Here are some thoughts I was having while considering what
to put on the second midterm.
The core of your studying should be the assigned homework problems: make sure
understand those well before moving on to other things ( like the old midterms on
– You should be able to determine the maximum and minimum value of a quadratic
function on a specified interval.
– You should be able find the maximum or minimum possible value of a quantity by
expressing it as a quadratic function of some other quantity (e.g., area of a
enclosure as a function of the width of the enclosure, etc.). Problem 7.10-7.14
involve this idea.
– You should understand what f(g(x)) means, and how to express a rule for f (g(x))
given rules for f(x) and g(x).
– I especially like problems 8.3, 8.4, 8.5.
• Chapter 9 - Three Construction Tools
– You should understand horizontal and vertical shifting, and horizontal
and vertical scaling (aka dilating)
– You should understand how to derive the graph of g(x) = af(bx + c) + d from
graph of f(x) (see, e.g., problem 9.2)
– I especially like problems 9.2, 9.3, 9.4, and 9.7
• Chapter 11 - Inverse Functions
– Another very short chapter.
– You should understand what an inverse function is, what conditions a
must satisfy in order to have an inverse (do all functions have inverses? can
if a function has an inverse by looking at its graph?), and how to find the
a given function
You should understand what a one-to-one function
is, and what is special about the
graph of a one-to-one function
– Problems 11.2, 11.6, and 11.8 are particularly good.
– An important chapter, it introduced a new class of functions for modeling.
– You should be able to find the asymptotes (horizontal and vertical) of
a linear-to-linear rational function, and be able to sketch the graph of a rational function
those in problem 12.1(a) or (b).
– You should be able to model with linear-to- linear rational functions.
down to finding a rational function of the form
1. passes through three given points
2. has a given asymptote and passes through two given points
3. has two given asymptotes and passes throuh one given point
You will need to translate the language of the modeling problem. Take a look at
midterm 2 exams from the archive for examples to work on.
Pay particularly close attention to the words “linear-to-linear”.
Note that a linear-to- linear function is not a linear function.
– I especially like problems 12.6, 12.7, 12.8, 12.10, and 12.11.
– You should be able to solve problems using the idea of trigonometric functions
ratios of sides of right triangles (e.g., problems 15.4, 15.7, 15.8) and some
– You should understand the definitions of sin x and cos x using the unit
should be able to determine certain simple properties of the functions sin x and
from this definition (e.g., the range, the domain, the graph, the values at
value of x, like x = 5π /2)
– You should be able to determine the location of an object moving circularly
about its speed and starting location (e.g., problems 15.2, 15.5, 15.9, 15.15,
and the three non-textbook Chapter 15 problems)
• Chapter 16 - Trigonometric Functions
– This is a short chapter which adds some final touches to our knowledge of the
sin x and cos x and related functions.
– I like problems 16.3 and 16.4.
• Chapters 17 - Sinusoidal Functions
– You should understand the notion of a sinusoidal function as a
version of the function sin x.
– You should understand the effect of the four parameters A,B,C and D on the
– You should be able to model with sinusoidal functions.
In particular, you should
be able to determine the parameters A,B,C, and D from a verbal description of a
quantity that varies sinusoidally with time (see problems 17.2, 17.3, 17.4,17.6)
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