Differential Calculus

Course Number:

MTH 251Z

Transcript Title:

Differential Calculus

Created:

Aug 15, 2022

Updated:

May 31, 2025

Total Credits:

4

Lecture Hours:

30

Lecture / Lab Hours:

20

Lab Hours:

0

Satisfies Cultural Literacy requirement:

No

Satisfies General Education requirement:

Yes

Grading Options

A-F, P/NP, Audit

Default Grading Options

A-F

Repeats available for credit:

0

Alignment with Institutional Learning Outcomes

Major

1. Communicate effectively using appropriate reading, writing, listening, and speaking skills. (Communication)

Major

2. Creatively solve problems by using relevant methods of research, personal reflection, reasoning, and evaluation of information. (Critical thinking and Problem-Solving)

Major

3. Extract, interpret, evaluate, communicate, and apply quantitative information and methods to solve problems, evaluate claims, and support decisions in their academic, professional and private lives. (Quantitative Literacy)

Not Addressed

4. Use an understanding of cultural differences to constructively address issues that arise in the workplace and community. (Cultural Awareness)

Not Addressed

5. Recognize the consequences of human activity upon our social and natural world. (Community and Environmental Responsibility)

To establish an intentional learning environment, Institutional Learning Outcomes (ILOs) require a clear definition of instructional strategies, evidence of recurrent instruction, and employment of several assessment modes.

Major Designation

1. The outcome is addressed recurrently in the curriculum, regularly enough to establish a thorough understanding.
2. Students can demonstrate and are assessed on a thorough understanding of the outcome.
   • The course includes at least one assignment that can be assessed by applying the appropriate CLO rubric.

Minor Designation

1. The outcome is addressed adequately in the curriculum, establishing fundamental understanding.
2. Students can demonstrate and are assessed on a fundamental understanding of the outcome.
   • The course includes at least one assignment that can be assessed by applying the appropriate CLO rubric.

Suggested Outcome Assessment Strategies

The determination of assessment strategies is generally left to the discretion of the instructor. Here are some strategies that you might consider when designing your course: writings (journals, self-reflections, pre writing exercises, essays), quizzes, tests, midterm and final exams, group projects, presentations (in person, videos, etc), self-assessments, experimentations, lab reports, peer critiques, responses (to texts, podcasts, videos, films, etc), student generated questions, Escape Room, interviews, and/or portfolios.

Course Activities and Design

The determination of teaching strategies used in the delivery of outcomes is generally left to the discretion of the instructor. Here are some strategies that you might consider when designing your course: lecture, small group/forum discussion, flipped classroom, dyads, oral presentation, role play, simulation scenarios, group projects, service learning projects, hands-on lab, peer review/workshops, cooperative learning (jigsaw, fishbowl), inquiry based instruction, differentiated instruction (learning centers), graphic organizers, etc.

The grade will include at least one project.

Course Content

Outcome #1: Calculate limits graphically, numerically, and symbolically; describe the behavior of functions using limits and continuity; and recognize indeterminate forms.

• One and two-sided limits; Squeeze Theorem

• Continuity and The Intermediate Value Theorem

• Limit Theorems and Evaluating Limits

• Limits at infinity and infinity as a limit

• Calculus and Graphing

• L'Hospital's Rule

Outcome #2: Apply the definition of the derivative and analyze average and instantaneous rates of change.

• Introduction – instantaneous rate of change and the need for limits

• Limit definition of derivative

• Derivatives as functions; Higher order derivatives

• Derivatives and the shape of graphs

• Calculus and Graphing

Outcome #3: Interpret and apply the concepts of the first and second derivative to describe and illustrate function features including the slopes of tangent lines, locations of extrema and inflection points, and intervals of increase, decrease, and concavity.

• Extreme Value Theorem and closed interval problems.

• First and Second Derivative Tests

• Calculus and Graphing

• Mean Value Theorem for Derivatives

Outcome #4: Apply product, quotient, chain, and function-specific rules to differentiate combinations of power, polynomial, rational, exponential, logarithmic, trigonometric, and inverse trigonometric functions, as well as functions defined implicitly.

• Derivatives of polynomials and the binomial expansion theorem

• Derivatives of power functions

• Derivative of the exponential function

• Derivatives of sums and differences

• Derivative Theorems; Product Rule – Quotient Rule

• Derivatives of Trig functions

• Chain Rule

• Implicit Differentiation

• Derivatives of inverse functions; Derivative of Cosh and Sinh 

Outcome #5: Apply derivatives to a variety of problems in mathematics and other disciplines, including related rates, optimization, and L’Hôpital’s rule.

• Continuity and The Intermediate Value Theorem

• Tangent Line approximations and differentials

• Related Rates

• Extreme Value Theorem and closed interval problems.

• First and Second Derivative Tests

• Calculus and Graphing

• Mean Value Theorem for Derivatives

• L'Hospital's Rule

• Newton's Method

• Optimization

Suggested Texts and Materials

Calculus Vol. 1, Openstax, Strang, Herman
This is an open textbook available at: https://openstax.org/details/books/calculus-volume-1

A graphing utility is required. We use Desmos; it is available as a free app for smartphones and tablets (from the app store) or through a browser on a laptop or desktop computer at: https://www.desmos.com

Department Notes

Answers to all application problems will be given in complete sentences with correct units. The grade will include at least one project.