Precalculus Midterm #3 Review

• §11.1: Systems of Equations
  • Solve systems of 2, 3, or 4 linear equations by substitution or elimination.
  • Determine if a system has many solutions (dependent), or no solution (inconsistent).
  • Write the many solutions of a dependent system in set-builder notation.


• §11.2: Augmented Matrices
  • Write coefficients of a system of equations in augmented matrix form.
  • Use row operations to manipulate augmented matrix.
  • Transform left side into an identity matrix (reduced row echelon form).
  • What remains in the right column will be the solution.
  • Recognize when the system is dependent or inconsistent.


• §11.4: Matrix Algebra
  • Matrix arithmetic: addition, scalar multiplication, matrix multiplication.
  • Given a square matrix, find its inverse (or determine that it has none)
  • by augmenting it with the identity matrix and using row operations.
  • Rewrite a system of equations as a single matrix equation, and solve
  • by left-multiplying both sides by the inverse of the coefficient matrix.


• §11.5: Partial Fraction Decomposition
  • Goal: to rewrite a rational expression as the sum of several simpler ones.
  • Factor the denominator, and set the whole thing equal to the sum of several
  • fractions each of which is a variable over one of those factors.
  • Multiply both sides by the original denominator, group together like terms,
  • and solve the resulting system of equations for A, B, C, etc.
  • Be careful with repeated factors and irreducible quadratic factors:
  • the number of variables should be the degree of the original denominator.


• §11.6: Nonlinear Systems of Equations
  • There's no universal algorithm as in linear systems, but similar methods often work.
  • Changing variables sometimes helps: replace every x² with u, for example.
  • Sketch a graph of both equations first to guess how many solutions there could be.


• §11.7: Systems of Inequalities
  • Represents the set of all points (shaded) that make ALL listed inequalities true.
  • Sketch each inequality as if it were an equation, then determine which region works.
  • Mainly useful for specifying a region of the plane.
  • Understand difference between "bounded" vs. "unbounded" regions.


• §12.1: Introduction to Sequences and Series
  • Index (subscript) notation {a_n}; sequence as a function on the natural numbers.
  • Recursive definition (a_n+1 as a function of a_n) vs. general formula (a_n as a function of n).
  • Difference between "sequence" (ordered list of numbers) and "series" (sum of these numbers).
  • Sigma notation for series; expanding ("splitting up") series, formulas for sum of c, k, k², k³.


• §12.2: Arithmetic Sequences and Series       §12.3: Geometric Sequences and Series
  • Be able to recognize whether a sequence is arithmetic, geometric, or neither.
  
  

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