Final Review : Chapters 4 and 5
N
10 Dec 2012
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About the review
This review is mainly concentrated on Chapters 4 and 5. Chapter 3 is also
very important, but a detailed midterm review II on it was given. So please
go over that. The sample problems given here are simple and will be worked
out in class to give you an idea as to how to attack these problems. The
practice problems from your textbook are probably closer to the level of
difficulty of the problems in your final (some of the practice problems are
harder). The solutions of sample problems will be up online for reference.
I’ll set up a poll over the weekend to find out which practice problems cause
most difficulties, and will try to put up solutions to those questions online
too.
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Important topics from Chapters 1,2 and 3
Finding domains of functions, finding limits of rational functions (by factoring and cancelling, by multiplying and dividing by conjugates) and using
limx→0 sinx x = 1, chain rule, related rates and implicit differentiation (finding
tangents for instance), increasing and decreasing tests, concavity tests, first
and second derivative tests, finding absolute and local extrema, optimization
problems, IVT, Rolle’s, MVT, EVT, Fermat’s fact. Finding antiderivatives
is of course essential !
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Practice problems for Chapters 1,2 and 3
1. Chapter 1 : (Pg 95-96) 5,6,7,8,23,26,27,30,35,37,49
2. Chapter 2 : (Pg 191-193) 13,16,18,26,27,30,36,41,45,48,50,73,74,78,79
3. Chapter 3 : (Pg 275-278)
(a) Concept check : 2,3,4,5,6
(b) True-False quiz : 1,5,8,10
(c) Exercises : 3,5,11,17,29,34,38,39,40,53,55,60.
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4.1
Chapter 4 : Integration
Definition and approximation by rectangles
Rb
We will only talk about continuous functions in this section. a f (x)dx refers
to the signed area under f for the interval [a, b]. You should know how to
divide the interval into n pieces and find the left and right approximations
by rectangles.
Sample problem : Look at the picture and find L5 and R5 (left and right
approximations by dividing the interval into 5 pieces).
4.2
Using geometry
Sometimes, you can find the integral by using geometry (i.e actually graphing
the curve and finding the area of the figure by known formulae). In this case,
you have to force the areas lying below the x-axis to be negative.
R7
Sample problem : Find −2 2xdx
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4.3
Basic properties of Integration
1. Integration behaves well with respect to addition, subtraction, multiplication by a constant. [Pg 303 in your book has these summarized in
a box].
RN
RP
RP
2. One very useful property is that M f (x)dx + N f (x)dx = M f (x)dx.
So if you already know two of the integrals, to find the third one you
just have to add !
3. This also tellsR you that if you
R a flip the end points, you get an extra
b
negative sign a f (x)dx = − b dx.
4. Sometimes finding integrals might be very hard, but we can give some
upper and lower bounds by making very simple comparisons. This is
Property 8 in Pg 305 if your book
Sample problems : Find upper and lower bounds for
R2 1
dx
0 1+x2
4.4
R π3
π
4
tan xdx and
Fundamental Theorem Of Calculus
The main theorem here is of course the Fundamental Theorem of Calculus.
You should know the statement of both the parts properly exactly as given
below
Rx
The Theorem : f is given to be continuous on [a, b]. Define g(x) = a f (t)dt
for a ≤ x ≤ b. Then
1. g 0 (x) = f (x)
Rb
2. a f (t)dt = F (b) − F (a) where F is any anti-derivative
[It is basically saying that the derivative of the area function is the function
itself, and that integration and differentiation are inverse processes. This
is only to help you remember the statement, if asked, you must state it
properly!]
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4.5
Applications
There are two very important applications.
1. Find derivatives of area functions or functions that resemble area functions
2. Find definite integrals by finding antiderivatives and evaluating at the
end points
Sample problems :
1. Find
d
dx
Rx
2. Find
d
dx
R x2
3. Find
d
dx
R7
4. Find
d
dx
R x2
5. Find
Rb
xn dx
6. Find
R1
7. Find
R5
8. Find
R9
a
7
tan(sect)dt
7
tan(sect)dt
sin x
sin x
−1
−5
tan(sect)dt
tan(sect)dt
x100 dx
πdx
√
s2 s−1+2s2
ds
s2
1
Page 322 of your book has a table of antiderivatives for the functions we
come across often. You should know this table !
4.6
Substitution for indefinite integrals
1. You will be asked to find
R
f (x)dx. This will look complicated.
2. Make a change of variables. Define u to be some suitable expression in
x
3. Find du. (It will be the (u0 )dx)
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4. Convert the integral into an integral in u. It will now look like
R
g(u)du
5. This integral will have an easy solution
6. The answer will be G(u) + C where G is an antiderivative of g and C
is an arbit constant.
7. Resubstitute back for x
Sample problems
1.
R
(3x − 1)50 dx
2.
R
cos(x−2 +3)
dx
x3
4.7
Substitution for definite integrals
1. You will be asked to find
Rb
a
f (x)dx. This will look complicated.
2. Make a change of variables. Define u to be some suitable expression in
x
3. Find du. (It will be the (u0 )dx)
4. Change the end points say to c and d
5. Convert the integral into an integral in u. It will now look like
Rd
c
g(u)du
6. This integral will have an easy solution.
7. Your answer will be G(d) − G(c), where G is an antiderivative of g
Sample problems
1.
R4
2.
Rπ
0
2
0
√ x
dx
1+2x
cos x sin(sin x)dx
Practice problems for this Chapter: (Pg 338-340)
1. True or False : 3,4,7,15,16,17,
2. Exercises : 5,8,9,11,12,15,16,17,25,26,35,37,39,49
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Chapter 5 : Areas and Volumes
5.1
Area between curves
1. (Integrating with respect to x ) : It is the integral of the top curve
minus the integral of the bottom with the end points which will be the
x coordinates of the intersection points.
2. (Integrating with respect to y ) : It is the integral of the right curve
minus the integral of the left with the end points which will be the y
coordinates of the intersection points.
Sample problems
1. Find the region bounded by y = 5x − x2 and y = x
2. Find the region bounded by x = 2y − y 2 and x = y 2 − 4y
5.2
Volumes
1. (Integrating with respect to x ) : Cut the solid into vertical slices. Find
Rb
the cross section area A(x) and evaluate a A(x)dx for appropriate end
points (figure out how x varies)
2. (Integrating with respect to y ) : Cut the solid into horizontal slices.
Rd
Find the cross section area A(x) and evaluate c A(y)dy for appropriate
end points (figure out how y varies)
Sample problems Find the volume of the solid obtained the given region
around the given axis
1. y =
√
x − 1, x = 5 and y = 0 ; about the x-axis
2. x = y − y 2 and x = 0 ; about the y-axis
3. Find the volume of a cone of radius r and height h
Practice problems for this Chapter: (Pg 378-379) 1,3,6,7,8,12,15 a) and
b), 16
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