Final Examination
Math 0120
Final Examination
Sample
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1. Show your University of Pittsburgh ID if requested.
2. Clearly print your name and social security n
Introduction
Probability Theory
Discrete Mathematics
Discrete Probability
Prof. Steven Evans
Prof. Steven Evans
Discrete Mathematics
Introduction
Probability Theory
7.1: An Introduction to Discrete Probability
Prof. Steven Evans
Discrete Mathematics
Intro
Discrete Probability
Rosen 5.1
Finite Probability
An experiment is a procedure that yields one
of a given set of possible outcomes.
The sample space of the experiment is the set
of possible outcomes.
An event is a subset of the sample space.
The proba
Review Exercises for Chapter 3
Review Exercises for Chapter 3 275
1. A numbei c in the domain of f is a critical number if f’(c) = 0 orf’ \-
is undefined at c.
At least six critical numbers on (~ 6, 6).
3. g(x) = 2x + 5 cos x, [0, 2177]
g’(x) = 2 —- 5
: B Quesfiarzﬁ +~ Amst "9 Daft/inﬂux (Ch Z)
IQ. A ball is thrown vertically upwards into the air. The height, [2 metres, of the bail above the
ground aﬁzer 1 seconds is given by .
‘ h= 2+20r—5t2, t2 0
(a) Find the initial height above the ground of the b
Chapter 4" TB pawl-ice
Q1. Using the substitution u = :17 x + 1, or otherwise, ﬁnd the
integralfxwigrx +1 dX.
Q2. Consider the function f : x |—> x — x2 for O S x s where 1,< k S 3,
(a) Sketch the graph of the function/I
(b) Find the total ﬁnite area. enc
Review Exercises for Chapter 4 349
“WWW
b
The Trapezoidal Rule overestimates the area if the graph The Trapezoidal Rule underestimates the area if the graph
of the integrand is concave up. {if the integrand is concave dawn.
4s. L(x) = flair
If
i
(a) LG)
8 8
CHAPTER 2 Déij’eremiation
Related Rates
4- AS a . h p 13
int 6 Shape Of 3 ‘; here is be' 0 i W L]. 6 VIII“ 6 3213 [g a] 1 [e [Eﬁe (} 4; “b h
-. in b 0 I1 ‘ ‘ ' l( “H: e
I I . I i a: s m 15 inc I h S
pal SECOIld. What {ate IS Iadlus the radius is 1
Chapter 3 Probability
We have completed our study of Descriptive Statistics and are headed for Statistical Inference. Before
we get there, we need to learn a little bit about probability.
3.1 Basic Concepts of Probability
1. Probability Probability is a n