Unformatted text preview: , EXEIEiSES SECIION 7 .3* THE NATURAL EXPOIIENTlAl FU’NCIIOII ’ 465 w 15 the number 2 deﬁned? at is an approximate Value for e ? . tab, by hand, the graph of the function f (x) = exwith
'cular attention to how the graph crosses the yaxis. ‘ (at fact allows you to do this? 'lify each expression. ,
(b) In J2
(b) e 3 1n 2 (b) eﬁ =5
(1)) ln(5_  2x) = —3 b" where a 3i b I=12 'd the solution of the equation correct to four decimal l4. 1n(l M/i) : 2
l6. NH) : 7 lve each inequality for x. '10 (b) lnx> —1 lnx < 9 (b) 22—“ > 4 r.’ u n n n “ ake a rough sketch of the graph of each function. Do
‘ eulatnr. Just use the graph'given in Figure 2’ and, if
e transformations of Section 1.3. 20.y=1‘
22. y=27 a n n with the graph of y = 6", write the equation of the
results from _' g2 units downward g 2 units to the right (c) reﬂecting about the xaxis
(d) reﬂecting about the yaxis‘ »
(e) reﬂecting about the x—axis and then" about the yaxis . Starting with the graph of y = e", ﬁnd the equation of the
graph that results from ‘ L
(a) reﬂecting about the line y = 4
' (b) reﬂecting about the line x = 2 25—30 Ill] Find the limit. 25. lim 8"” 26. lim em" xam _ xﬁ (11/2)1L 27 . 83x _ e’SX 23 1 63): __ 6A3;
. 11m ————3x ,3):  im 3, +‘ —3x me e + e _ we” e 29. lim 8/0"” 30. unisex/(2“) 1,—>Z‘ X—VZ" n u < n ~ n l: 3144 Ill] Differentiate the function. x e 3]; f(x) = xze" L ‘V y = 1+ x .33 y = 6"”3 y = e“(cos u + cu)
35. ﬂu) = M
31.1:(t) = am y = emu 39. y = 1/1 +33% yze‘lnx y = cos(e’”) Fin—+73 e‘ + 6" 4546 ml Find an equation of the tangent line to the curve at the
given point. 45. y = 62‘ cos 77x, ’(0, l) ‘46. y #— ex/x, (1, e) 47. Find y’ if 6ny = x +31. 48. Show that the function y ¥ Ae"" + Bxe” satisﬁes the differen
tial equation y” + Zy’ + y = 0. 49. [For what values of r does the function y = e” satisfy the equav
tion y” + 6y’ + 8y = 0? ' 50. Find the values of A for which y = 9’” satisﬁes the equation
y + y' = y”. 51; If f (x) = e“, ﬁnd a formula for f ("’(x).
52. Find the thousandth derivative of f (x) = xc". ...
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 Spring '06
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