Section 6.5 Approximate Integration
2010 Kiryl Tsishchanka
Approximate Integration
MIDPOINT RULE:
b
f (x)dx Mn = x[f (x1 ) + f (x2 ) + . . . + f (xn )]
a
where
x =
ba
n
1
and xi = (xi1 + xi ) = midpoint of [xi1 , xi ].
2
TRAPEZOIDAL RULE:
b
a
f (x)dx Tn =

Section 3.4 Exponential Growth and Decay
2010 Kiryl Tsishchanka
Exponential Growth and Decay
In many natural phenomena (such as population growth, radioactive decay, etc.), quantities
grow or decay at a rate proportional to their size. In other words, the

Calculus III Part 1
Name: Solutions
1. (a) u v = 2, 2, 0
(b) u v = 2
(c) Let [0, ] be the angle between the two vectors
cos =
(d)
uv
2
= = = 450
|u|v|
2 2
uv
v=v
|v|2
(e)
|(u v) 1, 0, 0 | =
2
2. (a) j
(b) i
(c) 0
(d) i
(e) j + k = 0, 1, 1
3. Unit tangen

Calculus III 2008 Spring
Instructor: Stechmann
Final Exam
100 points total.
1. (5 points) Find an equation for the plane tangent to the surface x2 + y 2 z 2 = 1 at the point (1, 1, 1).
2. (15 points) Find all critical points of the function f (x, y) = 6xy

Calculus III 2008 Spring
Midterm Exam 1
100 points total.
1. (15 points) Let u = 1, 1, 2 and v = 1, 1, 0 .
(a) Compute u v.
(b) Compute u v.
(c) What is the angle between u and v?
(d) What is the vector projection of u onto v?
(e) What is the volume of th

Calculus III Part 2
Name: Solutions
1. (cf Cal3 exam part 1 problem 6) Normal 1, 1, 1 , so plane x + y + z = 1
2. (cf Cal3 exam part 1 problem 7) We solve
y 2 x2 = 0
y(x y 2 ) = 0
Critical points: (0, 0), (1, 1). Test
2
fxx fyy fxy = 144(3xy 2 x2 y 2 )
fx

Section 3.1 Exponential Functions
2010 Kiryl Tsishchanka
Exponential Functions
DEFINITION: An exponential function is a function of the form
f (x) = ax
where a is a positive constant.
1
10
1
10
1
10
1
10
1
10
1
10
1
10
1
10
x
-3
-2
-1
0
1
2
3
x
x
-3
-2
-1

Section 4.3 Derivatives and the Shapes of Graphs
2010 Kiryl Tsishchanka
Derivatives and the Shapes of Graphs
INCREASING/DECREASING TEST:
(a) If f (x) > 0 on an (open) interval I, then f is increasing on I.
(b) If f (x) < 0 on an (open) interval I, then f

V63.0123-1 : Calculus III. Midterm2
Mon Apr 7. You have 60 minutes. Non-graphing calculators and a single side of
letter paper equations are allowed. 6 questions, continued on reverse. You do not
have to attempt them in order.
1. [6 points]
(a) Comput

V63.0123-1 : Calculus III. Midterm1
Wed Feb 19. You have 60 minutes. Potentially useful equations are on back.
Find the equation for the plane which includes the point
the line
,
,
.
1. [10 points]
and
"0 "& "
1#!% ) ('% $ #!
2. [14 points]
A point pa

Section 7.1 Areas Between Curves
2010 Kiryl Tsishchanka
Areas Between Curves
We dene the area A of S as the limiting value of the sum of the areas of the above approximating
rectangles:
n
A = lim
n
i=1
[f (x ) g(x )] x
i
i
From this it follows that the ar

Section 5.1 Areas and Distances
2010 Kiryl Tsishchanka
Areas and Distances
We can easily nd areas of certain geometric gures using well-known formulas:
However, it isnt easy to nd the area of a region with curved sides:
METHOD: To evaluate the area of the

