Objective: To Solve Optimization Problems
Which is greatest? Which is least? Minimize, Maximize.
Used in Business for Cost, Production, Volume, Area
Environment for Volume, Area, Cost of Pollution
Energy for Fuel Consumption, gas mileage
Traffic Flow, Med
Oblique Asymptotes (€26 maimed)
Let f(x) be a rational function, i.e.,
f<xr= :83)
If deg P(x) = deg S(x) +1 , then the graph of f(x) has an oblique
or slant line asymptote.
This means that, for large magnitude of x, the behavior of f(x)
is similar to
3‘3 Differentiation Ruies
Derivative of a Constant Function f(x)=o
'( =t/i ’VXH‘B'RX): Linn C " C- :hwi O = O
4‘ X) [rm E k-VO k k—vd
”if m: (at; O ‘1
RQWGUUIG for positive integers: Let n be a positive integer,
we’llﬂvfind the derivative of f(x)= xﬂ
Name:
MATH 165 Section_ Exam 1 2/5/2016
SHOW ALL YOUR WORK to avoid loss of points.
tan(5;‘c)
1. Redeﬁne the function f(x) 2 ‘ so that it is continuous at x : 0., and hence for all real cc.
x
[Hint]: For the required lirnit7 express f (:23) in terms
3.2 The Derivative as a Function
Last time we defined the derivative of y=f(x) at the point x: x0
¥\(2¢D= um $(x.+k)~ gm)
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Definiﬁgn The derivative of the function f(x) with respect
to the variable x is the function f’ whose value at x is:
\ _ 1m 1
2‘6 Limits lnvolvin
lottery
We are interested in the behavior of a function when the
magnitude of the independent variable x becomes
increasingly large, we write: X —+ 00 or X a _ co
plxl loge X <0
* The symbol 00 W represent a number, we use it to
desc
Continuity Test
A function f(x) is continuous at an interior
point x=c of its domain if and only if:
1. f(x) is
2.
3.
We have three kinds of discontinuities:
Discontiniuty
Discontiniuty
Discontiniuty
A function is continuous if it is continuous at every
p
2.4 One-Sided Limits
Definitions: (Side limits) Left-hand side limit; we write
This is the value that f(x) approaches, if it exists, when x approaches c
from the left, that is we consider values
Similarly, right- hand side limit; we write
This is the valu