimedia project using text, graphics, moving images, and sound B. Application of Productivity Tools 9. Create and manipulate information, independently and/or collaboratively, to solve problems and design and develop products B. Application of Productivity Tools 11. Identify a problem in a content area and formulate a strategy to solve the problem using brainstorming, flowcharting, and appropriate resources B. Design Process and Impact Assessment 3. Develop methods for creating possible solutions, modeling and testing solutions, and modifying proposed design in the solution of a technological problem using hands-on activities C.5 Assume a leadership position by guiding the thinking of peers in a direction that leads to successful completion of a challenging task or project F.2 Establish and follow performance goals to guide progress in assigned areas of responsibility and accountability during classroom projects and extra-curricular activities A.1 Analyze the relationship between various careers and personal earning goals A.2 Identify a career goal and develop a plan and timetable for achieving it, including educational/training requirements, costs, and possible debt A.5 Evaluate current advances in technology that apply to a selected occupational career cluster C.2 Characterize education and skills needed to achieve career goals, and take steps to prepare for postsecondary options, including making course selections, preparing for and taking assessments, and participating in extra-curricular activities.

4 Calculus Course Title: Calculus Grade Level: 12 th Overarching What is a limit and what does it tell us about the behavior of a function? What is a derivative and what purpose does it serve? How can we use the derivative in real-world applications? What is an integral? How can we use the definite integral in real-world applications? How can technology be used to aid in the applications of calculus? Can we use varied methods of representations of function? Overarching Enduring Understandings Students in Calculus will learn about limits, continuity, differentiability, and integration. A large focus of this course is also the multi-representation of functions graphically, numerically, algebraically, and analytically. Additionally, a large part of the course will focus on applications of both the derivative and the definite integral. Students will also gain a thorough understanding of how to use technology, specifically graphing calculators, to aid in the applications of calculus. Course Description This course is designed to meet the needs of those advanced math students who have demonstrated a thorough proficiency in all previous college prep courses. Topics include limits, derivatives, integration and applications of such. Limits and Continuity What is a limit? What techniques can be used to calculate a limit? What is the difference between a one-sided limit and a two-sided limit? How is calculating a limit related to the continuity of a function? What do limits tell us about the asymptotes of the graph of a function? Limit, vertical asymptote, horizontal asymptote, one-sided limit, two-sided limit, end behavior, continuity Students will be able to: Have a basic understanding of the intuitive idea of a limit. Understand the difference between a one-sided limit and a two-sided limit. Calculate simple limits using the properties of limits. Calculate limits at infinity and describe the end behavior of a graph.

5 Determine if a given function is continuous at a point or on an interval. Find limits of trigonometric functions and inverse functions. Suggested Lesson Activities Have students sketch graphs with given specific characteristics. Use the graphing calculator to analyze tables and graphs of difficult functions to determine a function s limit. Complete the continuity worksheet from the University of Delaware AP Calculus course binder. Use ipad technology to have students create mini-lessons that can be presented to Have students create a poster/song/graphical representation of a topic from the Have students create a wiki/blog based on a topic from the unit of study. Derivatives and Differentiability What is the difference between an average rate of change and an instantaneous rate of change? How can we find the instantaneous rate of change of a function? How are rates of change related to tangent lines on a graph? What is a derivative? How can we compute derivatives? If a function is continuous is it necessarily differentiable? average rate of change, instantaneous rate of change, slope of a tangent line, derivative, power rule, product rule, quotient rule, chain rule, derivative notation, differentiable Students will be able to: Calculate both an average rate of change and an instantaneous rate of change. Use the definition of the derivative (limit of the difference quotient) to find the derivative of a function. Find the derivative of a given function using the power rule, product rule, quotient rule and chain rule. Find higher order derivatives. Write the equation of a line tangent to a graph at a given point. Determine the derivative of the six trigonometric functions.

6 Determine if a continuous function is also differentiable at a given point or on a given interval. Suggested Lesson Activities Use Geometer s Sketchpad to animate how the slope of the tangent line (derivative) of a function changes depending on what x-value the tangent line is drawn at. Have students research each of the techniques of differentiation and present a minilesson on their findings. Use ipad technology to have students create mini-lessons that can be presented to Have students create a poster/song/graphical representation of a topic from the Have students create a wiki/blog based on a topic from the unit of study. Topics in Differentiation How can we find the derivative of an implicitly defined function? What is the process for finding the derivatives of logarithmic, exponential, and inverse trigonometric functions? How can we use derivatives to find the rate at which a quantity is changing with respect to time? Given that functions are locally linear, how can we use differentiation to estimate the value of a function at a given point? What is an indeterminate form? Explicitly defined function, implicitly defined functions, implicit differentiation, related rate, local linear approximation, differential, indeterminate form, l Hopital s Rule, Students will be able to Find the derivative of implicitly defined functions. Determine the derivatives of logarithmic, exponential, and inverse trigonometric functions. Calculate related rates. Use local linear approximations and differentials to estimate values of functions at a given point. Apply l Hopital s Rule to limits resulting in indeterminate forms.