Section 4.4 Curve Sketching
2010 Kiryl Tsishchanka
Curve Sketching
GUIDELINES FOR SKETCHING A CURVE:
A. Domain.
B. Intercepts: x- and y-intercepts.
C. Symmetry: even (f (x) = f (x) or odd (f (x) = f (x) function or neither, periodic
function.
D. Asymptote

Section 1.1 Functions and their Representations
2010 Kiryl Tsishchanka
Functions and their Representations
DEFINITION: A function f is a rule that assigns to each element x in a set A exactly one
element, called f (x), in a set B. Its graph is the set of

List of problems for the nal exam
Complex Variables
Instructor: Irina Nenciu
1. Suppose f is analytic in a region , and D . Further assume
that |f (z)| > 2 if |z| = 1 and f (0) = 1. Must f have a zero in the unit
disc?
2. Suppose f and g are analytic in D

Section 3.7 Indeterminate Forms and LHospitals Rule
2010 Kiryl Tsishchanka
Indeterminate Forms and LHospitals Rule
THEOREM (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
xa
l

Section 5.2 The Denite Integral
2010 Kiryl Tsishchanka
The Denite Integral
DEFINITION: The area A of the region S that lies under the graph of the continuous nonnegative function
f is the limit of the sum of the areas of approximating rectangles:
n
A = li

Section 5.5 The Substitution Rule
2010 Kiryl Tsishchanka
The Substitution Rule
THEOREM (The Fundamental Theorem Of Calculus, Part II): If f is continuous on [a, b], then
b
b
f (x)dx = F (b) F (a) = F (x)
a
a
where F is any antiderivative of f, that is F =

Section 7.2 Volumes
2010 Kiryl Tsishchanka
Volumes
DEFINITION OF A DEFINITE INTEGRAL: If f is a function dened on [a, b], the denite integral
of f from a to b is a number
n
b
f (x)dx =
a
lim
max xi 0
f (x )xi
i
i=1
provided that this limit exists.
DEFINIT

Section 6.2 Trigonometric Integrals and Substitutions
2010 Kiryl Tsishchanka
Trigonometric Integrals and Substitutions
This section consists of two parts:
Part I: Trigonometric Integrals. We will distinguish three main cases:
Case A: Integrals of type
sin

Section 8.3 The Integral and Comparison Tests
2010 Kiryl Tsishchanka
The Integral and Comparison Tests
The Integral Test
THE INTEGRAL TEST: Suppose f is a continuous, positive, decreasing function on [1, )
and let an = f (n). Then the series
an is converg

Section 4.5 Optimization Problems
2010 Kiryl Tsishchanka
Optimization Problems
EXAMPLE 1: A farmer has 2400 ft of fencing and wants to fence o a rectangular eld that
borders a straight river. He needs no fence along the river. What are the dimensions of t

Section 6.6 Improper Integrals
2010 Kiryl Tsishchanka
Improper Integrals
Type 1: Innite Intervals
Consider the innite region S that lies under the curve y = 1/x2 , above the x-axis, and to the right of the
line x = 1. You might think that, since S is inni

Section 3.2 Inverse Functions and Logarithms
2010 Kiryl Tsishchanka
Inverse Functions and Logarithms
DEFINITION: A function f is called a one-to-one function if it never takes on the same
value twice; that is,
f (x1 ) = f (x2 ) whenever x1 = x2
one-to-one

Section 3.5 Inverse Trigonometric Functions
2010 Kiryl Tsishchanka
Inverse Trigonometric Functions
DEFINITION: The inverse sine function, denoted by sin1 x (or arcsin x), is dened to be
the inverse of the restricted sine function
sin x, x
2
2
DEFINITION:

Section 4.1 Maximum and Minimum Values
2010 Kiryl Tsishchanka
Maximum and Minimum Values
DEFINITION: A function f has an absolute maximum (or global maximum) at c if
f (c) f (x) for all x in D, where D is the domain of f. The number f (c) is called the
ma

Section 4.2 The Mean Value Theorem
2010 Kiryl Tsishchanka
The Mean Value Theorem
THEOREM (The Extreme Value Theorem): If f is continuous on a closed interval [a, b], then f
attains an absolute maximum value f (c) and an absolute minimum value f (d) at som