Precalculus Midterm #2 Review

• §A.4: Synthetic Division (dividing by x-c)


• §A.5: Rational Expressions (simplifying and arithmetic)


• §A.7: Complex Numbers (meaning and arithmetic)

• §4.1: Polynomial Basics
  • degree, power functions, zeroes/roots, multiplicity, end behavior, turning points
  • effect of degree, roots, and multiplicity on graph of function


• §4.5: Real Roots of a Polynomial
  • use the Rational Roots Theorem to find possible roots; calculate to see what works
  • divide out a factor to determine what's left
  • fully factor to find all roots, then graph


• §4.6: Complex Roots of a Polynomial
  • factor out any real roots as usual
  • if what's left is a quadratic, use the quadratic formula
  • complex roots should come in conjugate pairs


• §4.2: Rational Function Basics
  • general appearance, asymptotes, gaps, end behavior


• §4.3: Graphing Rational Functions
  • factor everything, identify gaps, cancel out factors as needed
  • find vertical asymptotes (denominator) and x-intercepts (numerator)
  • find horizontal or oblique asymptote and determine if it is ever crossed
  • investigate end behavior and behavior near vertical asymptotes


• §4.4: Polynomial and Rational Inequalities
  • manipulate inequality to put everything on one side and 0 on the other
  • find all roots and vertical asymptotes: these are potential boundary points
  • test sample points between them to determine where function is positive or negative

• §5.1: Function Composition
  • composition as a binary operation on functions
  • notation; finding fog(x) and gof(x) given f(x) and g(x)


• §5.2: Inverse Functions
  • definition of inverse function; checking if functions are inverses of each other
  • one-to-one functions; horizontal line test; some inverses aren't functions
  • finding inverse of a function by interchanging input and output
  • graphing a function and its inverse


• §5.3: Exponential Functions
  • meaning and properties of exponents (including negatives, fractions, irrationals)
  • graphing exponential functions: domain, range, asymptote, standard transformations
  • solving simple exponential equations (rewrite with a common base)
  • the number e and its use as a universal base


• §5.4: Logarithmic Functions
  • definition of a logarithm as the inverse function of exponentiation
  • converting between exponential form and logarithmic form
  • common logarithms (base 10) and natural logarithms (base e)
  • graphing logarithmic functions: domain, range, asymptote, standard transformations
  • solving simple logarithmic equations (rewrite in exponential form)


• §5.5: Properties of Logarithms
  • product rule, quotient rule, power rule, change-of-base formula
  • condensing or expanding logarithmic expressions


• §5.6: Solving Exponential and Logarithmic Equations
  • if variable is in exponent, taking the logarithm of both sides is often useful
  • if variable is in logarithm, try rewriting in exponential form


• §5.8: Applications in Physics and Biology
  • exponential growth (decay): P(t) = P₀·e^rt, where r is a positive (negative) constant
  • population growth: predict when a population will reach a certain size
  • radioactive decay: understand meaning of "half-life" and how to calculate it
  • heating and cooling: T(t) = T_room + (T₀ - T_room)·e^kt, where k is a negative constant
  • in general: plug in known data and solve for unknown constants
  • pay close attention to asymptotes!

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