Active Outline

General Information

Course ID (CB01A and CB01B)
MATHD001A
Course Title (CB02)
Calculus I
Course Credit Status
Credit - Degree Applicable
Effective Term
Fall 2024
Course Description
This course covers the fundamentals of differential calculus.
Faculty Requirements
Course Family
Not Applicable

Course Justification

This course satisfies the CSU GE/Breadth requirement Area B4, Mathematics and Quantitative Reasoning, and also satisfies the IGETC requirement, Area 2 Mathematical Concepts, and Quantitative Reasoning. This course is CSU and UC transferable. This course is a required core course for the A.S.-T degree in Mathematics. This is the first course in a sequence of four courses in calculus. This course emphasizes single-variable differential calculus.

Foothill Equivalency

Does the course have a Foothill equivalent?
No
Foothill Course ID

Course Philosophy

Course Philosophy
This is the first course in the calculus sequence. As such, it develops the foundations from which MATH 1B-1D are built. Functions of two variables are a vital component in the development of the calculus. Thus, multiple representations of functions are stressed: numerical, symbolic, verbal and graphical. The exponential function is introduced since it is so fundamental to many natural phenomena. Also, the idea of a limit of a function is thoroughly studied, emphasizing the classical tangent and velocity problems -- the study of the derivative. These two quantities are the two most compelling representations of the material covered in this course -- that of the differential calculus.



Applications are very much a part of this course. Problems pertaining to the graphing of functions, related rates, and minimum and maximum are studied. The course concludes with a brief introduction to the second major branch of the calculus: the problem of finding antiderivatives. This sets the stage for the development of the integral calculus in MATH 1B.



This outline contains historical notes relevant to the material covered in the sections they appear. They are not part of the actual course outline. These historical notes are to be used at the discretion of the instructor.

Formerly Statement

Course Development Options

Basic Skill Status (CB08)
Course is not a basic skills course.
Grade Options
  • Letter Grade
  • Pass/No Pass
Repeat Limit
0

Transferability & Gen. Ed. Options

Transferability
Transferable to both UC and CSU
CSU GEArea(s)StatusDetails
CGB4CSU GE Area B4 - Mathematics/Quantitative ReasoningApproved
IGETCArea(s)StatusDetails
IG2XIGETC Area 2 - Mathematical Concepts and Quantitative ReasoningApproved
C-IDArea(s)StatusDetails
MATHMathematicsApprovedMATH D001A & MATH D001B required for C-ID MATH 210 MATH D001A & MATH D001B & MATH D001C required for C-ID MATH 900 S

Units and Hours

Summary

Minimum Credit Units
5.0
Maximum Credit Units
5.0

Weekly Student Hours

TypeIn ClassOut of Class
Lecture Hours5.010.0
Laboratory Hours0.00.0

Course Student Hours

Course Duration (Weeks)
12.0
Hours per unit divisor
36.0
Course In-Class (Contact) Hours
Lecture
60.0
Laboratory
0.0
Total
60.0
Course Out-of-Class Hours
Lecture
120.0
Laboratory
0.0
NA
0.0
Total
120.0

Prerequisite(s)

MATH D032., MATH D032H, MATH D043. or MATH D043H (with a grade of C or better), or appropriate score on Calculus Placement Test within the past calendar year

Corequisite(s)

Advisory(ies)

ESL D272. and ESL D273., or ESL D472. and ESL D473., or eligibility for EWRT D001A or EWRT D01AH or ESL D005.

Limitation(s) on Enrollment

(Not open to students with credit in the Honors Program related course.)

Entrance Skill(s)

General Course Statement(s)

(See general education pages for the requirements this course meets.)

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

Assignments

  1. Required readings from text
  2. Problem solving exercises
  3. A selection of homework, quizzes, group projects and exploratory worksheets
  4. Optional project synthesizing various concepts and skills from course content

Methods of Evaluation

  1. 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.
  2. 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.
  3. 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.
  4. 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.

Essential Student Materials/Essential College Facilities

Essential Student Materials: 
  • Students will use technology (computers or graphing calculators) to explore mathematical concepts graphically and numerically
Essential College Facilities:
  • None.

Examples of Primary Texts and References

AuthorTitlePublisherDate/EditionISBN
James Stewart, Daniel Clegg & Saleem Watson "Calculus: Early Transcendentals", 9th Ed. Cengage 2021.

Examples of Supporting Texts and References

AuthorTitlePublisher
Hass, Weir, and Thomas, "University Calculus, Early Transcendentals", 4th Ed. Pearson 2020.
Anton, Bivens, and Davis, "Calculus, Early Transcendentals Combined", 10th Ed. Wiley 2011.
Hughes-Hallett, et al., "Calculus, Single and Multivariable", 7th Ed. Wiley 2017.
Mathematics Multicultural Bibliography, available on the De Anza Math department resources website.

Learning Outcomes and Objectives

Course Objectives

  • Analyze and explore aspects of the differential calculus.
  • Compute and interpret limits of functions using analytic and other methods, including L'Hospital's Rule.
  • Apply the definition of continuity using limits to analyze the behavior of functions.
  • Find the derivative of a function as a limit.
  • 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.
  • Graph functions using methods of calculus
  • Apply the derivative to solve applications including related rate problems and optimization problems;
  • Define the antiderivative and determine antiderivatives of simple functions.

CSLOs

  • 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.

Outline

  1. Analyze and explore aspects of the differential 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, piece-wise, parametric, implicit, and inverse functions.
  2. Compute and interpret limits of functions using analytic and other methods, including L'Hospital's Rule.
    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.
    4. Limits at infinity.
    5. Limits of sums, differences, products, quotients, and composition of functions.
    6. The number e as a limit.
    7. Historical note:
      1. Zeno's Paradox
      2. The Marquis de La'Hospital's and John Bernoulli.
      3. The number e: Napier and Euler.
      4. Cauchy: Definition of a limit.
  3. 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.
      1. Applications to continuous functions.
      2. Examples of discontinuous functions.
  4. 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.
  5. 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.
    1. The derivative as the slope of a tangent line.
      1. Find an equation for the tangent line to the graph of a function.
      2. 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.
    8. Historical note:
      1. The geometry of Newton's Method and tangent lines.
      2. Applications in finding roots of functions.
      3. The importance of the initial approximation.
      4. Newton only used his own method once.
      5. Connections to chaos theory via the function f(x) = 1 + x^2.
  6. Graph functions using methods of calculus
    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.
      1. Examine the connection between instantaneous and average rate of change.
      2. Examine explicit connection between a function and its first derivative.
    6. Historical notes:
      1. Uses of derivatives in CAD to explore Bezier curves.
      2. Joseph Lagrange


  7. Apply the derivative to solve applications including related rate problems and optimization problems;
    1. Tangents and velocity.
    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;
      1. Model the mathematical relationship between changing quantities.
      2. Apply differentiation techniques, including the chain rule, to express a rate of change.
      3. 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.
      1. Apply first derivatives to find local minima, local maxima, global minima and global maxima.
      2. 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
      1. Projectile Motion
      2. Circular motion
      3. Cycloids
  8. 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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