Portland Community College | Portland, Oregon

Upon completion of the course students should be able to:

• Application – Analyze real world scenarios to: recognize when limits and derivatives are appropriate, formulate and model these scenarios (using technology, if appropriate) in order to find solutions using multiple approaches, judge if the results are reasonable, and then interpret these results. 

• Concept – Recognize the underlying mathematical concepts of limits and derivatives.

• Computation – Use limits and derivatives with correct mathematical terminology, notation, and symbolic processes.

• Communication – Communicate mathematical applications, concepts, computations, and results with classmates and colleagues in the fields of science, technology, engineering, and mathematics.

Enjoy a life enriched by exposure to one of humankind's greatest achievements.

1. Demonstrate an understanding of the concepts of derivatives and limits, and their application to real world problems in:

   • at least two proctored exams, one of which is a comprehensive final

   • proctored exams should be worth at least 50% of the overall grade

   • a closed book/closed note/no technology exam over derivative formulae

   • laboratory reports (graded homework with an emphasis on proper notation and proper documentation)

   • And at least one of the following:

     • Take-home examinations

     • Quizzes

     • Graded homework problems

2. The Lab component will account for at least 15% of the grade.

3. Consistently demonstrate proper notation, documentation, and use of language throughout all assessments and assignments. For proper documentation standards, see Addendum.

4. Demonstrate an ability to work and communicate with colleagues on the topics of derivatives and limits in at least two of the following:

   • A team project with a written report and/or in-class presentation

   • Participation in discussions

   • In-class group activities

• Students will learn to evaluate limits graphically, numerically, and symbolically.
• Students will learn the formal definition of the derivative and find algebraic derivatives using both this definition and the traditional shortcut formulas associated with derivatives.
• Students will learn and apply the relationships between functional behavior and first and second derivative behaviors.
• Students will learn to model and solve several types of applications using derivatives.

• Classroom activities will include lecture/discussion and group work.
• Students will communicate their results in oral and written form.
• Students will apply concepts to real world problems.
• The use of technology should be demonstrated and encouraged by the instructor where appropriate. 

1. Limits and Continuity The goal is to understand the limit at a number, infinite limits, limits at infinity, continuity at a number and continuity over an interval.

   1. Determine the limit at a number (from the left, from the right, and two sided) for functions presented in graphical form.

   2. Estimate the limit at a number (from the left, from the right, and two sided) for functions presented in symbolic form numerically using an appropriate table.

   3. Determine the limit at a number (from the left, from the right, and two sided) for functions, including piecewise-defined functions, presented in symbolic form using limit laws.

   4. Determine the limit at a number that results in the indeterminate form (\(\frac{0}{0}\) or \(\frac{\infty}{\infty}\)) numerically, graphically, and symbolically using algebraic techniques and limit laws.

   5. Be exposed to determining the limit at a number using the Squeeze Theorem.

   6. Estimate the limits at both positive and negative infinity for functions numerically, graphically, and symbolically and state the corresponding horizontal asymptote(s).

   7. Determine the one-sided limits at a number that results in the form \(\frac{ \text{non-zero} }{0}\) as either \(\infty\) or \(-\infty\) numerically, graphically, and symbolically.

   8. Identify types of discontinuities at a number for functions presented in graphical and symbolic form, including piecewise-defined functions.

   9. Identify one-sided continuity at a number for functions in graphical and symbolic form, including piecewise-defined functions.

2. The Derivative and an introduction to the Antiderivative The goal is to understand the definition of the derivative, the meaning of instantaneous rate of change, the practical meaning of the derivative as a rate of change, and the concept of the antiderivative.

   1. Find average velocities for objects whose position functions are presented in graphical, tabular, and symbolic form.

   2. Estimate instantaneous velocities for objects whose position functions are presented in graphical, tabular, and symbolic form.

   3. Estimate instantaneous accelerations for objects whose position functions and/or velocity functions are presented in symbolic form.

   4. Estimate derivative values for functions presented in graphical, tabular, and symbolic form.

   5. Understand the formal definitions of the derivative representing tangent line slope as the limit of a secant line slope:

      1. \(f'(a) = \mathop {\lim }\limits_{x \to a} \frac{{f\left( {x} \right) - f\left( a \right)}}{x-a}\)

      2. \(f'(a) = \mathop {\lim }\limits_{h \to 0} \frac{{f\left( {a + h } \right) - f\left( a \right)}}{h}\)

   6. Use the formal definitions of the derivative to find derivative values for functions presented in symbolic form.  

   7. Find the equations of tangent lines to a given curve. 

   8. Sketch the graph of the derivative for functions presented in graphical form.

   9. Use the formal definition of the derivative to find the derivative as a function for functions presented in symbolic form.

