Title of Unit Pre-Calculus Integrals Grade Level HS (11-12) Curriculum Area Calculus (Mathematics) Time Frame 3-4 weeks January February Materials Calculus AP* Edition Finney, Demana, Waits, Kennedy Chapter 6 and 7.1 to 7.3 Handouts Learning Targets Additional support Khan Academy https://www.khanacademy.org/ o Indefinite integral as antiderivative o Riemann sums and definite integration o Integration by parts o U-substitution o Definite integrals o Trigonometric substitution o Fundamental theorem of calculus Calculus videos: http://www.online.math.uh.edu/houstonact/videocalculus/
o The Area Under a Curve o The Integral o The Fundamental Theorem of Calculus o Antidifferentiation and Indefinite Integrals o Change of Variables (Substitution) o The Natural Logarithm o The Exponential Function o The Inverse Trigonometric Functions o Integration by Parts o Integration of Powers and Products of Sine and Cosine o Integration of Powers and Products of Secant and Tangent, Cosecant and Cotangent o Trigonometric Substitutions o Partial Fraction Expansions o Numerical Integration AP Calculus homepage at AP Central: http://apcentral.collegeboard.com Developed By Jim Bice
Overview Integral calculus is the study of the area under the curve. The connection between differential calculus and integral calculus is the fundamental theorem of calculus. This unit will help explain that connection and how to use it to solve problems. Essential Questions What is the integral, how do I get it, and how is it used to solve problems? How is the derivative related to the definite interval as found in the fundamental theorem of calculus? Knowledge (Content Understanding) Students will know Students will know Absolute change Area under a curve Bounded function Definite integral Dummy variable Integrable function Integral of f from a to b Integrand Mean value theorem Net area Partition Rectangular Approximation method Skills (Processes and Understandings) Students will be able to I can understand the concept of area under a curve using a Riemann sum over equal subdivisions I can use the limit of a Riemann sum to calculate a definite integral I can find definite integrals and Antiderivatives I can describe and use the Fundamental Theorem of Calculus I can use the graphing calculator to compute definite integrals numerically I can use techniques of antidifferentiation including integration by parts and simple partial fractions I can integrate the trigonometric functions I can differentiate and integrate inverse trigonometric functions I can use numerical Integration: Trapezoidal Rule and Simpson s Rule I can find general and particular solutions to differential equations with separable variables
(RAM) LRAM, MRAM,RRAM Riemann sum Simpson s rule Subinterval Trapezoidal rule Upper bound Antidifferentiation by parts Antidifferentiation by substitution Arbitrary constant of integration Euler s method Indefinite integral Leibniz notation for integrals Numerical solution Slope fields I can create Slope fields Plans for Adjusting Core to Meet the Needs of All Students Support Challenge Provide in class support Strategic partnering Chunking of assignments Practice problems vary by level of expertise Assessment Evidence (Stage 2) Challenge problems during class Self-assessment Formative Summative
Learning targets for unit Reflections Objective checks daily Classroom Practice Homework 6 AP Questions AP Study notes (prepared by student) Unit tests in AP Format Semester Exam
Content Standards Florida Standards MAFS.912.C.4.1: Use rectangle approximations to find approximate values of integrals. MAFS.912.C.4.2: Calculate the values of Riemann Sums over equal subdivisions using left, right, and midpoint evaluation points. MAFS.912.C.4.3: Interpret a definite integral as a limit of Riemann sums. MAFS.912.C.4.4: Interpret a definite integral of the rate of change of a quantity over an interval as the change of the quantity over the interval. That is, f'(x)dx = f(b) - f(a) (Fundamental Theorem of Calculus). MAFS.912.C.4.5: Use the Fundamental Theorem of Calculus to evaluate definite and indefinite integrals and to represent particular antiderivatives. Perform analytical and graphical analysis of functions so defined. MAFS.912.C.4.6: Use these properties of definite integrals: [f(x) + g(x)]dx = f(x)dx + g(x)dx k f(x)dx = k f(x)dx f(x)dx = 0 f(x)dx = - f(x)dx f(x)dx + f(x)dx = f(x)dx If f(x) g(x) on [a, b], then f(x)dx g(x)dx MAFS.912.C.4.7: Use integration by substitution (or change of variable) to find values of integrals. MAFS.912.C.4.8: Use Riemann Sums, the Trapezoidal Rule, and technology to approximate definite integrals of functions represented algebraically, geometrically, and by tables of values. MAFS.912.C.5.1: Find specific antiderivatives using initial conditions, including finding velocity functions from acceleration functions, finding position functions from velocity functions, and solving applications MAFS.912.C.5.2: Solve separable differential equations, and use them in modeling. MAFS.912.C.5.4: Use slope fields to display a graphic representation of the solution to a differential equation, and locate particular solutions to the equation. Florida Math Standards MAFS.K12.MP.1.1 Make sense of problems and persevere in solving them. MAFS.K12.MP.2.1 Reason abstractly and quantitatively.
MAFS.K12.MP.3.1 Construct viable arguments and critique the reasoning of others. MAFS.K12.MP.4.1 Model with mathematics. MAFS.K12.MP.5.1 Use appropriate tools strategically. MAFS.K12.MP.6.1 Attend to precision. MAFS.K12.MP.7.1 Look for and make use of structure. MAFS.K12.MP.8.1 Look for and express regularity in repeated reasoning. Florida Language Standards LAFS.1112.RST.1.3 Follow precisely a complex multistep procedure when carrying out experiments, taking measurements, or performing technical tasks; analyze the specific results based on explanations in the text. LAFS.1112.RST.2.4 Determine the meaning of symbols, key terms, and other domain-specific words and phrases as they are used in a specific scientific or technical context relevant to grades 1112 texts and topics. LAFS.1112.RST.3.7 Integrate and evaluate multiple sources of information presented in diverse formats and media (e.g., quantitative data, video, multimedia) in order to address a question or solve a problem. LAFS.1112.SL.1.1 Initiate and participate effectively in a range of collaborative discussions (one-on-one, in groups, and teacher-led) with diverse partners on grades 1112 topics, texts, and issues, building on others ideas and expressing their own clearly and persuasively. LAFS.1112.SL.1.2 Integrate multiple sources of information presented in diverse formats and media (e.g., visually, quantitatively, orally) in order to make informed decisions and solve problems, evaluating the credibility and accuracy of each source and noting any discrepancies among the data. LAFS.1112.SL.1.3 Evaluate a speakers point of view, reasoning, and use of evidence and rhetoric, assessing the stance, premises, links among ideas, word choice, points of emphasis, and tone used. LAFS.1112.SL.2.4 Present information, findings, and supporting evidence, conveying a clear and distinct perspective, such that listeners can follow the line of reasoning, alternative or opposing perspectives are addressed, and the organization, development, substance, and style are appropriate to purpose, audience, and a range of formal and informal tasks. LAFS.1112.WHST.1.1 Write arguments focused on discipline-specific content. LAFS.1112.WHST.2.4 Produce clear and coherent writing in which the development, organization, and style are appropriate to task, purpose, and audience. LAFS.1112.WHST.3.9 Draw evidence from informational texts to support analysis, reflection, and research.