| Credit- Degree applicable | | Effective Quarter: Fall 2021 | I. Catalog Information
| MATH 1A | Calculus | 5 Unit(s) |
| | (See general education pages for the requirement this course meets.) (Not open to students with credit in MATH 1AH.)
Prerequisite: MATH 32, 43 or 43H (with a grade of C or better), or appropriate score on Calculus Placement Test within the past calendar year.
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 differential calculus. |
| Student Learning Outcome Statements (SLO)
| | | Analyze and synthesize the concepts of limits, continuity, and differentiation from a graphical, numerical, analytical and verbal approach, using correct notation and mathematical precision. |
| | | Evaluate the behavior of graphs in the context of limits, continuity and differentiability. |
| | | Recognize, diagnose, and decide on the appropriate method for solving applied real world problems in optimization, related rates and numerical approximation. |
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II. Course Objectives | A. | Analyze and explore aspects of the differential calculus. |
| B. | Compute and interpret limits of functions using analytic and other methods, including L'Hospital's Rule. |
| C. | Apply the definition of continuity using limits to analyze the behavior of functions. |
| D. | Find the derivative of a function as a limit. |
| E. | Derive and use the power, quotient, product, and chain rules to differentiate functions, including implicit and parametric functions, and find the equation of a tangent line to a function. |
| F. | Graph functions using methods of calculus |
| G. | Apply the derivative to solve applications including related rate problems and optimization problems; |
| H. | Define the antiderivative and determine antiderivatives of simple functions. |
III. Essential Student Materials | | Students will use technology (computers or graphing calculators) to explore mathematical concepts graphically and numerically. |
IV. Essential College Facilities V. Expanded Description: Content and Form | A. | Analyze and explore aspects of the differential calculus.
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| 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, piece-wise, parametric, implicit, and inverse functions. |
| B. | Compute and interpret limits of functions using analytic and other methods, including L'Hospital's Rule.
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| 1. | Define, compute and interpret two sided limits. |
| 2. | Define and apply left and right hand limits. |
| 3. | Use L'Hospital's Rule to calculate limits of the form 0/0 or infinity/infinity and other indeterminate types. |
| 5. | Limits of sums, differences, products, quotients, and composition of functions. |
| 6. | The number e as a limit. |
| b. | The Marquis de La'Hospital's and John Bernoulli. |
| c. | The number e: Napier and Euler. |
| d. | Cauchy: Definition of a limit. |
| C. | Apply the definition of continuity using limits to analyze the behavior of functions. |
| 1. | Definition of continuity at a point using limits. |
| 2. | Continuity from the left or right. |
| 3. | Singularities, including removable singularities. |
| 4. | Continuity of sums, differences, products, quotients, and composition of functions. |
| 5. | Apply the Intermediate Value Theorem when locating roots of functions. |
| a. | Applications to continuous functions. |
| b. | Examples of discontinuous functions. |
| D. | Find the derivative of a function as a limit. |
| 1. | Derivative of a function at a point using limits. |
| 2. | Derivatives of power functions and trigonometric functions using limits. |
| E. | Derive and use the power, quotient, product, and chain rules to differentiate functions, including implicit and parametric functions, and find the equation of a tangent line to a function.
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| 1. | The derivative as the slope of a tangent line. |
| a. | Find an equation for the tangent line to the graph of a function. |
| b. | Use the chain rule to find the slope of the tangent line for curves defined parametrically |
| 2. | The derivative as a function. |
| 3. | Derivatives of logarithmic and inverse trigonometric functions by implicit differentiation. |
| 4. | Functions that are not differentiable. |
| 5. | Continuity and differentiability. |
| 6. | Derivatives, linear approximations, and differentials. |
| 7. | Apply Newton's Method to find roots of functions. |
| a. | The geometry of Newton's Method and tangent lines. |
| b. | Applications in finding roots of functions. |
| c. | The importance of the initial approximation. |
| d. | Newton only used his own method once. |
| e. | Connections to chaos theory via the function f(x) = 1 + x^2. |
| F. | Graph functions using methods of calculus
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| 1. | The first derivative: increasing and decreasing functions. |
| 2. | The second derivative: concavity and the shape of curves. |
| 3. | Critical values and inflection points. |
| 4. | Function graphing, including using asymptotes. |
| 5. | Interpret and apply the Mean Value Theorem for derivatives in relation to average and instantaneous rate of change. |
| a. | Examine the connection between instantaneous and average rate of change. |
| b. | Examine explicit connection between a function and its first derivative. |
| a. | Uses of derivatives in CAD to explore Bezier curves. |
| G. |
Apply the derivative to solve applications including related rate problems and optimization problems; |
| 2. | Average and instantaneous rate of change. |
| 3. | The instantaneous rate of change as a marginal rate of change. |
| 4. | Growth/decay rates of change. |
| 5. | Use differentiation to solve applications such as related rate problems and optimization problems; |
| a. | Model the mathematical relationship between changing quantities. |
| b. | Apply differentiation techniques, including the chain rule, to express a rate of change. |
| c. | Appropriate applications from a variety of fields such as: physics, sports, chemistry. |
| 6. | Formulate equations to model minimum/maximum problems and use derivatives to arrive at plausible solutions. |
| a. | Apply first derivatives to find local minima, local maxima, global minima and global maxima. |
| b. | Apply second derivatives to find local minima, local maxima, global minima and global maxima. |
| 7. | Investigate application problems using parametric equations in two dimensions such as but not limited to |
| H. | Define the antiderivative and determine antiderivatives of simple functions. |
| 1. | Practice finding the most general antiderivative of a function. |
| 2. | Examine the position function as the antiderivative of the velocity function, and velocity as the antiderivative of acceleration.
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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 students' 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 examinations 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.
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| B. | Examples of Supporting Texts and References |
| 1. | Hass, Weir, and Thomas, "University Calculus, Early Transcendentals", 4th Ed. Pearson 2020.
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| 2. | Anton, Bivens, and Davis, "Calculus, Early Transcendentals Combined", 10th Ed. Wiley 2011.
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| 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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