   10. Find the units for, and interpret the meaning of, derivative values for applied functions presented in graphical, tabular, symbolic, and written form.

   11. Identify points of non-differentiability for functions presented in graphical form.

   12. Define and identify the concavity and points of inflection for functions presented in graphical form.

   13. Determine the shape of a function from numerical and graphical information about that function’s first and/or second derivatives.

   14. Given a function in graphical form, determine the shape of its antiderivative.

3. Symbolic Differentiation The goal is to find derivative formulas for functions presented in symbolic form using both Leibniz notation and prime notation and to interpret the formulas in applied contexts.

   1. Recognize the difference between the name of a derivative (\(\frac{dy}{dx}\), \(\frac{dz}{dt}\), ...) and the derivative operator (\(\frac{d}{dx}\), \(\frac{d}{dt}\), ...).

   2. Utilize the rules of differentiation for power, exponential, logarithmic, trigonometric, and inverse trigonometric functions.

   3. Differentiate the sum, difference, product, and/or quotient of two or more functions, emphasizing the use of Leibniz notation for the derivative operator.

   4. Differentiate the composition of two or more functions, emphasizing the use of Leibniz notation for the derivative operator.

   5. Differentiate implicit equations, emphasizing the use of Leibniz notation for the derivative operator and for the name of a derivative.

   6. Differentiate functions using logarithmic differentiation.

4. Applications of the Derivative The goal is to use derivatives to solve real-world problems involving rates and use the first and second derivatives to analyze the behavior of families of functions.

   1. Solve applications involving related rates.

   2. Find the critical numbers for a function.

   3. Understand the Mean Value Theorem for derivatives graphically.

   4. Use the First Derivative Test to identify intervals where the function is increasing and decreasing, and to identify maxima and minima.

   5. Use the Concavity Test to identify intervals where the function is concave up or concave down, and identify points of inflection.

   6. Graph a function by hand after identifying the increasing/decreasing behavior, concavity, asymptotes and intercepts.

This course is a required prerequisite for MTH 252. The topics related to applications of derivatives (such as critical numbers and the first derivative test) should be given adequate time for coverage. These topics will only be briefly reviewed in MTH 252 in preparation for optimization.

All work in this course will be evaluated for your ability to meet the following writing objectives as well as for mathematical content.

1. Every solution must be written in such a way that the question that was asked is clear simply by reading the submitted solution.
2. Any table or graph that appears in the original problem must also appear somewhere in your solution.
3. All graphs that appear in your solution must contain axis names and scales. All graphs must be accompanied by a figure number and caption. When the graph is referenced in your written work, the reference must be by figure number. Additionally, graphs for applied problems must have units on each axis and the explicit meaning of each axis must be self-apparent either by the axis names or by the figure caption.
4. All tables that appear in your solution must have well defined column headings as well as an assigned table number accompanied by a brief caption (description). When the table is referenced in your written work, the reference must be by table number.
5. A brief introduction to the problem is almost always appropriate.
6. In applied problems, all variables and constants must be defined.
7. If you used the graph or table feature of your calculator in the problem solving process, you must include the graph or table in your written solution.
8. If you used some other non-trivial feature of your calculator (e.g., SOLVER), you must state this in your solution.
9. All (relevant) information given in the problem must be stated somewhere in your solution.
10. A sentence that orients the reader to the purpose of the mathematics should usually precede symbol pushing.
11. Your conclusion shall not be encased in a box, but rather stated at the end of your solution in complete sentence form.
12. Remember to line up your equal signs.
13. If work is word-processed all mathematical symbols must be generated with a math equation editor.

Lab time shall be used by students to work on group activities—the activities to be used during lab are on the mathematics department home page at https://www.pcc.edu/programs/math/course-downloads.html

Emphasis should be placed on using technology such as Desmos and GeoGebra appropriately; such as when computing approximations, graphing curves, or visualizing or checking answers.  Technology should not be used as a substitute for meeting the outcomes and skills for the course that are expected to be done by hand.