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solutions-t2-1501-Fa07

# solutions-t2-1501-Fa07 - KEY Student Name and ID Number...

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Unformatted text preview: KEY Student Name and ID Number 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 Newton-Raphson 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 2-5 : 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* 4-4 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 strictly-increasing 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 Kb-9(\$)=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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solutions-t2-1501-Fa07 - KEY Student Name and ID Number...

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