Department of Mathematical Sciences
CARNEGIE MELLON UNIVERSITY
21-120 Dierential and Integral Calculus, 10 units, Fall, 2012
Instructor: Dr. Russ Walker, Wean Hall 6219, ext. 8-9657, rw1k@andrew.cmu.edu
Text: Stewart, Essential Calculus: Early Transcenden
An application of the Intermediate Value Theorem:
The Bisection Method:
If f is a continuous function and f (a)f (b) < 0 and b > a, then to
approximate a solution to f (x) = 0 in the interval [a, b] to within
:
a+b
. If f (x1) = 0, STOP.
2
2. If f (x1)f (
Denition. Let f be a function dened on an open interval that contains
the number a, except possibly at a itself. Then we say that the limit of
f (x) as x approaches a is L, and write
lim f (x) = L
xa
if for every
> 0 there is a > 0 such that if 0 < |x a|
Linearization. A portion of the graph of f and a tangent line to
the graph of f are shown below.
There are two approximations over the interval from [x0, x0 + x]:
One to the change y in y and one to the value of f (x0 + x).
.
.
. .
. .
.
. .
. .
.
.
.
.
.
Proposition. If f (x) = 0 for all x in an interval (a, b), then f is constant
on (a, b).
A function F is called an antiderivative of f on the interval I if F (x) =
f (x) for all x in I.
Proposition. If F is an antiderivative of f on an interval I, then th
Dierentiation rules: Let f and g be dierentiable functions
and c a constant:
Algebraic functions:
3.1-2
d
(c) = 0
dx
d
(x) = 1
dx
d
d
(cf (x) = c (f (x)
dx
dx
d
d
d
(f (x) + g(x) =
(f (x) +
(g(x)
dx
dx
dx
d
d
d
(f (x)g(x) = f (x) (g(x) + g(x) (f (x)
dx
dx
21-120 Dierential and Integral Calculus
Quiz questions for Fall 2014:
Questions for Quiz 6:
1. Suppose that a dierentiable function f satises f (8) = 24 and |f (x)| 3. What is the
smallest value that f (2) can have?
2. Is it possible for a dierentiable fu
ht AND HUHIZCIHTAL SHIFTS Suppose I." h [1. Te ehtiiti the graph t
ﬁrm I is a rule that assigns In each element r in a set D exactly cne
t, celled ﬁx}. in a set E. {(t] + e, shift the graph efp = {(1) stlistence e units upward
llt] - e. shift the graph ti
E DEFINITION The area A of the region S that lies under the graph of the contin-
uous function f is the limit of the sum of the areas of approximating rectangles:
A: lim Rn: lim [f(x1)Ax + f(x2)Ax + + f(xn)Ax]
naw n+m
E DEFINITION OF A DEFINITE INTEGR
CALC STUDY NOTES
f Concave Up
f Increasing
Concave Down
Decreasing
f Positive
Negative
Maximum
Critical
point sign
change
+/f(c) = 0
f(c) < 0
Minimum
Critical
point sign
change
-/+
f(c) = 0
f(c) > 0
Point of inflection
Maximum
or
minimum
Critical point si
Review problems for the Final
The Final will be Monday, December 10, 8:30-11:30 AM
The 8:30 Lecture will be in DH 2210.
The 9:30 Lecture will be in DH 2315.
Assuming that I get the space requested, I will oer a review Saturday, December 8 at 1:00 in
DH 23
Suppose that c is a constant and that the limits
lim f (x) and
xa
lim g (x)
xa
exist. Then:
The rst six rules say the limits behave as expected under algebraic combination:
1. lim [f (x) + g (x)] = lim f (x) + lim g (x)
xa
xa
xa
2. lim [f (x) g (x)] = lim
Example. The radius of a circular oil slick on a body of water from
a spill is t + 4 meters t minutes after the spill, show that the rate
of change of the area of the slick is its circumference.
Example. Suppose that the length of a steel beam depends on
Everything you need about exponentials and logarithms:
An exponential function:
For a a positive constant, an = a a a a (n factors)
a0 =
a1 =
an =
an =
ax+y =
axy =
(ax)y =
(ab)x =
m
If a > 1, then lim ax =
and lim ax =
If 0 < a < 1, then lim ax =
and lim
Examples involving dierentiation of logs and exponentials:
Example 1. y = ln(x2 + 4)
Example 2. y = x ln(2x + 1)
Example 3. y = (x2 + 3)e2x
2
Example 4. y = ex Sketch the graph, determine the second derivative, show the inection
points, and connect to sta
LHospitals Rule. Suppose f and g are dierentiable and
g (x) = 0 near a except possibly at a. Suppose that
lim f (x) = 0 and lim g (x) = 0
xa
xa
or that
lim f (x) = and lim g (x) = .
xa
xa
Then
f (x )
f (x )
= lim
x a g ( x )
xa g (x)
if the limit on the r
An approach to solving applied extrema problems:
1. Read the problem carefully to gain an understanding of the
problem and the quantities involved. Decide what is being asked.
2. Draw a diagram if possible, and identify given and unknown
quantities on the
Integration with inverse trigonometric functions:
We previously have derived the dierentiation formulas for the inverse trigonometric functions. Here they are, coupled with the Chain Rule, together with more general versions of
their anti-dierentiation fo
21-120 Dierential and Integral Calculus
Quiz questions for Fall 2012:
Questions for Quiz 2:
5
1. If sec(x) = , determine sin(2x).
3
2. Determine the equation of the line through (2, 5) and perpendicular to the line
2x + y = 6, and express it in the form A
21-122 Integration & Approximation, Spring, 2016
Weeks 1 2:
Chapter 8 Applications of Integration
January 10 24
8.1 Arc Length
8.2 Area of a Surface of Revolution
8.3 Applications to Physics and Engineering
8.4 Applications to Economics and Biology
Bonus