Study Guide for MATH 1451, Spring 2018

The final exam will be Monday, May 7 from 1-3PM in our usual classroom.  You are allowed to use a calculator and to have one 4"x6" card (or smaller) of notes (both sides).

The exam will cover the material from the following sections of the textbook:

Chapter 6, Sections 7.1, 7.2, 7.5-7.7, 8.1, 8.2, 8.4,
Chapter 9, and Sections 10.1-10.4, 11.1-11.6

There will NOT be any questions about Sections 7.3, 7.4, 8.3, 8.5, or 10.5.  There will also NOT be any questions about Error behavior in 7.5 (those are the sections headed "Error in ..."), Simpson's rule in 7.5, known cross-sections from 8.2, arc length in 8.2, Error Bounds for Alternating Series in 9.4, Error Bounds Using the Alternating Series Estimate in 10.4, or Equilibrium Solutions in 11.5.

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Approximately half of the questions will cover material since Exam 3.  That is, Sections 10.4, 11.1-11.6.  The main topics in those sections are:

- Lagrange error bound for Taylor polynomials
- Test if a function is a solution to a differential equation
- Draw a slope field for a differential equation
- Sketch solutions using a slope field
- Use Euler's method to numerically approximate a solution
  to a differential equation
- Identify separable differential equations and use Separation
  of variables to find solutions.
- Exponential growth and decay problems
    Find constants, doubling times, half-lives
- Applications and Modeling
    Write a differential equation to describe a given problem

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The other half (roughly) of the questions will be on material from earlier in the course.  I didn't post a list of topics for Exam 1, so here is such a list:

- Sketch antiderivative given graph of function
- Compute values of antiderivatives using definite integrals
- Indefinite integral
- Find general antiderivative
- Properties of antiderivatives
- Use antiderivatives to compute definite integrals
- Equations of motion/ falling objects
- Construction of antiderivatives using the definite integral
 (Second Fundamental Theorem of Calculus)
- Integration by substitution
- Integration by parts

For completeness, here are the topics from the Exam 2 and Exam 3 study guides:

- Numerical methods of integration(LEFT, RIGHT, MID, TRAP)
- Improper integrals with infinite limits
- Improper integrals with infinite integrads
- Comparison of integrals to determine convergence or divergence
   (incl. good functions like 1/x^p and e^{-a x} as given in
    the "Useful Integrals for Comparison" box in section 7.7)
- Areas by slicing
- Volumes by slicing
- Volumes of solids of revolution
- Finding a total quantity given a density function (section 8.4)
- Finding total mass of a 1-dimensional rod, 2-dimensional region,
   or 3-dimensional solid given a mass density function
- Find the center of mass with point masses, a rod with given
   mass density function, or a 2D region with constant density.

- Sequences: recursive definitions, formulas, convergence
- Convergence of a monotone, bounded sequence
- Sum of finite geometric series
- Sum of infinite geometric series
- Partial sums and convergence of infinite series
- Convergence properties of series
- Harmonic series and alternating harmonic series
- The Integral Test for series convergence
- The Comparison Test for series convergence
- The Limit Comparison Test for series convergence
- The Ratio Test for series convergence
- The Alternating Series Test for series convergence
- Deciding if a series is absolutely convergent or if it is
   conditionally convergent
- Definition of a power series about x = a
- Finding the Radius of Convergence for a power series
- Finding the Interval of Convergence for a power series
- Computing the Taylor Polynomial of degree n approximating
   f(x) about x = a
- Computing the Taylor Series for f(x) about x = a
- Know and recognize the Taylor Series about x = 0 for sin x,
   cos x, and e^x
- The Binomial Series for (1 + x)^p
- Computing new Taylor Series from old by: substitution,
   differentiation, integration, multiplying