Portland Community College | Portland, Oregon

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’Hospital’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 anti-derivatives.  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 the “expand” command on their CAS for more complicated problems. Because this course is also a pre-requisite for MTH 261, logic and correct application of theorems should be emphasized.

Upon completion of the course students should be able to:

• Analyze real world scenarios to recognize when derivatives or integrals are appropriate, formulate problems about the scenarios, creatively model these scenarios (using technology, if appropriate) in order to solve the problems using multiple approaches, judge if the results are reasonable, and then interpret and clearly communicate the results.
• Recognize derivative and integral concepts that are encountered in the real world, understand and be able to communicate the underlying mathematics involved to help another person gain insight into the situation.
• Work with derivatives and integrals in various situations and use correct mathematical terminology, notation, and symbolic processes in order to engage in work, study, and conversation on topics involving derivatives and integrals with colleagues in the field of mathematics, science or engineering.

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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
   • at least one of the exams should require the use of technology
   • 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
     • Attendance
   The Lab component will account for at least 15% of the grade 
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 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 the formal definition of the definite integral and several estimation techniques rooted in this definition.
• Students will learn to antidifferentiate 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 definitive integrals.
• Students will learn to evaluate indeterminate form limits using L'Hospital's Rule.

• 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 calculators and/or computers will be demonstrated and encouraged by the instructor where appropriate. Technology will be used (at least) when graphing curves, evaluating derivatives, and evaluating definite integrals. Students should be encouraged to check their work for limits, derivatives and integrals using the CAS features of their caclulator, but this should not be used as a substitute for completing these operations by hand.

1. Applications of the Derivative The goal is to 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'Hospital'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 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 antidifferentiation.
      1. Antidifferentiate power, exponential, and trigonometric functions.
      2. Antidifferentiate a function using substitution.
      3. Antidifferentiate a function using integration by parts.

      4. Antidifferentiate 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 antidifferentiation using trigonometric substitution.

      6. Find antiderivatives using combinations of the above techniques.

      7. Evaluate integrals using a computer algebra system
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 totals 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.
      1. Construct and evaluate a Riemann Sum using left-hand endpoints, right-hand endpoints, or midpoints.
      2. Use the trapezoid rule to approximate a definite integral.
      3. Use Simpson's rule to approximate a definite integral.
      4. Estimate the errors accrued in the approximation techniques listed above.
   10. Evaluate improper integrals.
       1. Determine divergence or convergence utilizing 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. Discs

         2. Washers

         3. Shells

      4. Calculate arc length of a curve (explicit or parametric) over a closed interval.
   2. Study applications to physics, engineering, and other sciences
      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.
   3. Find the average value of a continuous function over a given interval.
   4. Understad the mean value theorem for integrals.

5. Optional Topics

   1. Antidifferentiation using tables.

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

   3. Further applications of the definite integral to physics, engineering and other sciences such as center of mass, hydrostatic force, cumulative distribution functions, or applications to biology and economics.

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.