Review Questions - Sections 4.1 to 5.3, 7.5, 7.6
- What are the definitions of global maximum, global minimum, local minimum,
local maximum, global extremum, and local extremum? Give examples in graphs
and with simple function formulas.
- What is meant by a family of functions? Give examples. What are some methods
used to select a particular member of such a family if conditions of the member
are specified?
- What is the difference between the statement, "The maximum value of
the function f is 5." and the statement, "The function f has a maximum
at 5."?
- Criticize the following statement, "If f '(a)=0 then f has a maximum
or minimum at a". Do the same for the statement, "If f has a maximum
at the number a then f '(a)=0." How many different ways can these
statements mislead you?
- What written evidence would be sufficient to convince a reader that a function
f has a global maximum at a particular point?
- What are upper bounds of a function? What are lower bounds? What is meant
by "best possible bounds"?
- Give some example applications of the maximum and minimum theory.
- What are revenue functions, cost functions and profit functions? What is
the relationship between these? Give examples. What are marginal revenue,
marginal cost, and marginal profit functions? How does one find maximum profit?
- Give an example of an optimization modeling problem which can be solved
using the closed interval method. Give an example of one which requires another
technique.
- How can one use a velocity function to estimate total distance traveled?
- What is the general technique used in this course for approximating the
area between a curve and the x-axis over some interval? What are the specific
instances of this technique that we used and how did we implement these techniques
on a calculator?
- What is a Riemann sum? An integrand? The limits of a definite integral?
- What is sigma notation? Give an example of a use of sigma notation in area
estimation.
- For the left-hand rule, how does one calculate the width of each subinterval?
Be able to write out the terms of a left endpoint sum in specific examples.
How does one express such a result in sigma notation? Does the left-hand rule
produce an underestimate of the area? Be able to explain your answer.
- Consider the questions from the preceding item when applied to the right-hand
and midpoint rules.
- For the approximations just discussed, what factors affect the accuracy
of the computed approximation? Explain how one can see this in the geometry
of a drawing. Why is an understanding of this idea important in order to understand
the definition of the definite integral?
- What is the trapezoid rule and what is its purpose? Draw a sketch explaining
this method. When does it produce an overestimate and when does it produce
an underestimate? How do the values produced by the trapezoid rule compare
to those produced by the left-hand, right-hand, and midpoint rules?
- What is Simpson's rule and what is its purpose? Draw a sketch explaining
this method. Can one predict if it will produce an overestimate or an underestimate?
How do the values produced by Simpson's rule compare to those produced by
the left-hand, right-hand, midpoint and trapezoid rules?
- How does one use integration to measure total change of a function over
an interval?
- For what functions and intervals is the value of a definite integral of
the function always the same as an area measurement? Explain and be able to
show examples supporting your answer.
- A TI calculator has a built in routine named fnInt. What is its purpose,
how do you use it, and what theory forms the basis of its operation?
- How does one use integration to measure total change of a function over
an interval?
- What is meant by the average value of a function and how is it calculated?
How does one visualize the average value of a function?
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