Test 1 Review
(Note: You can detach this page to be a separate review document. Section 2.5 notes are on the other page.)
Useful Algebra Skills (also go over the algebra handout on website):
• Distributing: don’t forget minus signs!
• Expanding a square: (a + b)2 = a2 + 2ab + b2 (the 2ab is the “cross term”)
– Completing the square: Given x2 + ax, add (a/2)2 to each side. x2 + ax + (a/2)2 = (x + a/2)2 .
We do this to eliminate ax terms we don’t want in circle equations.
• Factoring: useful for canceling and for solving equations with 0 on right side
– It ONLY helps with 0 on the right side.
For example, x(x − 1) = 1 does NOT mean x = 1 or x − 1 = 1.
• Using ± appropriately: with even roots, you get TWO answers!
√
• Domain concerns: Denominators can’t be zero, bodies of
must be ≥ 0
– Number lines are a really important tool for finding the intervals you want.
– To do inequalities with x2 , it may help to write: x2 ≤ a2 ⇒ |x| ≤ a ⇒ −a ≤ x ≤ a.
• Nested fractions: My recommendation is to write plenty of parentheses and use “invert and multiply” for
the main denominator. For instance,
1
1
−
x+h
x
1
1
1
1
x 1 (x + h)
1
=
−
·
=
· − ·
·
(h)
x+h x
(h)
(x + h) x x (x + h)
h
where the last step makes a common denominator for that difference.
Studying for Tests:
• Be familiar with your calculator.
• Practice solving questions at the speed you’ll need. (Expect 5 min for short questions, 10 min for long
ones.) The “Practice Another Version” button in HW can be great for this!
• Don’t just memorize formulas: memorize formulas AND pictures AND reasons. In other words, when you
learn a formula, learn also how to recognize it and why people use it.
– Similarly, don’t memorize problem types: recognize the skills you use. In tests, the wording can
change, but the skill set is what remains the same!
Test-Taking Strategy:
• Leave answers unsimplified: WebAssign accepts it, and you don’t want to mess up the simplification.
• Submit answers one at a time. Don’t accidentally submit a correct answer twice!
– Scan over questions again at the end to make sure you’re answering exactly how the question asks.
• Skip over problems and return to them if necessary. It’s important to manage time and not get too
stressed on early problems.
Selected WebQuiz 1 and 2 Problems
Some numbers are changed in these problems. There are multiple versions of problems.
For a brief overview of every WQ problem, see the handout provided in my site’s schedule.
WebQuiz 1
#1: “Find all points on the y-axis that are a distance 7 from P (4, 3).”
This question is straight from homework. Note that another way to phrase this is “Find all points on the
y-axis which are on the circle with center (4, 3) and radius 7”... why?
#6: “A triangular region in the first quadrant is formed by the coordinate axes and a line with negative
slope. Given that the x-intercept is 4 and the triangle area is 18 square units, find an equation of the line.”
Draw a picture in the first quadrant: you see that the intercepts give you the two legs of the triangle! Say
the x-intercept is a and the y-intercept is b, so Area = ab/2. Since you know a and Area, you can get b and
hence a second point on the line.
#10: “In exercise physiology, aerobic power P is defined in terms of maximum oxygen intake. For altitudes
up to 1800 meters, aerobic power is optimal, that is, 100%. Beyond 1800 meters, P decreases linearly from the
maximum of 100% to a value near 40% at 5000 meters.
(a) Use the values of 100 and 40 to express aerobic power P in terms of altitude h (in meters) for domain
1800 ≤ h ≤ 5000.
(b) Estimate aerobic power in Mexico City (Mexico) (altitude: 2400 meters).”
The main challenge here is identifying what the roles of the variables are! Look at the phrase “express P in
terms of h”: that tells you that P is the output and h is the input. You’re also told that P takes the values 100
and 40, so in other words, they do not want you writing the % symbol with your values of P .
Now, read the problem again: you’re given two points! When h is 1800, power P is 100, so the point is
(1800, 100). Get the second point as well. How do you get a line equation from two points?
WebQuiz 2
#5: “Find the domain of f .”
√
f (x) =
x2
2x − 5
− 7x + 6
This has both a square root and a denominator! The root requires 2x − 5 ≥ 0 (solves to x ≥ 5/2). The
denominator can’t be zero; it factors as (x − 1)(x − 6). Draw a number line with two holes and also shade
x ≥ 5/2; how many intervals do you get?
#7: “It has been estimated that 1000 curies of a radioactive substance introduced at a point on the surface
of the open sea would spread over an area of 30, 000 km2 in 30 days. Assuming that the area covered by the
radioactive substance is a linear function of time t and is always circular in shape, express the radius r (in km)
of the contamination as a function of t (in days).”
The key thing to note is that area is described in two ways, so you will write two equations for it. The
spread is circular, so A = πr2 , but you also are told A is linear in t, so A = mt + b as well. You need to figure
out m and b.
The phrase “at a point” is important: a point has 0 area, so the intercept is b = 0. Find one other point
(t, A) and use it to get the slope m. Now, A = πr2 = mt + 0, and you want to solve for r!
#8: “A bowl in the shape of a hemisphere with
radius R inches is filled with water to a depth h inches
as shown. The area of the surface of the water is
A = 23πin2 . Express the radius R of the bowl as a
function of the depth of water h.”
In this problem, there is a sphere and a circle! Let’s say the circle of water on the surface has radius r. Area
of that circle is given: A = 23π = πr2 . Use this to get r, the horizontal leg of the right triangle. Can you figure
out the vertical leg using h? If you can do that, then you know two sides of the right triangle, and Pythagoras
gives you the third side R.
NOTE: The last problems on WebQuiz 2 (the word problems) should be similar to the word problems on
the test, but they WON’T be the same. You may not be asked for the same things you’re asked for in the
WebQuizzes! That’s why it’s important that you make sure to identify which variable is the unknown, so you
know what to solve for. Don’t memorize every question in 2.4C; instead, ask yourself how you recognize the
cues for which formulas to use!