Portland Community College | Portland, Oregon

Upon successful completion of the course, students should be able to:

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

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

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.

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.

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

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

   • at least two proctored, closed-book, no-notes exams worth at least 50% of the overall grade, one of which is a comprehensive final that is worth at least 25% of the overall grade, consisting primarily of free response questions (although a limited number of multiple choice and/or fill in the blank questions may be used where appropriate)

   • a proctored, closed-book, no-notes, no-calculator proctored exam over derivative formulae

   • laboratory reports (graded with an emphasis on proper notation and proper documentation) that account for at least 15% of the overall grade

   • And at least one of the following:

     • Take-home examinations

     • Quizzes

     • Graded homework problems

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

3. 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

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

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

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

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

   1. Determine 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. Apply the formal definitions of the derivative representing tangent line slope as the limit of a secant line slope:

      1. \(f’(a) = \displaystyle\lim_{x \to a} \frac{f(x) - f(a)}{x - a}\)

      2. \(f’(a) = \displaystyle\lim_{h \to 0} \frac{f(a + h) - f(a)}{h}\)

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

   7. Find the equation of the tangent line at a given point for a function presented in symbolic form.

   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. Use appropriate 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.

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. Utilize the name of a derivative (\(\frac{dy}{dx}, \frac{dz}{dt}, \ldots\)) and the derivative operator (\(\frac{d}{dx}, \frac{d}{dt}, \ldots\)) appropriately.

   2. Find the derivative of power, polynomial, rational, exponential, logarithmic, trigonometric, and inverse trigonometric functions. Utilize the constant multiple, sum, difference, product, and quotient rules of derivatives; emphasizing the use of Leibniz notation for the derivative operator.

   3. Utilize the chain rule to differentiate the composition of two or more functions, emphasizing the use of Leibniz notation for the derivative operator.

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

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 applied problems involving related rates.

   2. Use the concepts of the first and second derivatives symbolically to identify key function features.

      1. Find the critical numbers of a function.

      2. Use the first derivative to identify intervals where the function is increasing and decreasing.

      3. Use the First Derivative Test to identify the relative/local maximum and minimum values of a function and where they occur.

      4. Find absolute/global maximum and minimum values of a function over a closed interval and where they occur.

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

      6. Use the Second Derivative Test to identify relative/local maximum and minimum values and where they occur.

   3. Solve applied problems involving optimization.

   4. Use L'Hôpital's Rule to evaluate limits for indeterminate forms such as \(\frac{\infty}{\infty}, \frac{0}{0}\), and \(0 \cdot \infty\).

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.

This class is a foundational course for many STEM majors. Some topics are of particular importance for students continuing into MTH 253Z including: using L'Hôpital's Rule to evaluate limits, improper integrals, and error estimates for definite integrals. Students may be taking this course concurrently with calculus-based physics courses. It can be beneficial for these students if the integral symbol is introduced early on to represent antiderivatives.  Partial fractions are a particularly important technique for engineering students (which will be revisited in MTH 253Z and MTH 256). Students should be able to do simple partial fraction expansions by hand, but may use technology for more complicated problems. Since this course is also a prerequisite for MTH 261, logic and correct application of theorems should be emphasized.

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.