MATH 147 006 - Review Notes for Test #3 -- 4/1/05

Under construction: this is NOT the final form of this list.

Last Update: Wed Mar 30 15:56:18 MST 2005

The test will cover the material of assignments #25 - #34, and #35, roughly.

This means it covers, roughly, sections 3.4 - 3.6, 4.1 - 4.5, and 5.1 - 5.3.

Calculator Portion.

Be prepared to use your calculator in connection with logarithms and exponential models of growth and decay.

Old Tests and Review Helps

• For the chapter-three practice test, pages 328, problems 5 and 9a-d are relevant as of last-update date above.
• For the chapter-four practice test, page 393, problems 1 - 8 are relevant as of last-update date above.
• For the chapter-five practice test, page 454, problems 1 - 10 are relevant as of last-update date above.
• Summer 2001 147 #1: problems 6, 9
• Summer 2001 147 #2: problems 1, 2, 11
• Fall 2002 143 #2: problems 8, 9
• Summer 2001 147 #3: problems 2, 4, 8, 9, 10d
• Summer 2001 147 #4: problems 1, 2, 10, 11, 12, 13

Chapter 1, 2, and 3: Review Stuff to Know:

• The sign-chart method of solving non-linear inequalities. Sections 1.7 and 3.1, and 3.6.
  • Inequalities involving factored polynomials or fractional expressions (sign-change charts).
  • Factors which change sign.
  • Factors which do not change sign.
  A few of you have told me you don't "get" this. Now's the time to figure it out.
• Familiarity with the page-164 "Friendly Faces" (click here) will be assumed.
• Be able to recognize which flipped, shifted, or stretched friendly-face graph (page 164) goes with which equation.
• Polynomial division and interpretation, most recently in connection with graphing rational functions (section 3.6).
• Descartes The algebra version of "point (A,B) lies on the graph of the equation E(x,y)".
• Complex Arithmetic as per assignment #25.
• 3.4 - Complex Numbers:
  • standard a + bi form
  • real part of a + bi
  • imaginary part of a + bi
  • the complex conjugate of a + bi
  • the complex conjugate of a real number
  • adding complex numbers
  • multiplying complex numbers
  • using the complex-conjugate trick to write the quotient of two complex numbers in standard a + bi form
• 3.5 - Complex Zeros of Polynomials:

  We mostly considered real-coefficient polynomials, for which the complex zeros appear in complex-conjugate pairs. This section was a continuation of section 3.3.

• 3.6 - Graphing Rational Functions: Rough graphs of factored rational functions. A sample grading rubric for a possible rational function:
  1. Shape
  2. HA y = 2
  3. HA approach
  4. VA x = 1
  5. VA approach
  6. VA x = 2
  7. VA approach
  8. x-intercept (0,0)
  9. x-intercept (-4,0)
  10. HA-intercept (2/7,2)
  11. Labels: equations for asymptotes and coordinates for important points.
  Five points for each of these.

  On other problems I'll be looking for

  • sign-chart construction
  • sign-chart interpretation
  • long-division interpretation
  • horizontal-asymptote thinking
  • oblique-asymptote thinking
  • labeling asymptotes with equations rather than numbers

Chapter 4: Review Stuff to Know:

• 4.1 - Exponential Functions:
  • Appearance of the graph of y = a^x for the two cases a > 1 and 0 < a < 1
  • Applying the section-2.5 moves to exponential graphs (see (4.1: 13-18) and (4.1: 21, 22, 23, 32, 33, 34))
  • The compound-interest formula.
  • Decoding the APR depending on how many time in one year the interest is compounded.
  • Euler's number e.
  • The continuously-compounded-interest formula.
• 4.2 - Logarithmic Functions:
  • The function log_A is the inverse of which function?
  • The problems (4.2: 1-6) and (4.2: 7-12) go right to the heart of what a logarithm is.
  • Appearance of the graph of y = log_A for the two cases A > 1 and 0 < A < 1
  • Applying the section-2.5 moves to logarithmic graphs (see (4.2: 39-44) and (4.1: 47-54)
• 4.3 - Logarithmic Identities:
  • Know BLUE353 cold.
  • Know BLUE359 cold. These are the famous Three Laws of Logarithmics
  • The logarithmic-explode game: (4.3: 1-26)
  • The logarithmic-collect game: (4.3: 39-48), and how to handle it if we add -3 to any of these problems (how does it change their BOB answers?)
  • 4.3: 62 constitutes an imaginative collection of dud log laws.
• 4.4 - Logarithmic Algebraic Manipulations:
  • Using logarithms to solve exponential equations.
  • The BLUE362 change-of-base formula is superfluous. Know what a logarithm is and how to solve exponential equations, and you won't need to burden your memory with BLUE362. Find a calculator approximation to log_2/3(19).
  • Solving logarithmic equations by algebra.
  • Solving logarithmic equations approximately by looking at graphs.
  • Extirpating extraneous roots of logarithmic equations.
• 4.5 Exponential Models
  • "Mommy, what's half-life?"
  • If N(t) denotes the amount of exponentially changing stuff present at time t and N(a) = A with N(b) = B, then

    N(t) = A (B/A)^(t-a)/(b-a)

Chapter-5 Stuff

• The degree and radian names for the clock-face angles. Click here for a sample exam question about this.
• SOH-CAH-TOA: you need to know the right-triangle trigonometric functions.
• You need to be conversant with the unit-circle idea of the trigonometric functions. For θ in standard position, the relevant items in terms of θ:
  • x =
  • y =
  • m =
• The BLUE-423 even-odd identities.
• The BLUE-424 identities.
  • express all the trig functions in terms of the sine and cosine.
  • Mom and the other Pythagorean Identities (5.2: odds 61-67, for instance)
• Graph of one cycle of the sine function.
• The length of one cycle of the sine function.
• The period of the sine function.
• The amplitude of the sine function.
• Graph of one cycle of the cosine function.
• The length of one cycle of the cosine function.
• The period of the cosine function.
• The amplitude of the cosine function.
• Graph, amplitude, period, and "phase shift" for

  f(x) = Asin(Bx + C) + D

  or

  f(x) = Acos(Bx + C) + D.