Fact 2 [The Closed therval Method). 0n the intervat [a1 5], the function f always has at least one
absolute maximum and absatate minimum, and theyI can only occur at either
I the critical numbers that are inside that interval, or
I the endpaints rs = a an
II. Important Corollaries
The following two facts follow from the Mean Value Theorem.
Fact 1. If 3‘13] 2 i] for at]! x in an interval ([1,5], then f is constant on the interval.
Fact 2. If f’{:t) = g’[:t) for every point a: in an interval [(1,5)], than th
Example 4. Let ’s sketch. a graph of a function ﬁx] that satisﬁes ait of the given conditions:
I f’fm] > 0 on {012), f’[a:] <2 0 on (—oo,ﬂ) and [2,00]
I f”[£} > 0 011 {—3.2} and [2100), f"{a:] < I} on [—00,—3]
I vertical asymptote a: : 2, lim f[a:) = 3, l
Exanlple 5. Finally we put! same geemet'ry and trige'iiemet'ry out of the dusty reeeeeee ef erur
mind. We have a ﬁxed meteng'te with one side if tenyth L and the ether .ef length W. I’ve draw
that in red in the picture betcha. We’re geihg te eimumeeribe
III. Hint on nﬂnlmialng or maxinﬂalug distances
The value that optin‘iiaes
“something
and the value that optimizes
something
are the same value, and the seoond expression is a lot easier to get the derivative of.
So for our previous example1 when it comes
Question: How to ﬁnd the velocity of the particle at a given moment of time?
We can deﬁne the average velocity}.r of the particle on a particular time interval. Fix a moment
of time t and consider the time interval [t. t + ﬂit].
all + A!)

,l 
t Hat
d
Exanlple 3. We’tt ﬁnd the equation of the tangent tine to the ennle y = er one: at :1: = I}. First
we ﬁnd the slope of the tangent fine. The slope is just the derivative evaluated at 1' = H, 50 we ﬁnst
ﬁnd the derivative using the pmduet mte:
y" =e1'{eeem
II. Tangent lines
The tangent line to a graph at a point turns out to be useful in all kinds of applied prohlerus1
some of which we’ll touch on in this course. How can you get the tangent line equation from the
derivative? The quickest and easiest way is
Theorem 2 (Rolle’s Theorem]. Let [(1, b] be a given eiosed interval. Let f be a. function such that
a f is continuous on [(1, b],
a f it diﬁerentiabie on (a, b],
I ffa}=ffb}
Then there exists; a. number C such that a. < c < b and f'{c} = '3.
Geometricall
ict 2 [The First Derivative Test). Say the number I : c is a criticai numher of a continuous
function f. Then:
I if the derivative f“ changes sign from “+” to “—” at I = c, then f has a iocat marcimum
there,
a if the derivative f’ changes sign from “— ” t
Describe a scenario when conditional formatting can help you be more efficient.

Ranking basketball players
Monitor money made in a company using ranges (red range) to contact them etc.
When would you normally use R/Python instead of Excel

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h FE 543 Intro to Stochastic Calculus
for Finance Midterm, Alternate Exam
October 23, 2015.
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Daniela Cardona
Professor T. Koeller
BT 244 Assignment 3
5 December 2013
Terror Insurance Battle Heats Up
This article explains the talk over the Terrorism Risk Insurance Act (TRIA) of 2002
which provides economic protection by the government against terr
Daniela Cardona
Professor T. Koeller
BT 244 Assignment1
24 September 2013
Weak Demand Keeps Singapore Home Prices Low
In this article about Singapores private housing market plummeting for the past five
years, a factor that can pose a change in demand is
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BT 440 Review Third Exam
Chapter 7: Tools
The Market for Reserves and the Fed Funds Rate:
Federal Funds Rate: rate at which the banks lend to each other overnight using the reserves held by the FED.
Note: what the FED actually alters is the target rate, n
Value Investing
The Individual Fundamentals that drive the Valuation of Each Company are known as alpha
The overall trends in the market are known as beta
Selecting the companies with the worst financial attributes yields higher returns than selecting
tho
Example 5. The derivative of any straight tine is the siope of that tine. Why? The equation of a
line is
ﬁx) 2 mm + b,
where m is the stope of the tine and b is the yintereept. By deﬁnition of the derivative
f[$+h)—f[.r} m[:t+h]+b—{m:t+b}
:lim
f’[:t) : l
ict 4 {The Second Derivative Test). Suppose f" is continuous near :1: : c.
I If f'lc} = i} and fl'lc} > 0, then f has a local minimum at I : c.
a If file} = ﬂ anal file} «:1 0, then f has a local maximum at a; = c.
The Second Derivative Test is often the
Example 1. Suppose ffO] = —3 and f’[a‘] $ 5 for every 11:. Hour targe ean f(2] possibly be?
The assumption “f’[$) g 5 for every 11'” implies that f is diﬁ'erentiabte (hence, continuities,I
everywhere. Consider the interval [0.2]. f is continuous on [0,2]
Exanlple 1. Say we‘ve asked ta ﬁnd the dimensians of a rectangle with perimeter lﬂﬂ in whose
asea is as large as passihle. What da they want me ta madEmits? The area of a rectangle, which
l’lt call A for area. AnyI rectangle Js area is a fanctian cf its
Example 3. Find the dimensions of the rectangie of iargest area that has its base on the xaxis
and its other two uertices above the xaxis, and tying on the paraboia y = 9 —x2.
Let (x, 3,!) denote the upperright corner of the rectangle, that is, we assu
Example 4. Sag we he asked to ﬁnd the point on the parahota :1: + y2 = 0 that is ciosest to the
point [I], —3}. What are we asked to minimize? The distance from a point to [0. —3), which I’ll: catt
ti. The distance of any point (my) to the [1], —3] is
e =
Example 2. A boa: with a square base and open top must have a volume of 321001} (31113. Find the
dimensions of the box that minimize the amount of material.I used.
Denote by a: the side of the base and lay it the height of the box. We have to optimize th
Fact 3 (Derivative of an exponential function). For any allowaﬁle base a,
d I I
Em ]—l_na+a
By “allowable base” I mean that a must be positive and :1 ;E 1.
Example 3. Here are some illustrations of the formula alloys:
3
dr
salineer
One particular case of