Spring2014 Instructor: Nelson Phouphakone
Math 151 — Test 3 ’
Name: Class Time: ID #:
Show appropriate work on all problems below. Total point value is 110 oints. Place answers in the blanks provided. Give
. exact answers and simplify your answers to all
‘4.9 Antiderivatives
Example: Suppose: F (x) = 3x4 + 2x - 12, then F ’(x) = 12963 + 2
If we have F ’(x) and we want to ﬁnd F (x), this process is called “integration or antidifferentiation.”
An antiderivative of f is simply a ﬁmction whose derivative is f
Spring 2014 Instructor: Nelson Phouphakone
Math&151 — Test 2
_ Name: Section: ID #:
Show appropriate work on all problems below. Total point value is 110 oints. Place answers in the blanks provided. Give
exact answers and simplify your answers to all prob
Recommended Homework Exercises
This is a list of exercises from the textbook. You are strongly advised to do
all of these problems for practice, even though you will not be asked to turn
any of them in. The answers to all of these questions can be foundin
4.4 Optimization Problems
In this Section, we want to ﬁnd extreme values of some applications in many area of life. We called optimization problems.
For examples, we want to maximize area, volumes, and proﬁts and to minimize distance, times and costs.
Gui
Spring 2014 lnstructor: Nelson Phouphakone
Math&151 - Test 2b
Name: 7 ' Section: ID #:
Show appropriate work on all problems below. Total point value is I 10 oints. Place answers in the blanks provided. Give »
exact answers and simplify your answers to a
/S’ ring 2014
p Instructor: Nelson Phouphakone
- , Math&151 - Test 1
Name: V L _ 7 Section: ID #: .V , V V
Show appropriate work on all problems below. > Total point value is 110 oints. Place answers in the blanks provided. Give
exact answers and simplify
M; a.“ ‘
4.7 L’Hospita'l’s Rule
III-lespitai's Rule Suppose f and g are differentiableand gf x) aé l} on an open I;
interval I that contains a. (except possibly at a). Suppose that“ " f L ' '
limeX) = 0
3—?“ ’
limsﬂx} = G
x—a-a
and
or that a
I ' and the appr xim Ion";
4.5 Linear Approximation and Differentials
If f is differentiable atq and x is'rcloseto‘al', then the tangent line L(x) is close to f(x).
(Fig.1) Thus, we could use thet'ang'ent line at (a,f(a) .as an approximation to
the curve
J
4.1 Maxima and Minima
Absolute or global maximum or minimum
Beiiuitieu Let c be a number in the domain D of a function f. Thenﬂc) is the
I absolute maximum value of f on D ifﬂc) % ﬁx) for ail x in D.
I absolute minimum value. offon I? if f
/ .;
4.2 What Derivatives Tell Us 9
DEFINITION er- INCREASJNGAND DECREAS-lNGfUHCTIQNS' , _
fais on an interval I ﬂxl) _< f(x2) whenener x1: (15:2, in].
fisgecreasingon fan intervalﬂxl) > f(x2) wheneiier x1 _<_; a; in I;
Ez’fisvincreas' V ing'
i.
3.10 Related Rates 3
In a related rates problem, the idea is to compute the rate of change of one quantity in terms of the rate of
change of another quantity (Which may be more easily measured). The procedure is to ﬁnd an equation that
relates the two qua
Q" . %K[w”]=llna(0\¥) ﬁauzimﬂaq) ‘4’ Me“: emu
3.8 Derivatives of Logarithmic and Exponential Functions ' -
' Ex. 23 = 8 <=> log2(8) = 3
Lo arithm E uivalent to an Ex onential ‘ ,
g ‘1 - P 2.3429) — s
The statement by = x is equivalent to the statement 10g
4.3 Graphing Functions — C,
Here is the message of this section:
“Graphing utilities are valuable for exploring functions, producing preliminary graphs, and checking your
‘work. But they should not be relied on exclusively because they cannot explain why
é) , i
3.6 The Chain Rule 6 ' .A x R m
We use Chain Rule to differentiate composites of functions,€°g)(x) = f (g (90).
The Chain Rule is the MOST IMPORTANT AND MOST USED OF THE DIFFERENTIATION PATTERNS.
EX. Find two functions f and g so that the given fun
A 3.9 Derivatives of Inverse Trigonometrie Functions
Trigonometry of Right Triangles. ' SC ‘4. r ' ’, L '\ s
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3.5 Derivatives as Rates of Change
5 ﬁabiﬂﬁimﬁ Average and Instantaneous velocity
x Let s m ﬁr) be the position functionof an object. moving aiong a line. The average
' velocity of the object over the time interval [(2, a. + m3 is the slope of the man:
. J .
3.7 Implicit Differentiation
Implicit Function
If we cannot solve for x explicitly in terms ofy, we say that the equation determines by y as an implicit function of x.”
Ex.y5+y+x=3; y+y3=x2+2 I
dy ) A
Implicit differentiation is the procedure of dif
0' x+>¥to
3.4 Derlvatlves of Trlgonometrlc Functlons CW L8 4 gmw : (
sin 6
' ' V : 2. 5 2
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cos 6 — 1
1) Show that lim . = 1 2) Show that lim =
- 6—»0 sm 6 6—»0 6
. - cu &+ l )
LHS : ﬂew V Lug
AZAInﬁqﬂeLhnﬁs
:ﬁiﬁﬁféiﬁﬁﬁ inﬁnite Limits
81113130313 f is deﬁnad for :13 x near :2. If ﬁx) grows arbitrarily huge for ail x sufﬁ—
cientiy close. (but net equal) to a (Fég'zzm' a), we. write.
.an
« ﬁm ﬁx) : cc.
12—012
z~
We say the 11:83.: of f as X app
i3 Techniques for Computing Limits
In this section we use the following properties of limits, called the Limit Laws, to calculate limits.
ﬁ-EE‘GREM 2.3 Limit Laws .
Assume lim ﬁx) and lim g(x) exist. The following properties hold, where c is a
v—w '
x
/§.1 The Idea of Limits
AVERAGE RATE OF (ﬂimﬂﬁ
The average rate of change of the fuiictiony = f(x) Between x = a and x = b is
Change in? 2. Kb) “ fta)
avera rats fchan e = , a ,
ge 60 g ehangemx 13-5;
The average rate of change is the slope of the secant
2.2 Deﬁnitions of Limits
DEFINITION Limit of a Function (Preliminary)
Suppose the function f is deﬁned for all x near a except possibly at a. If f (x) is arbitrarily close to L (as
close to L as we like) for all x sufﬁciently close (but not equal) to a,