Math 120
Outline of material in Chapters 4 & 5 for Final Exam
§4.1 Maximum and Minimum Values of Functions
Absolute (a.k.a. global) max & min
Local max & min
Extreme Value Theorem
Critical number, point
Local max & min occurs at critical number
Closed Interval Method to find absolute max & min
§4.3 How Derivatives Affect the Shape of a Graph
What does f ’ say about f ?
Increasing/Decreasing Test: sign of f ’
First Derivative Test for local max & min: where f ’ sign changes sign
What does f ’’ say about f ?
Concave upward/downward: f ’ increasing/decreasing, respectively
Concavity Test: sign of f ’’
Inflection point: where f ’’ sign changes sign
Second Derivative Test for local max & min: sign of f ’’
Inconclusive when f ’’(c) = 0 or does not exist
§4.7 Optimization Problems
To solve an optimization problem, follow the 6-step procedure outlined on the first page of
the section.
Steps 1–5: Set up the mathematical max/minimization problem, making sure to specify
the domain of the objective function.
Step 6
: Find the absolute max/min of the objective function using methods of §4.1 or
4.3, extended:
•
For a finite domain interval, apply the Closed Interval Method of §4.1. A finite
non-closed domain interval arising from the “story problem” can usually be
extended to the corresponding closed interval in order to apply this method, as in
Example 1. (If easier, can also apply one of the following methods.)
•
For an infinite domain interval, apply the First Derivative Test for Absolute Extrema,
introduced in Example 2, or the Second Derivative Test for Absolute Extrema (not
named in text) introduced as an alternative solution in Example 1 (near the end),
i.e., f ’’ being of one sign throughout the domain. These are extensions of the
1st/2nd Derivative Tests for Local Extrema of §4.3, respectively.
Finally, having solved the mathematical max/minimization problem, make sure to
answer the original question exactly as stated in the “story problem”.
Study the examples carefully.
continued…
Math 120
Outline of material in Chapters 4 & 5 for Final Exam
(continued)
§4.4 Indeterminate forms and L’Hôpital’s Rule
Indeterminate forms
L’Hôpital’s Rule for indeterminate forms of “type 0/0” and “type ∞/∞”
Make sure that the conditions of L’Hôpital’s Rule apply
L’Hôpital’s Rule can be applied successively
§4.9 Antiderivatives
Definition
Table of anti-derivatives
Condition f(a) = b to determine the “constant of anti-differentiation” C
Application: Rectilinear motion: position from velocity or acceleration, with “initial
condition(s)” like the above to completely determine it. Important special case of
const. acceleration, for example, vertical motion under the influence of gravity (only).
§5.1 Areas and Distances
The Area Problem
Area under the graph of a continuous function as a limit of sums of areas of rectangles
Sample points: left endpoints, right endpoints, midpoints; can be arbitrary
For increasing function, left/right endpoints over/under-estimate area, respectively; and
vice-versa for decreasing function
The Distance Problem
Find distance traveled during a certain time from velocity: mathematically same as the above
§5.2 The Definite Integral
Definition, from Area and Distance Problems of previous section
Integrand, limits of integration
Riemann sum: that sum of areas of rectangles set up above
Net area
Evaluating integrals:
As net area
By actually calculating limit (involved; once is enough)
(Can use computer to calculate Riemann sums for large n)
(Better way to come…)
Properties 1-5 of the definite integral
§5.3 The Fundamental Theorem of Calculus
FTC2 for calculating the definite integral (difficult using limit definition) – very useful:
Suggested by observations of the previous two sections.
Roughly says the definite integral of the derivative of a function equals
the net change of the original function.