Portland Community College | Portland, Oregon

Upon completion of the course students should be able to:

• Application – Analyze real world scenarios to: recognize when limits, derivatives and integrals 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, derivatives, and integrals.

• Computation – Use limits, derivatives, and integrals 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 great achievements.

1. Demonstrate an understanding of the concepts of integrals 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 antiderivative 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 integrals, 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 use the first and second derivatives of a function to find extreme function values and to solve applied maximum/minimum problems.

• Students will learn to evaluate indeterminate form limits using L'Hôpital's Rule.

• Students will learn the formal definition of the definite integral and several estimation techniques rooted in this definition.

• Students will learn to integrate function formulas and use the Fundamental Theorem of Calculus to evaluate definite integrals.

• Students will learn to model and solve several types of applications using definite integrals.

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

   1. Use the first derivative to help find absolute extreme function values over a closed interval.

   2. Use the Second Derivative Test to classify the behavior of a function at appropriate critical points.

   3. Solve applied problems involving optimization.

   4. Use L'Hôpital's Rule to evaluate limits for indeterminate forms including \(\frac{\infty}{\infty}\), \(\frac{0}{0}\), \(0 \cdot \infty\), \(0^0\), \(1^\infty\) and \(\infty^0\).
2. The Antiderivative and the Indefinite Integral The goal is to find the antiderivative(s) of a function expressed in graphical or symbolic form.

   1. Draw the family of antiderivative curves given the graph of the derivative.

   2. Understand the role of the constant of integration in creating a "family” of antiderivative functions.

   3. Estimate values of an antiderivative given the graph of the derivative and initial conditions for the antiderivative.

   4. Utilize the rules of integration.

      1. Integrate power, exponential, and trigonometric functions.

      2. Integrate a function using substitution.

      3. Integrate a function using integration by parts.

      4. Integrate using partial fractions with linear factors (e.g. \(\frac{1}{(x-1)(x+2)}\) and \(\frac{1}{(x-1)(x+2)^2}\)).

      5. Be exposed to integration using trigonometric substitution.

      6. Find integrals using combinations of the above techniques.

      7. Evaluate integrals using technology.
3. The Definite Integral The goal is to develop a practical understanding of the definite integral, to make connections between the derivative and the definite integral, and to understand multiple techniques for definite integral evaluation and estimation.

   1. Understand the limit definition of the definite integral,  \(\int_a^b f(x)dx=\lim_{n \to \infty} \sum_{i=1}^n f(x_i^*)\Delta x\).

   2. Be exposed to sigma notation.

   3. Find left-hand, right-hand, and midpoint Riemann sums for functions presented in graphical, tabular, and/or symbolic form.

   4. Interpret the practical meaning of Riemann sums of rate functions.

   5. Determine/estimate the net change and total change in a function when the derivative of the function is presented in graphical, tabular, and/or symbolic form.

   6. Understand the connection between definite integrals and net area between a function and the horizontal axis.

   7. Evaluate definite integrals using the Fundamental Theorem of Calculus.

   8. Apply the properties of the definite integral.

   9. Approximate definite integrals numerically using the following techniques:

      1. Riemann Sums

      2. Trapezoid Rule

      3. Simpson's Rule

   10. Estimate the errors accrued in approximating definite integrals numerically.

   11. Evaluate improper integrals.

       1. Determine divergence or convergence using limits and the Fundamental Theorem of Calculus.

       2. Determine divergence or convergence by comparison.

4. Using the Integral The goal is to use the definite integral to solve application problems.

   1. Study applications to geometry.

      1. Find areas of planar regions using definite integrals.

      2. Calculate volumes of a solid using cross-sectional slices.

      3. Calculate volumes of revolution generated by revolving a planar region around a line or axis using:

         1. Disks

         2. Washers

         3. Shells

      4. Calculate arc length of a curve (explicit or parametric) over a closed interval.

   2. Find the average value of a continuous function over a closed interval.

   3. Understand the Mean Value Theorem for Integrals.

   4. Cover at least one of the following applications in physics, engineering, and/or other sciences in similar depth to the required topics of this course, such as:

      1. Work

         1. Discuss unit analysis and emphasize the difference between mass and weight.

         2. Calculate work done by a force applied to an object to move it a given distance.

      2. Center of mass

      3. Hydrostatic force

      4. Cumulative distribution functions

      5. Probability

      6. Applications to biology and economics

5. Optional Topics

   1. Integrate using tables.

   2. Integration with partial fractions that have irreducible quadratics (e.g. \(\frac{1}{x^2+x+1}\)).

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 253 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 253 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.