Mathematics 133 -- Intensive Calculus for Science 1

Exam 1 -- Things to Know

September 21, 2005

General Information

The first full-period exam of the semester will be given in class on Wednesday, September 28. It will cover the material from sections 1 - 6 of Chapter 1. There will be seven or eight questions (maybe grouped together as parts of larger questions) similar to problems from the quizzes so far. Calculators will not be allowed on this exam.

We will review for the exam in class on Tuesday, September 27. If there is interest I can also see if Rosie Arcuri is available for an evening ``last minute question'' session on Tuesday.

Material To Know

You should know the following material.

• The material on rules for exponents, fractions, etc. that we reviewed the first week.
• Functions, linear functions and their properties (Section 1.1)
  1. The slope-intercept (y = mx + b) and point-slope (y - y₀ = m(x - x₀)) forms for linear functions
  2. The meaning of the slope and how to determine it from either a formula for the function, or from a table of values
• Exponential functions and their properties (Section 1.2)
  The general formula for exponential functions f(x) = ca^x (or using different letters, P(t) = P₀ a^t, as on page 11 of the text).
  1. Exponential growth versus exponential decay (which values of a give which case)
  2. Be able to determine an equation for an exponential function, given a graph, a table of values, or a % growth rate per unit time.
  3. How to tell exponential functions apart from linear functions
• New functions from old, via horizontal and vertical shifting, stretching/shrinking (Section 1.3)
• Inverse functions. (Section 1.3) Know:
  1. How to tell whether or not a function is invertible from its graph,
  2. How to derive a formula for the inverse function f^{-1} from a formula for f,
  3. How to sketch the graph of the inverse function from the graph of f.
• The natural logarithm function f(x) = ln(x) and its properties (Section 1.4)
  1. g(x) = ln(x) is the inverse function of the exponential function f(x) = e^x.
  2. Formulas for logs of products, quotients, powers and how to apply them
  3. The shape of the graph y = ln(x)
  4. Using logarithms to solve equations involving exponentials
• Trigonometric functions. Know:
  1. Radian measure for angles and how to determine the values of sin(t), cos(t), tan(t) for an angle t in radians
  2. How to sketch graphs for sinusoidal oscillations y = A sin(Bx) + C or y = A cos(Bx) + C and the meanings of A,B,C
  3. How to find a formula for a sinusoidal oscillation, given the graph.
• Polynomial functions: f(x) = a_nxⁿ + a_n-1x^n-1 + ... + a₁x + a₀. Know:
  1. How to find x-axis intercepts by factoring
  2. How the sign of a_n determines the overall shape of the graph
  3. How to determine the smallest possible n and/or a possible equation from a graph (like Section 1.6/14)
• Rational functions: f(x) = p(x)/q(x), where p(x), q(x) are polynomials. Know:
  1. How to determine x-axis intercepts, vertical asymtotes, horizontal asymptotes from the equation (like Section 1.6/6)

Good Review Problems:

From the Review problems at the end of Chapter 1: 1-21, 38-41.

Also see the Sample Exam 1.