MATH 170 008 - Review Notes for Test #1 -- 9/9/04

This is the final form of this list. Last-minute additions in red.

The test will cover the material of assignments #1 - #6, roughly.

There will be a calculator portion.

The exam will be in two parts:

1. The colored-page part at the beginning for which you MUST have a working calculator,
2. The white-pages part: no calculators -- just you and your pencil and eraser. This part will be worth more points than the colored-paper portion.

We will start with part 1. When you have finished part 1, you put away your calculator, raise your hand, and your instructor will swap your part 1 for part 2.

Definitions and formulas to know cold. For these, one must be prepared to write out the definitions ("reGURG").

• Difference Quotient.
• Two forms of difference quotient.
• m_seq
• m_tan

Calculator tricks to know how to do:

• Evaluate a function f(x) at several different x values.
• Graph a function in several appropriate windows (??ZOOMfit??).
• Graph a difference quotient.
• Graph a difference quotient and use the graph to determine m_tan. If you are careful, things like TI-83 Nderiv, although they don't calculate the difference quotient, may help with the value of m_tan.
• Use your calculator to estimate the value of a limit.

Non-Calculator Algebra, Trigonometry, and Calculus Things to Know:

1. Graph a piecewise function. And get a function graph.
2. Set up a difference quotient.
3. Simplify a difference quotient.
4. Combine complex fractions.
5. The least-common-denominator shtik.
6. The page-39 transformations.
7. Computing domains for functions involving square roots (1.3: 40) or natural logarithms.
8. Two forms of difference quotient.
9. How a tangent-line equation relates to a difference quotient.
10. Limit notation and values: x -> a+, x -> a-, and x -> a, both via algebra (as when absolute value is around) and from pictures (2.2: 4, 6, 8 from #5).
11. The limits and one-sided limits of piecewise functions.
12. Absolute value gives rise to piecewise functions.
13. There are limits which are not obvious difference quotients (for instance 2.3: 11, 13, 15, 29 -- see assignment #6).
14. The Old Conjugate Trick: (2.3: 22, 23) and (2.6: 23)

Grading Notes on Assignment #1
Updated: Thu Aug 26 11:14:45 MDT 2004

• A MATH-147-review assignment.
• 90 points possible for this assignment:

  Problem	Points
  2	30 (5 each)
  22 f(2+h)	5
  22 f(x+h)	5
  22 DQ	10
  38	15
  40	15
  50	10

  The average score was 56.11.
• Don't be too terse in your answers. For instance a 1.1: 2a answer like this

  -2,       4

  depends on the reader to fill in the right thing. This makes the reader feel insecure and hostile. Better to have some lead-ins:

  f(-4) = -2,       g(3) = 4.

• Lead-ins: this shows up several places, but especially in 1.1:22. You need to tell the reader where your f(2 + h) starts, and ...

  In chapters 4 and 5, we'll have to keep track of several items in a computation. So you'll need to have lead-ins for the various items, both for the reader and for your own self.

• Interval Notation for Endpoints: use standard end-point notations for intervals: (, ), [, and ] all make a difference.
• Arrows instead of Equal Signs: avoid using arrows, such as ==>, where = is the thing. In other courses you will find that ==> has been universally reserved to mean something other than =.
• The answers to 1.1: 38 and 1.1:40 are function graphs. They should pass the famous Vertical-Line Test.
• In graphs, such as the one in 1.1: 38, important points must be labeled with their coordinates. A label must be right up beside the thing it's labeling.

  You can't rely on the reader to count tick marks correctly to figure out for you what the labels should be. So write down your own correct labels.

  In 1.1: 38, the points (-1,1) and (-1,4) rate these adjacent salient-feature labels (ASFL).

• Which one of

  A = B² - 3/B

  B = sin(3A) + A³

  expresses A as a function of B?

Grading Notes on Assignment #2
Updated: Sun Aug 29 07:50:15 MDT 2004

• 130 points possible for this assignment:

  Problem	Points
  1.1: 54	30
  1.3: 4abc	15
  1.3: 4d	10
  1.3: 6	5
  1.3: 14	10
  1.3: 18	10
  1.3: 24	10
  1.3: 40	40
  1.3: 10	5

  The average score was about 70.
• 1.1: 54: The graph consists of horizontal line segments: think of the fare for 1/4 mile and for 3/4 mile.
• 1.1: 54: For trips longer than 1 mile, the graph consists of 1/10-mile steps.
• 1.3: 4c: A vertical stretch away from the horizontal axis.
• 1.3: 4d: The problem uses
  • reflect through x-axis
  • squeeze vertically
  • shift up 3 units.
  The order in which these are applied is important:
  • y = f(x)
  • y = (1/2)f(x)
  • y = -(1/2)f(x)
  • y = -(1/2)f(x) + 3
• 1.3: 6: g(x) = 2f(x-2)
• 1.3: 24: Without the absolute-value bars the graph is a parabola. The effect of the absolute-value bars is to bend up the below-x-axis part of the parabola so it lies above the x-axis.
• 1.3: 40: To come up with the domain of f, you have to guarantee that 2x + 3 is non-negative by solving an inequality.

