| Credit- Degree applicable | | Effective Quarter: Fall 2021 | I. Catalog Information
| MATH 1B | Calculus | 5 Unit(s) |
| | (See general education pages for the requirement this course meets.) (Not open to students with credit in MATH 1BH.)
Prerequisite: MATH 1A or MATH 1AH.
Advisory: EWRT 211 and READ 211, or ESL 272 and 273. Lec Hrs: 60.00
Out of Class Hrs: 120.00
Total Student Learning Hrs: 180.00 Fundamentals of integral calculus. |
| Student Learning Outcome Statements (SLO)
| | | Analyze the definite integral from a graphical, numerical, analytical, and verbal approach, using correct notation and mathematical precision. |
| | | Formulate and use the Fundamental Theorem of Calculus. |
| | | Apply the definite integral in solving problems in analytical geometry and the sciences. |
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II. Course Objectives | A. | Analyze and explore aspects of the integral calculus. |
| B. | Analyze and evaluate the definite integral as a limit of a Riemann sum and examine its properties |
| C. | Examine the Fundamental Theorem of Calculus |
| D. | Find definite, indefinite, and improper integrals using various techniques |
| E. | Apply the definite integral to applications |
| F. | Examine differential equations |
III. Essential Student Materials | | Students will use technology (computers and graphing calculators) to examine mathematical concepts graphically and numerically. |
IV. Essential College Facilities V. Expanded Description: Content and Form | A. | Analyze and explore aspects of the integral calculus. |
| 1. | Using a variety of perspectives: verbally, graphically, numerically, and symbolically. |
| 2. | The following functions are incorporated in this development: linear, polynomial, rational, exponential, logarithmic, power, trigonometric, inverse trigonometric, piece-wise, and parametric functions. |
| B. | Analyze and evaluate the definite integral as a limit of a Riemann sum and examine its properties |
| 1. | Apply Riemann Sums to approximate area under the graph of a function |
| 2. | Define and evaluate the definite integral as a limit of the Riemann sum |
| 3. | Apply the definition of the definite integral to calculate the area under the graph of a function exactly |
| 4. | Examine the properties of the definite integral |
| a. | Georg F. B. Riemann (1826-1866, Germany) |
| b. | Gottfried Leibniz (1646-1716, Germany): The indefinite integral symbol |
| c. | Jean Baptiste Joseph Fourier (1768 - 1830, France): The definite integral symbol |
| C. | Examine the Fundamental Theorem of Calculus |
| 1. | Use the Fundamental Theorem of Calculus to calculate definite integrals using antiderivatives |
| 2. | Use the Fundamental Theorem of Calculus to obtain an antiderivative function |
| D. | Find definite, indefinite, and improper integrals using various techniques |
| 1. | Integrate by using substitution |
| 3. | Integrate by using trigonometric substitution |
| 4. | Integrate after performing partial fraction decomposition |
| 5. | Integrate curves that are defined parametrically |
| 6. | Use the table of integrals to evaluate integrals |
| 7. | Develop techniques for approximating definite integrals and associated errors |
| a. | Approximate definite integrals by left sums, right sums, midpoint rule, trapezoidal rule and Simpson's rule |
| b. | Evaluate errors to numerical integration |
| c. | Estimate the value of definite integrals within given error bounds |
| 8. | Define and evaluate improper integrals |
| a. | Thomas Simpson (1710-1761, England): he didn't develop Simpson's Rule; it was developed before he was even born. He wrote texts and worked on probability! |
| b. | Joseph Louis Lagrange (1736-1813, France): Developed integration by parts |
| c. | Maria Agnesi (1718-1799, Italy): story of "Curve of Agnesi" |
| E. | Apply the definite integral to applications |
| 1. | Find the area bounded by two or more curves |
| 2. | Find the volume of solids with known cross section |
| 3. | Find the volume of solids of revolution |
| 5. | Examine some applications of the definite integral to other areas in Mathematics such as but not limited to |
| a. | Find the surface area of a solid of revolution |
| b. | Find the average values of a function. |
| c. | Present a logical development of the natural log as a definite integral and the corresponding development of exponential function as the inverse of the natural log |
| 6. | Examine some applications of the definite integral to Physics such as but not limited to |
| 7. | Examine some applications of the definite integral to other subjects such as but not limited to |
