Portland Community College | Portland, Oregon

This course is required for all students in the Civil and Mechanical Engineering Technology program, and is a prerequisite for many other CMET courses. Full-time students will generally take this course in their third term of the program.

The student will be able to:

1. Use differential and integral calculus to solve engineering problems involving rates of change.
2. Apply knowledge gained in this course in subsequent engineering courses such as Dynamics, Thermodynamics, or others.
3. Communicate analysis and results clearly: orally, in writing, and through diagrams and calculations.
4. Work in small groups with individuals of diverse cultural backgrounds

Individual, small group, and full class discussion; homework problems; examinations; and small group problem-solving sessions may be used to assess outcomes.
Lecture, homework, and in-class group activities will be coordinated.
Specific evaluation procedures will be defined during the first week of class. In general, grading will depend on weekly tests, homework, class participation, and a comprehensive final exam.

1. Engineering problems often involve quantities that are changing, and calculus is needed to describe and model that change.
2. A mathematical relationship may be described verbally, graphically, numerically, or symbolically; each representation has advantages and disadvantages. A complete analysis of an engineering problem may include several of these approaches.
3. Calculus is an essential mathematical tool for engineers and technicians. Even though many people in the engineering field do not frequently solve equations that require derivatives and integrals, a basic knowledge of calculus is necessary for understanding the mathematical background of many relationships that are used.
4. Studying calculus opens doors for engineering technicians. Many technical books, articles, and references use symbols and procedures that involve calculus. People who have not studied calculus will often avoid such material because they are intimidated by the unfamiliar symbols and language.
5. Studying calculus greatly improves a student’s algebra skills.
6. A large complex problem consists of many inter-related smaller problems which must be solved in a logical order.
7. Solution of an engineering problem is not useful unless communicated clearly and completely.
8. There is often more than one correct approach to the solution of an engineering problem. Sharing ideas with others will often lead to the most efficient or clearest solution.

CONTENT:
1. Review of functions:
            a. Function notationb
            b. Evaluating a function
            c. Domain and range
            d. Graphing with and without using a graphing calculator
2. Limits and Continuity
            a. Determine if a function is continuous at a point, or continuous over an interval
            b. Find the limit of a function as the independent variable approaches a given value or as it approaches infinity, by using a table of                 values and by using pre-programmed operations on a scientific calculator.
3. Derivative of a function
            a. Find the derivative of a function using the definition of a derivative (the delta-process).
            b. Find the derivative of a polynomial function using the rules for the derivative of a constant, a power of the independent variable, a             sum or difference, a constant times a function.
            c. Find the derivative of a product and a quotient of functions.
            d. Find the derivative of a composite function by using the chain rule.
            e. Find the derivative of a power of a function.
            f. Find the derivative of a trigonometric or inverse trigonometric function.
            g. Find the derivative of a logarithmic or exponential function.
            h. Find the derivative of an implicit function.
            i. Find the time derivative of a function.
            j. Find second derivatives and higher derivatives of a function.
4. Applications of differentiation
            a. Find the equation of a line tangent or normal to a given function.
            b. Use Newton’s method for solving for zeros of a function.
            c. Find the average rate of change and the instantaneous rate of change of a function.
            d. Use semi-graphical methods to solve problems involving average rate of change and the instantaneous rate of change.
            e. Solve kinematics problems involving instantaneous velocity and acceleration of an object undergoing rectilinear motion or                         curvilinear motion.
            f. Solve applied problems involving two or three variables with related rates of change.
            g. Solve applied problems involving optimization, including applications from statics, physics, and strength of materials.
            h. Solve applied problems involving beam relationships of shear, bending moment, and deflection.
            i. Use derivatives in curve sketching to determine concavity and find local extrema and points of inflection.
            j. Find the differential of a function.
            k. Use pre-programmed operations of a scientific calculator to find the derivative of a function.
5. Integration of polynomial, trigonometric, logarithmic, exponential, and composite functions.
            a. Find an antiderivative of a given function.
            b. Integrate a function, using indefinite integration.
            c. Evaluate the definite integral of a function.
            d. Approximate the value of a definite integral by using the trapezoidal rule and Simpson’s rule.
            e. Integrate a function using integration by parts.
            f. Integrate a function using trigonometric substitution.
            g. Integrate a function using integration tables.
            h. Solve first order differential equations using separation of variables.
            i. Perform definite and indefinite integration using pre-programmed operations of a scientific calculator.
6. Applications of Indefinite Integrals
            a. Solve kinematics problems involving instantaneous position, velocity and acceleration of an object undergoing rectilinear motion             or curvilinear motion.
            b. Solve thermodynamics problems involving work done by an ideal gas in a isothermal or adiabatic process.
            c. Solve beam problems involving relationships of load, shear, bending moment, slope, and deflection. Write moment equation                     using singularity functions to describe discontinuous loads, then differentiate and integrate to produce the other equations.
7. Applications of Definite Integrals
            a. Find the area between two curves, using vertical or horizontal elements.
            b. Find the volume of a solid of revolution, using disks, washers, and shells.
            c. Find the centroid of an area, and the centroid of a solid of revolution.
            d. Find the moment of inertia and radius of gyration of an area with respect to a reference axis, and with respect to a centroidal axis.             Find the polar moment of inertia with respect to the centroid.
            e. Find the mass moment of inertia and radius of gyration of an solid of revolution with respect to a reference axis, and with respect             to a centroidal axis. Find the polar moment of inertia with respect to the centroid.
            f. Solve problems involving the average value of a function, including the average specific heat in a thermodynamic process.
            g. Solve applied problems involving work by a variable force.
            h. Solve applied problems involving a force due to hydrostatic pressure.
COMPETENCIES AND SKILLS:
The student will be able to:

1. Symbolically differentiate and integrate many types of function.
2. Show relationships between symbolic and graphical representations of functions.
3. Apply differentiation to problems involving optimization and related rates of change.
4. Apply integration and differentiation to problems of kinematics and beam loadings.
5. Communicate analysis and results clearly: orally, in writing, and through diagrams and calculations.
6. Read technical material that includes references to derivatives and integrals.