Unformatted text preview: 30. 31. SECTION 4.6 GRAPHING WITH CALCULUS AND CALCULATORS 351 and if c < 0. there are no 1P. For c > 0. there are [P at (i 2c/3, (2—3/2). Note that the y—coordinate of the IP is constant. We see that ifc 5 0. ﬂat) 2 ln(w2 + c) is only deﬁned for m2 > —c : lml > \/—c. and lim f(m) = lim f(x) 2 —00. since lny —> —00 as y —> 0. Thus. for c < 0. there are vertical
m—>\/——C+ z—>—\/——c" asymptotes at a: : :l:\/E. and as 0 decreases (that is. lc increases). the asymptotes get further apart. For c = 0. lim f (x) : —00. so there is a vertical asymptote at m = 0. If c > 0. there are no asymptotes. To ﬁnd the extrema 11—»0 1
m2+c and inﬂection points. we differentiate: f (cc) 2 in (m2 + c) :> f’ (m) = (216), so by the First Derivative Test there is a local and absolute minimum at :E : 0. Differentiating again. we get 1
x2+c 2(0 — 3:2) mm) : ($2 + Q2 . (2) + 2a:[— (m2 + c)_2 (212)] : Now if c g 0. f” is always negative, so f is concave down on both of
the intervals on which it is deﬁned. If c > 0, then f” changes Sign when
0 : 0:2 4:) a: : :lq/E. So for c > 0 there are inﬂection points at as : :l:\/5, and as c increases. the inﬂection points get further apart. Note that c : 0 is a transitional value at which the graph consists of the maxis. Also. we can see that if we substitute ~c for c, the functlon f (w) = W Will be reﬂected in the m—axrs. so we investigate only posrtive
c a:
values of 0 (except c : 71. as a demonstration of this reﬂective property). Also. f is an odd function. lim (9:) : 0, so 3; = 0 is a horizontal asymptote for all c. We calculate $—>::OO 1 + 022:2)c — cm(2c2:c) C(62$2 — 1)
I ( 7 I 2 2 _ _
f (x) : (1 c2zc2)2 — (14—02302)? f (1:) — 0 > c :c — 1 — 0 4:} m — :lzl/ci So there is an absolute maximum value of f (1 / c) : % and an absolute minimum value of f(—1/c) : ~ %. These extrema have the same value regardless of c. but the maximum points move closer to the y—axis as c increases. (—2c3m) (1 + c2m2)2 — (~03z2 + 0) [2(1 + c2m2) (2c2$)] f (1') (1+ C2332)4
_ (—2032?) (1 + (22332) + (631132 — C) (402m) 203m(c2m2 — 3)
— (1 + c2272)3 — (1 + c2m2)3 f"(x) : 0 <:> :c = 0 or :tx/g/c. so there are inﬂection points at (0.0) and at ( x/g/c, x/g/4). Again. the ycoordinate of the inﬂection points does not depend on c, but as 0 increases. both inﬂection points approach the yaxis. ...
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 Spring '10
 Ban
 Calculus

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