Exam 1 Study Guide

Exam 1 Topics

  • The first exam covers weeks 1 through 6 of class. See the Daily Lessons for a complete list of topics.

  • Each daily lesson contains a collection of resources you can use to review the lessons. In particular pay attention to:

    • Learning Goals: Each lesson has a list of learning goals. Make sure you know/are able to do everything listed in those goals.

    • Worksheet Solutions: The worksheets have both written and video solutions. These can be useful when going over the worksheets.

    • Notes: The course notes contain the main information needed for each lesson. Most of the notes contain colored boxes which contain key information for that lesson.

Exam 1 Summary

Here is a summary (partial list) of things you should know and be able to do. Use the above links for the full list.

  • Matrix Algebra: Know matrix addition/subtraction, scalar multiplication, and matrix multiplication. Know when it is possible and not possible to do the previous operations.

  • Be able to solve Linear Systems by writing the system as an augmented matrix and preforming elementary row operations to reduce the system to a simpler system (Gaussian elimination). Be able to solve systems a unique solutions, and an infinite number of solutions. Be able to determine if a system has no solution.

  • Be able to compute the determinant of square matrices using cofactor expansion.

  • Know that an eigenvector \(\vec{\bf v}\) is any vector such that \(A\vec{\bf v} = \lambda \vec{\bf v}\) and \(\lambda\) is the eigenvalue. Be able to find Eigenvalues and Eigenvectors and Non-Real Eigenvalues and Eigenvectors.

  • Know the Derivative and Antiderivative Rules, the notation, and what they represent from Calculus.

  • Know the Terminology, classifications, and vocabulary used in differential equations.

  • Know what it means to be a solution to a differential equation and how you can verify a given formula is a solution by substituting it into the both RHS and LHS of the equation, then use algebra and calculus to show they both simplify to the same (equivalent) expression.

  • Be able to determine the order of a differential equation and determine if it is linear or non-linear. In addition if a differential equation is linear, be able to state if it is homogeneous or nonhomogeneous.

  • Be able to identify if a first order differential equation is autonomous.

  • Be able to to find and classify equilibrium solutions and draw a phase line graph for autonomous DEs. Be able to use the phase line graph to draw integral curves and answer questions about solution behavior.

  • Be able to use slope (directional) fields to graph integral curves and answer questions about solution behavior.

  • Be able to both recognize which method to use and to solve first order DEs using one of the three methods from class. In all three cases know how to calculus and algebra to rewrite the DE in the form:

    \[\frac{d}{dt}\Big(\text{formula}\Big) = \text{formula with only }t\]

  • Be able to work with models:

    • The examples that have come up so far have been exponential growth/decay, Newton’s Law of Cooling, population dynamics (logistic growth), second order chemical reactions, and tank mixing problems.

    • Be able to setup an initial valued problem (IVP) for tank mixing problems.

    • Given an IVP for one of the above situations, be able to:

      • Use qualitative analysis to answer questions about solution behavior and limiting values.
      • Be able to solve the DE for an explicit solution.
      • Be able to use the solution to answer questions about the model.

Review / Study Tips

Here are some tips to help you study for the exam in order of importance:

  • Review the old quizzes. Rework any problem you had trouble with.

  • Read over all the learning goals on the Daily Lessons page.

  • Complete the four exam review worksheets (these worksheets contain old exam questions).

  • Review the worksheets and their solutions.

  • Review the WeBWorK assignments. Rework the problems you had trouble with.