Grading Notes on Assignment #3
Updated: Tue Aug 31 12:05:12 MDT 2004

• 70 points possible for this assignment:

  Problem	Points
  2.1: 4a as amended	15
  2.1: 4b	10
  2.1: 4c	10
  2.1: 4d	10
  2.1: 6a as amended	15
  2.1: 6b	10

  The average score was about 26.
• In the amended 4a and 6a, many wrote down the relevant difference quotient correctly, but did not offer a graph of it.
• 4a difference quotient: Y₁ = (ln(X) - ln(2))/(X - 2) (we're looking for the y-coordinate of the point where X = 2) or Y₁ = (ln(2 + X) - ln(2))/X (we're looking for the y-coordinate of the point where X = 0).
• 6a average velocity over [1, X] is a difference quotient after all: Y₁ = (58X - 0.83X²)/(X - 1). (we're looking for the y-coordinate of the point where X = 1).
• TI calculators have a ZoomFit command which is frequently helpful. In 4a and 6a the difference-quotient graph may look nearly horizontal because the range of y coordinates on the graph is set too tall. ZoomFit shrinks the plot window vertically, thereby emphasizing the variation in the function. The 4a and 6a graphs, after ZoomFit, both run from the upper-left corner of the window to the lower-right corner.
• Sometimes one's first y-range choice is off the beam: the plot comes up blank. ZoomFit will find the graph's y range for one.

Grading Notes on Assignment #4
Updated: Thu Sep 2 09:50:59 MDT 2004

• 90 points possible for this assignment:

  Problem	Points
  1	15
  2	15
  3	15
  4	15
  5	15
  6	15

  Eleven papers were graded. The average score is about 66. This includes 5 papers with scores 80 or better.
• Stick with the difference-quotient form you start with: if you start with the h -> 0 form, don't do t -> A at the end.
• Write stuff with lead-ins (ask the instructor if you aren't sure what this means).
• Use lead-ins to avoid having your reader get the idea that you think f(x) is the same as its difference quotient.
• In problem 4, one cannot simplify the difference quotient by squaring it. Maybe the "old conjugate trick"?
• In problem 5, synthetic division works like a champ to find f(4/3), the y-coordinate of the point of tangency.
• In problem 5, even though m_tan = 0, we still have a tangent line. It is a line with slope zero.
• In problem 6, you can use your problem-1 result -- you do not have to do problem 1 over again as part of your problem-6 solution.

Grading Notes on Assignment #5
Updated: Sun Sep 5 08:47:04 MDT 2004

• 103 points possible for this assignment:

  Problem	Points
  2.2: 4	10
  2.2: 6	24
  2.2: 8	14
  2.2: 12 graph	15
  2.2: 12 limit existence	10
  2.2: 20	10
  2.2: 22	10
  2.2: 26	10

  Again, eleven papers were graded. The average score is about 86. This includes 6 papers with scores of 90 or better.
• 2.2: 6(i): the authors intend that you see the graph as infinitely oscillatory to the right of x = 4. Thus the limit from the right does not exist.
• 2.2: 12: there were 15 points for the graph. On such problems, it's important to label important features of the graph with labels right up on the feature being labeled. In problem 12, the crucial salient features are the endpoints of the graph segments -- they need to be labled with their coordinates. Don't even bother with tickmarks on the axes -- the grader cannot count tickmarks reliably. If it says "ASFL" on your paper, you need to label stuff.
• 2.2: 12: there were 10 points for answering the question as to where the limit of the function exits.
• 2.2: 20 and 22: neither of the functions is defined at the point in question. However, the limits do exist. Try values of x near the point in question to get an inkling of the value of the limit. In chapter 3, we'll do an exact non-calculator job on problem-20-type limits. Problem-22-type limits will be a PROBLEM.
• 2.2: 26: this problem draws on your rational-function graphing studies from MATH 147 (in chapter 3 of the 147 text, or see google stuff). The question is how the graph looks around its vertical asymptote at x = 0.

Grading Notes on Assignment #6
Last Update: Wed Sep 8 12:57:43 MDT 2004

• 138 points possible for this assignment:

  Problem	Points
  2.3: 2	18
  2.3: 12	10
  2.3: 14	10
  2.3: 16	10
  2.3: 18	10
  2.3: 20	10
  2.3: 22	10
  2.3: 24	10
  2.3: 26	10
  2.3: 28	10
  2.3: 30	10
  2.3: 43	10
  2.3: 44	10

  The average score was about 98.
• These are the "DNE" problems: 2b, 2d, 16, 43. All the rest have finite limits.
• Problem 43 is one of those "DNE" problems about which we can say "-infinity", which means that the limit fails to exist because the function becomes huger and huger negatively in the limit.
• Problems 12-18: factor the numerator and look to "cancel out" the singularity.
• Problems 20-24 and 28 are m_tan problems. Combine and factor, looking the while to "cancel out" the singularity.
• Problem 26 teaches us that

  infinity - infinity = 1

  just as 44 teaches us that

  infinity - infinity = 0

  and 2.6: 23 teaches us that

  infinity - infinity = 1/6.

  The trick is to combine, then try to do some algebra that will illuminate it for you.