Guidelines in reviewing for Exam 1
Exam 1 will cover material we have gone over in the course from
Sections 2.1-2.2, 2.4-2.7, and 3.1-3.4.
Following is a list amassing together the daily goals
for days covering material tested on Exam 1. In studying for
this exam, you are of course encouraged to look over class notes
and examples, homework (don't just look over your past write-ups;
try to re-work a smattering of assigned problems “from
scratch”) and problems from the “Practice Exercises”
sections at the ends of chapters. You should also assess your
readiness for the exam based on the list of goals.
Goals for Class Date: Sept. 4, 2007
Students should be able to
- write an expression for the average
rate of change of a function given its formula.
- identify the graphical meaning of an average
and instantaneous rate of change of a function.
- numerically (using a calculator) investigate the limit
of average rates of change of a function at a point on
its graph.
- understand what is meant by the terms secant line,
average rate of change of a function, instantaneous
rate of change of a function, slope of a curve,
tangent line to a curve
Goals for Class Date: Sept. 5, 2007
Students should be able to
- recognize from graphs instances in which a limit at
at a point exists, along with instances when no limit
exists.
- learn the limit laws, both what they say and
situations in which they apply.
- make a reasonable explanation of what the statement
limx → a f(x) = L
really means, perhaps using
this applet as a visual aid.
Goals for Class Date: Sept. 6, 2007
Students should be able to
- recognize situations in which the evaluation
of a limit comes down to plugging in a number.
- provide reasonably complete statements of the limit laws.
- use algebra and limit laws to obtain limits of
functions, particularly when the function has the
form 0/0 at the point in question.
Goals for Class Date: Sept. 10, 2007
Students should be able to
- provide a reasonably complete statement of the sandwich theorem.
- apply the sandwich theorem to evaluate the limit of
of function.
- explain the difference between limits (two-sided) and
one-sided limits.
- determine the existence and values of one-sided limits.
Goals for Class Date: Sept. 11, 2007
Students should be able to
- explain what is meant by the term horizontal asymptote.
- explain what is meant by limx → -∞
f(x), and provide graphs of situations in which such a limit
does/does not exist.
- evaluate limx → ±∞
f(x) (or correctly state that these limits do not exist)
when f is given by ratios of sums/differences of powers
of x.
Goals for Class Date: Sept. 12, 2007
Students should be able to
- Determine from a given formula whether a rational function
has a vertical or horizontal asymptote.
- Use limit notation to describe the behavior of a
function near a vertical asymptote or “at infinity”.
- Give a definition for vertical/horizontal asymptotes that
involves a limit.
Note: The discussion of oblique asymptotes (Example 9,
p. 101) may be ignored.
Goals for Class Date: Sept. 13, 2007
Students should be able to
- relate the criteria necessary for the statement “the function
f is continuous at x = a” to hold.
- recognize types of functions which are continuous, and make
use of continuity when evaluating limits.
- identify the various types of discontinuity.
- state laws concerning the continuity of combinations of
continuous functions.
Goals for Class Date: Sept. 17, 2007
Students should be able to
- state the intermediate value theorem.
- recognize situations in which existence of a
solution is all that is sought.
- apply the intermediate value theorem correctly.
Goals for Class Date: Sept. 18, 2007
Students should be able to
- write down an appropriate formula for the derivative of
a function at a point, and explain what it means in
terms of slope.
- recognize the various notations for the derivative,
along with their meanings.
- use the definition of derivative to compute the
derivative function, for a given f.
- find the slope of a given curve at any specified
location, or explain why no such slope exists.
They should be able to write equations of tangent
lines when it is desired.
- recognize (from among a list of graphs) the graph
of the derivative function, when f is known.
Goals for Class Date: Sept. 19, 2007
Students should be able to
- state the following derivative rules: sum rule, difference rule,
constant multiple rule, product rule, quotient rule, power rule
- find derivatives for sums, differences, products and quotients
of monomials using these rules.
Goals for Class Date: Sept. 20, 2007
Students should be able to
- recognize blatant points of non-differentiability on a graph.
- determine the general shape of the graph of f '
from the graph of f.
- learn how to differentiate ex, and
various functions made from sums, differences, products
and quotients of monomials and exponentials.
Goals for Class Date: Sept. 24, 2007
Students should be able to
- interpret the derivative f ′ as slope on the
graph of f.
- interpret the derivative physically as a rate of change,
particularly when the independent variable of the function
is not time.
- interpret the 2nd derivative as a rate of change of the
rate of change; in particular, when the independent variable
is time and the dependent variable is position, the 2nd
derivative is acceleration.
Goals for Class Date: Sept. 25, 2007
Students should be able to
- explain that limt→0(sin t)/t
is evaluated using area considerations and the sandwich theorem.
- show how to evaluate limt→0
(cos t - 1)/t knowing the limit above.
- state the derivatives of the six trigonometric functions.
- find, using basic trigonometric identities, differentiation rules,
and knowledge of the derivatives of sin x and cos x,
the derivatives of the other four trigonometric functions.
- find derivatives for expressions involving sums, differences,
products and quotients of trigonometric functions, polynomials,
and exponential functions.
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