7 Suggested Lesson Activities Have students graph both the exponential function and its derivative using the nderiv option (Math 8) on the calculator and have them determine what the derivative of the exponential function is based on its graph. Do the same with the natural logarithmic function. Have students work in groups on related rate problems and present their solutions to the class. Ask students to calculate f (3.025) in a particularly difficult function. Then show how using a local linear approximation is a better way to estimate its value. Use ipad technology to have students create mini-lessons that can be presented to Have students create a poster/song/graphical representation of a topic from the Have students create a wiki/blog based on a topic from the unit of study. Applications of the Derivative What can the first and second derivatives tell us about the graph of a function? How can we apply the first and second derivatives to real-world situations? What can derivatives tell us about the motion of a particle along a line? Critical points, maximum, minimum, relative extrema, points of inflection, concavity, increasing, decreasing, first and second derivative tests, absolute extrema, Intermediate- Value Theorem, Extreme Value Theorem, Rolle s Theorem, Mean-Value Theorem, rectilinear motion, velocity, speed, acceleration Students will be able to: Find the critical points of a function by using the first derivative test. Find the points of inflection of a function by using the second derivative test. Determine where a function is increasing, decreasing, concave up, concave down, has extrema, and has points of inflection. Apply the Extreme-Value Theorem to determine where a function has absolute extrema on a closed interval. Apply the first derivative test to real-world problems to determine absolute criteria (i.e. maximum profit, minimum cost, etc.) Determine the motion (velocity, speed and acceleration) of a particle in rectilinear motion using the derivative tests. Use both Rolle s Theorem and the Mean-Value Theorem to estimate velocities and instantaneous rates of change of given functions.

8 Suggested Lesson Activities Have students draw a curvilinear graph and label the parts of the graph: increasing, decreasing, concave up, concave down, maximum, minimum, point of inflection. Have students work optimization problems in groups and present their solutions to the class. Use ipad technology to have students create mini-lessons that can be presented to Have students create a poster/song/graphical representation of a topic from the Have students create a wiki/blog based on a topic from the unit of study. Integration Can we find the area of irregularly-shaped figures? What is an integral? How calculate we find an integral? What is the difference between a definite integral and an indefinite integral? Can integration be used in relation to rectilinear motion? Can we find the average value of a function over a given interval? Can we find a function if we are given its derivative? Antidifferentiation, integral, integrand, variable of integration, indefinite integral, constant of integration, Riemann Sum, inscribed rectangles, circumscribed rectangles, left-hand Riemann Sum, right-hand Riemann Sum, midpoint Riemann Sum, trapezoidal approximation, definite integral, Fundamental Theorem of Calculus, limits of integration, average value, differential equation, initial condition Students will be able to: Explain the basics of the area problem in calculus. Identify the parts of an indefinite integral. Calculate integrals using the power rule, u-substitution, and definitions of the trigonometric, logarithmic and exponential integrals. Estimate the areas of plane figures with curvilinear boundaries using various Riemann Sums. Find exact areas of plane figures with curvilinear boundaries using the definite integral and the Fundamental Theorem of Calculus. Find the position and velocity of a particle in rectilinear motion given its acceleration as well as find the total distance traveled by a particle. Understand and apply the fact that the definite integral acts as an accumulation

9 function in real-world problems. Calculate the average value of a function over a given interval. Suggested Lesson Activities Have students find the area of known shapes first, then irregular shapes that they must break into known shapes. Finally, have them estimate the areas of shapes that cannot be divided into known shapes easily (suggest maybe skinny rectangles) as a way to introduce Riemann Sums. Write a short paragraph comparing the graphs of the position, velocity and speed to decipher the difference among the three. Use ipad technology to have students create mini-lessons that can be presented to Have students create a poster/song/graphical representation of a topic from the Have students create a wiki/blog based on a topic from the unit of study. Applications of the Definite Integral Is it possible to find the area of a plane figure with boundaries defined by two or more functions? Can we find the volume of irregularly-shaped figures? Boundary, dx rectangle, dy rectangle, solid of revolution, disk, washer Students will be able to: Determine the area of a plane figure with boundaries defined by two or more functions. Use the Disk Method to find the volume of a solid of revolution around either axis or an axis other than the x or y axis. Use the Washer Method to find the volume of a solid of revolution around either axis or an axis other than the x or y axis. Have students create a poster/song/graphical representation of a topic from the Suggested Lesson Activities Using a drill with dowel rods and brightly-colored shapes attached, quickly rotate the shapes to simulate a solid of revolution.

10 Use dowel rods, rolls of paper towels, toilet paper rolls, etc. to symbolize disk method and washer method. Use ipad technology to have students create mini-lessons that can be presented to Have students create a poster/song/graphical representation of a topic from the Have students create a wiki/blog based on a topic from the unit of study.