Guidelines in reviewing for Exam 4
Exam 4 is cumulative, but will focus upon class material in
Sections 4.8, 5.1-5.6, 6.1-6.3 and 6.6.
Following is a list amassing together the daily goals for days
covering material in the sections listed above. 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: Nov. 5, 2007
Students should be able to
- check whether one function is an antiderivative of
another.
- describe the meaning of the indefinite integral
of f.
- find antiderivatives of various given functions.
- identify various properties (sum and differences,
constant multiples) of the indefinite integral.
Goals for Class Date: Nov. 6, 2007
Students should be able to
- set up an approximating sum of areas of rectangles for the
area under a curve using whatever rule desired: an
upper sum, a lower sum, a left or right-hand sum, or a midpoint
rule. After doing so, they should be able to calculate the
resulting value.
- set up rectangle sums in the same way as above when
the function is a rate of change of something, and use
them to approximate the total change of that thing.
- find the exact area under a non-negative function when
that function is piecewise linear.
Goals for Class Date: Nov. 7, 2007
Students should be able to
- turn a sum expressed involving sigma notation into one
where the individual summands all appear.
- take various sums and rewrite them in sigma notation.
Goals for Class Date: Nov. 9, 2007
Students should be able to
- state what is required for a definite integral of f
from a to b to exist.
- state and use the properties of the definite integral.
- explain what the definite integral tells us in terms of
areas of regions lying between the integrand (function)
and the x-axis.
- evaluate the definite integral when the integrand produces
regions involving simple geometric shapes (circles, rectangles
and triangles).
Goals for Class Date: Nov. 14, 2007
Students should be able to
- state clearly and concisely the Mean Value Theorem for
Definite Integrals, and both parts of the Fundamental
Theorem of Calculus. In particular, they should be able
to articulate sufficient conditions (the hypotheses) for
the results of these theorems to hold.
- articulate, in the case f is a nonnegative
function, why it is appropriate to call the function
F(x) defined in Theorem 4, p. 346, an area-so-far
function. Moreover, they should be able to employ
Part I of the FTC in order to differentiate such a function.
They should also be able to combine FTC, Part I with other
properties of the definite integral, and with the chain
rule, to differentiate other similarly-built functions
(like those found in Exercises 43, 45 on p. 352).
- employ FTC, Part II in order to evaluate many definite
integrals exactly (i.e., without any approximation or
roundoff error).
- articulate senses in which integration and differentiation
may be viewed as reverse processes of each other.
Goals for Class Date: Nov. 15, 2007
Students should be able to
- reverse various known formulas for derivatives (of power
functions, the natural log function, exponential functions,
trigonometric and arc-trigonometric functions) in order to
identify certain integrands which have immediate
antiderivatives.
- propose possible substitutions u = g(x) for integrands
that do not have immediate antiderivatives, and test out
whether such substitutions are useful in the quest for
finding indefinite integrals.
- see the substitution rule as a reverse of the chain rule, and
use it to find the antiderivatives of various functions.
Goals for Class Date: Nov. 16, 2007
Students should be able to
- employ the substitution rule to correctly evaluate definite
integrals. In particular, they should be able to find
appropriate limits of integration in u = g(x) so
that the original integral (say, in x) has the
same value as the new one in u.
- write down and evaluate definite integrals whose values give
the areas of regions in the plane between two curves. This
will sometimes involve finding appropriate formulas for
curves, and intersections between curves.
- identify situations in which it is easier to write a
y-integral for the given region than it is to write
an equivalent x-integral, and to do so correctly.
- give an appropriate interpretation of a definite integral
of a difference [f(x) - g(x)] in terms of areas.
Goals for Class Date: Nov. 19, 2007
Students should be able to
- write definite integrals whose values are the volumes of
solid regions of space, when cross-sections of these
solids are simple geometric shapes.
- apply this previous skill in the particular case when
the solid region is obtained from a region of the plane
through evolving it about an horizontal or vertical line.
In such a setting, cross-sections may be viewed as
discs or washers.
Goals for Class Date: Nov. 20, 2007
Students should be able to
- identify, for a given solid of revolution, which types
of slices (parallel to the x-axis or parallel to
the y-axis) through the un-revolved region yield,
upon revolution, discs/washers, and which yield cylindrical
shells.
- employ the disc/washer method in setting up a definite
integral (or sum of definite integrals) whose value
is the volume of the solid of revolution.
- employ the method of cylindrical shells in setting up a
definite integral (or sum of definite integrals) whose value
is the volume of the solid of revolution.
- discern situations in which, given the choice, the method
of cylindrical shells is more (or less) desirable than the
disc/washer method for finding the volume.
- evaluate the integrals set up via the disc/washer method
or the method of cylindrical shells.
Goals for Class Date: Nov. 27, 2007
Students should be able to
- set up integrals whose values equal the arc length
of various types of planar curves.
- evaluate arc length integrals.
Goals for Class Date: Nov. 28, 2007
Students should be able to
- describe work as force across a distance,
and identify appropriate units for measuring it.
- identify situations in which the standard integral formula
for work found in the box on p. 431 is inadequate. In
such settings, they should be able to write down a Riemann
sum approximating the desired work done, and then pass
from such a sum to an appropriate definite integral.
- use the alternate information sometimes provided in problems
involving springs in order to find the appropriate value of
the spring constant.
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