Calculus 1 Exam 2
Suggestions for Review
The following list is meant to supplement your studies for exam 2.
The list is not exhaustive in terms of material to review. It is meant
as a guide and includes common areas of struggle in each section. You do
not need to know the proofs of any theorems. However, you are
responsible
for knowledge of the previous sections as outlined in the guide for exam
1.
Section 2.4
-
Know what the theorems of this section state and their significance.
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Understand Examples 1, 2, and 5.
-
Know the derivatives of the six basic trig functions. (You can derive the
other four from the formulas for the derivatives of sine and cosine.)
Section 2.5
-
Understand the Chain Rule and the notation that goes along with it, and
understand how to use it, including its repeated use.
-
Learn the "Outside-Inside" rule on page 156.
Section 2.6
-
Understand how to compute derivatives by implicit differentiation.
-
Know how to use implicit differentiation to compute slopes, tangents, and
normals to curves.
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Know how to use implicit differentiation to compute higher derivatives.
-
Understand the Power Rule for rational powers and how to use it.
Section 2.7
-
Understand and be able to use the strategy described on page 174.
-
Understand the examples.
-
Know the formulas for (i) the areas and perimeters of
triangles, rectangles, and circles and (ii) volumes and surface areas for
cubes, spheres, and right circular cylinders.
Section 3.1
-
Make sure you clearly understand the definitions of absolute (global) and
local extreme values (maximum and minimum) and critical points.
-
Know the Max-Min and First Derivative Theorems. Remember to check endpoints!
Note that not every
critical point is necessarily a local maximum or local minimum (see Example
5, page 194).
-
Know how to compute absolute extreme values of functions on closed
intervals.
Section 3.2
-
Understand Rolle's and the Mean Value Theorems and the three important
corollaries. You are responsible for the statement and meaning of
these theorems, but not their proofs.
-
Be able to apply these theorems and corollaries: be able to find a function
from knowing its derivative, including being able to find velocity and
position from knowing acceleration, and know how to determine where a function
is increasing or decreasing.
Section 3.3
-
Be able to apply the First Derivative Test for local extreme values.
-
Understand the examples. Use the sign pattern of f'(x) over intervals to
determine where f(x) is increasing and decreasing.
Section 3.4
-
Know the definitions of concave up, concave down, point of inflection,
and cusps. Note that not all points where f''(x) is either zero
or does not exist are necessarily inflection points (see Example 5, page
212).
-
Know and be able to apply the two Second Derivative Tests, one for concavity
and one for local extreme values.
-
Use the sign pattern of f'' to determine where f(x) is concave up or concave
down.
-
Be able to graph a function using its first and second derivatives as described
by the strategy outlined on page 214.
Section 3.5
-
Know the definitions of the limit as x approaches plus or minus infinity
and the properties in Theorem 6.
-
Understand how to take limits of rational functions as x approaches plus
or minus infinity; remember while you study this that it is best to think
in terms of 1/x.
-
Know the definitions of horizontal, vertical, and oblique asymptotes and
how to compute them.
-
Be able to graph a function using its asymptotes and dominant terms.
Section 3.6
-
You need not remember the specifics of any of the examples or theorems
in this section, but you should be able to do similar problems using the
strategy described on page 236.
Section 3.7
-
Know the definitions of linearization and differentials.
-
Be able to compute the linearization of a function.
-
Understand the relationship between derivatives and differentials and how
to estimate absolute, relative, and percentage change using differentials.
Section 3.8
-
Understand Newton's method: know the formula and the geometric interpretation,
and be able to apply it.
-
Skip the last three subsections on convergence, things that go wrong,
and chaos.