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MATH 1501 Test 2, October 25, 2007, WTT 1. Complete the following to form a statement of the First Fundamental Theorem: Let f and G’
be continuous on [05,13] with G differentiable on (a, b). $150: fo) fwd? xjnam (4,5)} M zap;
“b
E
k
rt.
u
E
G
\a
l
c,\
'\
3 2. Complete the following to form a statement of the Second Fundamental Theorem: Let f be
continuous on [(1, b] and let c be any number in [a, 1)]. Also, let F be the function deﬁned on [(2, b]. 63 fax , F’C’K): ﬁat) 96w 4,66 x 754/». {4,4} @ 3. Use differentials to estimate V6403. 4%): JK HM) = 5’ I
, .
We : if; 57w = T} F 72; 4. The NewtonRaphson method uses the formula can“ = can — f/f’ If f : $2 — 35
and 3:2 = 5.0, ﬁnd 333. KB 2.“. X1...— .—/'4) miss ,, 5—! ml
._ [a
25 : gelrE 5. Consider the function f = $2, the closed interval [1, 10] and the partition P 2 [1, 2, 7, 10].
a. What is the mesh MP) of the partition P? M, { p) . may A x; :2
b. For the selection t1 = 2, t2 = 2, and t3 = 10, What is the value of the Reimann sum 23=ifCti)A($i)? Q,
7* 44 9 I) + set) ~= 9. _
9_«:i_+ nos +20~3 =
6. Complete the following to make correct statements. '3 5&0 '
FYKLJ @a @ K
, o The natural logarithm function 1113: is deﬁned by ln m = t
I l' o The derivative of lnx is x o lncc is strictly increasing because 0M6. pathf each gagged, of by 4M (gagmwmézma (—oo,0<>) lR
«aux rvaM‘a’l If x is positive and q is rational, then ln(2:9) = The range of ln m is If m and y are positive, then ln(;r;y) = The exponential function Exp(:c) is deﬁned as The domain of Exp(:r) is C  we 1%,) 01" R 0" L J flﬁu yﬁgpw col“
The range of Exp($) is C a , m ) o v 4’60 I796
The derivative of Exp(a:) is E x P (X) Exp($) is strictly increasing because ’EXPCKJ > O
Thenumbereis Ml?“ 4.0 (an x, Z l
£n(f%) ': If q is rational, then eq = Exp(q) because For all irrational values of 3:, we then set 6“ = E x f) (X J
X + ‘a
For all :13, y, 6363’ = e
For all my, (ex)? = e x 5
)9 .ﬂnAﬁL
When a is positive and a: is irrational, we deﬁne a": = e The function AT(a:) (which will genial—Elly be called the inverse tangent or arctangent function)
5 J «t is deﬁned by AT($) m 0 1H7.
The domain ofAT(a:) is has , to) Dr R a” W we“? “ﬂu/(«Lari
The derivative of AT(3:) is Til—ﬁg” '
AT($) is strictlyincreasing because ale Iv IVA/gum “£91791 J ‘r' 1’ “l?
The range of AT($) is an Open interval of the form (—b, b). In fact 2) = (“J/SL—
7. Deﬁne a function h by .
Mm) = [:2 sinh sees 33 633 d5
'3 X Find h’($) 69 £1,002 MJHL WETG' 8. Find the derivatives of the following functions: , a. f(m)=lnsinh(e$+%) x ,c
@ ﬁx): ‘ againfexwi) {‘9 ﬁg?)
MkCeK+Jxl
3. I (5,4214%)!A/lx
.n Kb9($)=x :— e a“
' X
E‘ﬁ 916x) ~.. 5369"“M ' [(4% 1") 3’4 + 02’”) WK} .. 225 9. Calculate; “‘3
(a + e
H 2/; a. fe“(1+e“)‘7/5 dy ':
@  “3
L .1 y,
J WBX 8 AK '2 I g1). oﬂmzeﬁdm : 3;!) «g:
_ Q5
L1; 10. . A radio—active substance of quantity 36 grams reduces to 12 grams after 2 years. Express its
half—life in years. ...
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This note was uploaded on 07/09/2008 for the course MATH 1501 taught by Professor N/a during the Fall '08 term at Georgia Institute of Technology.
 Fall '08
 N/A
 Calculus

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