Study Guide for Exam 3

This will be a closed-book exam. No calculators are permitted on this exam.

To do well on the exam, you should be able to do the following:

Section 4.1: Exponential Functions

  1. Evaluate an exponential function \(f(x) = a^{x}\) at integer and rational values of \(x\).
  2. Specify the domain and range of an exponential function.
  3. Determine whether a given function is an exponential function.
  4. Sketch the graph of an exponential function \(f(x) = a^{x}\), including its behavior as \(x\) approaches both positive and negative infinity.
  5. Given the graph of a function \(f(x)\), sketch the graphs of \(g(x) = f(x) + c\), \(g(x) = f(x - c)\), \(g(x) = -f(x)\), and \(g(x) = f(-x)\).

Section 4.2: The Natural Exponential Function

  1. Give the decimal expansion of the number \(e\) to 3 decimal places.
  2. Sketch the graph of an exponential function \(f(x) = e^{x}\), including its behavior as \(x\) approaches both positive and negative infinity.

Section 4.3: Logarithmic Functions

  1. Relate the logarithmic function with base \(a\) to the exponential function with base \(a\).
  2. Explain why the logarithm \(\log_{a} x\) of a number \(x\) is an exponent.
  3. Specify the domain and range of a logarithmic function.
  4. Evaluate logarithms \(\log_{a}x\) for a given value \(x\).
  5. Sketch the graph of a logarithmic function \(f(x) = \log_{a} x\).
  6. State the base of the common logarithm.
  7. State the base of the natural logarithm.

Section 4.4: Laws of Logarithms

  1. State the 3 “Laws of Logarithms” in the blue box on page 354 of Stewart, Redlin, and Watson.
  2. Use the 3 “Laws of Logarithms” to expand and combine logarithmic expressions.
  3. State common errors made when attempting to simplify logarithmic expressions, and identify them in a given attempted simplification.
  4. State the change of base formula for converting from a logarithm base-\(b\) to a logarithm base-\(a\).
  5. Use the change of base formula to convert from a logarithm base-\(a\) to a logarithm base-\(10\) or base-\(e\).

Section 4.6: Modeling with Exponential Functions

  1. Given the doubling / tripling / quadrupling time \(a\) of a population and the initial size of a population \(N_{0}\), develop a model for the population size as a function of time.
  2. Given the relative growth rate \(r\) of a population and the initial size of a population \(N_{0}\), develop a model for the population size as a function of time.
  3. Given an exponential model for a population, determine the amount of time it takes for the population to reach a certain population size.
  4. Given the half-life \(h\) of a radioactive material and the amount of radioactive material present, develop a model for the amount of radioactive material as a function of time.
  5. Given the proportion of radioactive material remaining in a sample relative to some starting time and the half-life \(h\) of the radioactive material, determine the amount of time that has passed since the starting time.