{[ promptMessage ]}

Bookmark it

{[ promptMessage ]}

Chapter 4 89

# Chapter 4 89 - 30 31 SECTION 4.6 GRAPHING WITH CALCULUS AND...

This preview shows page 1. Sign up to view the full content.

This is the end of the preview. Sign up to access the rest of the document.

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 m-axis. 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 y-coordinate of the inﬂection points does not depend on c, but as 0 increases. both inﬂection points approach the y-axis. ...
View Full Document

{[ snackBarMessage ]}

Ask a homework question - tutors are online