MATH138: Midterm ExamAID
SOS
Prepared by Vincent Chan
[email protected]
February 21, 2010
I include numbers after each section heading that correspond to the textbook section headings for your
convenience. Proofs of theorems are included in your textbook, and will be omitted. Finally, I strongly
recommend going through the examples
and
exercises in your textbook, as they are very helpful in prepar
ing for your exams and understanding the material. The material that follows will cover the most dif±cult
material that is presented in Math 138, while assuming knowledge from Math 137 (such as basic integration
and areas).
1
Review of Integration (5.1  5.5)
Recall: if a function
°
is de±ned on an interval
°
±²³
±
, we divide the interval into
´
subintervals of equal
length
²
µ
³´
³
°
±
µ
¶´
. Letting
±
³
µ
°
²µ
±
²···²µ
°
³
³
be the endpoints of these intervals and
µ
°
±
±
°
µ
±
±
±
±
±
,
we de±ne the
defnite integral oF
°
From
±
to
³
as
°
²
³
°
´
µ
µ
¸µ
³¶
·¸
°
²³
°
±
±
²±
°
´
µ
°
±
µ²
µ
provided this limit exists. If so, we say
°
is
integrable on
°
±
. The integral can be thought of as the area
under the graph
°
, if
°
is nonnegative. Some properties of integration:
(i) Linearity:
°
²
³
¹°
´
µ
µ¹
º
´
µ
µ
¸µ
³
¹
°
²
³
°
´
µ
µ
¸µ
¹
°
²
³
º
´
µ
µ
¸µ²
for any constant
¹
and integrable functions
°²º
.
(ii) Decomposition:
°
²
³
°
´
µ
µ
¸µ
³
°
´
³
°
´
µ
µ
¸µ
¹
°
²
´
°
´
µ
µ
¸µ²
for any
»
±
°
±
and integrable function
°
.
(iii) Monotonicity:
°
²
³
°
´
µ
µ
¸µ
²
°
²
³
º
´
µ
µ
¸µ²
if
°
´
µ
µ
²
º
´
µ
µ
on
µ
±
°
±
, for any integrable functions
.
(iv)
°
²
³
°
´
µ
µ
¸µ
³
°
°
³
²
°
´
µ
µ
¸µ²
for any integrable function
°
.
1
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View Full DocumentWATERLOO SOS EXAMAID: MATH138 MIDTERM
Of great importance is the Fundamental Theorem of Calculus, which shows us that integration is a sort
of antidifferentiation, to some extent:
THEOREM
1.1
[FUNDAMENTAL THEOREM OF CALCULUS]
.
Suppose
°
is continuous on
°
±²³
±
.
(i) Then function
´
deFned by
´
²
µ
³´
°
°
±
°
²
¶
³
·¶
for
µ
°
°
±
is continuous on
°
±
, differentiable on
²
³
, and
´
°
²
µ
°
²
µ
³
.
(ii) Then
°
²
±
°
²
µ
³
·µ
´
¸
²
³
³
±
¸
²
±
³
where
¸
is any antiderivative of
°
, i.e.
¸
°
´
°
.
Recall: since the Fundamental Theorem gives a connection between integrals and antiderivatives, we
use the notation
±
°
²
µ
³
·µ
to denote the antiderivative of
°
, and call it an
indeFnite integral
. Recall the most
general antiderivative of a given integral is obtained by
adding a constant to a particular antiderivative
, giving
rise to a term of the form “
µ
¹
” when evaluating inde±nite integrals. Furthermore, we will adopt the
convention that a formula for a general inde±nite integral is
only valid on some interval
, not necessarily the
whole real line. Based on the Fundamental Theorem and our knowledge of differentiation, we can produce
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