| a. | Probability - Use the definite integral to define and calculate probabilities and expected value, from a probability distribution function. |
| b. | Applications in Economics such as consumer and producer surplus |
| c. | Applications in Biology such as cardiac output and the flux of blood flow |
| a. | Earlier attempts all over the world at solving area, volume problems with a hint of calculus-like ideas: |
| 1. | Areas and Volumes: Eudoxus' (408-355 B.C., Asia Minor, modern day Turkey) method of exhaustion; Archimedes (287 - 212 B.C., Sicily) found sums of "infinite series" (e.g. area of circle) using double reductio ad absurdum that actually incorporated some of the technical details of limits |
| 2. | Area of circle: Liu Hui (220 - 280, China): Principle of exhaustion and Calvalieri's principle; Seki Kowa (1642-1708, Japan): inscribed rectangles - early "Riemann sum" |
| 3. | Ancient Egyptians: Volume of the frustum of pyramid |
| 4. | Ibn al-Haytham (965-1039, Persia), AKA Alhazen: Method of compression to find the volume of the solid formed by rotating the parabola around a line perpendicular to the axis of the curve. |
| b. | Evangelista Toricelli (1608-1647, Italy) and Gabriel's Horn |
| c. | Pappus (290-350 Egypt): Centroid and its relation to volume and surface area. |
| F. | Examine differential equations |
| 1. | Establish the concept of a differential equation and its slope field |
| 2. | Use Euler's method to approximate the solution to a differential equation |
| 3. | Use separation of variables to solve a differential equation |
| 4. | Establish exponential model of growth and decay |
| 5. | Establish logistic model of growth (optional) |
| a. | Leonhard Euler (1707-1783, Switzerland): Concept of differential equations |
| b. | Jakob Bernoulli (1654-1705, Switzerland): Development of separation of variables
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VI. Assignments | A. | Required readings from text |
| B. | Problem solving exercises |
| C. | A selection of homework, quizzes, group projects and exploratory worksheets |
| D. | Optional project synthesizing various concepts and skills from course content |
VII. Methods of Instruction | | Lecture and visual aids
Discussion of assigned reading
Discussion and problem-solving performed in class
In-class exploration of internet sites
Quiz and examination review performed in class
Homework and extended projects
Guest speakers
Collaborative learning and small group exercises
Collaborative projects
Problem solving and exploration activities using applications software
Problem solving and exploration activities using courseware
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VIII. Methods of Evaluating Objectives | A. | Periodic quizzes and/or assignments from sources related to the topics listed in the curriculum will be evaluated for completion and accuracy to assess studentsʼ comprehension and ability to communicate the course content as well as to provide timely feedback to students on their progress. |
| B. | At least three one-hour exams without projects or at least two one-hour exams with projects are required. Students are expected to provide complete and accurate solutions to problems that include both theory and applications by integrating methods and techniques studied in the course. Students shall receive timely feedback on their progress. |
| C. | One two-hour, comprehensive, final examination is required, in which students are expected to display comprehension of course content and be able to choose methods and techniques appropriate to the various types of problems that cover course content. |
| D. | Projects (optional)
Projects may be used to enhance students' understanding of topics studied in the course in group or individual formats, whether communicating their understanding orally through classroom presentation or in writing. Evaluation is to be based on completion and comprehension of course content, and students shall receive timely feedback on their progress. |
IX. Texts and Supporting References | A. | Examples of Primary Texts and References |
| 1. | James Stewart, Daniel Clegg & Saleem Watson "Calculus: Early Transcendentals", 9th Ed. Cengage 2021. |
| B. | Examples of Supporting Texts and References |
| 1. | Hass, Weir, and Thomas, "University Calculus, Early Transcendentals", 4th Ed. Pearson 2020. |
| 2. | Anton, Bivens, and Davis, "Calculus, Early Transcendentals Combined", 10th Ed. Wiley 2011. |
| 3. | Hughes-Hallett, et al., "Calculus, Single and Multivariable", 7th Ed. Wiley 2017. |
| 4. | Mathematics Multicultural Bibliography, available on the De Anza Math department resources website. |
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