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Unformatted text preview: Math 137 ASSIGNMENT 4 Fall 2008 Submit all boxed problems and all extra problems by 8:20 am. October 10.
All solutions must be clearly stated and fully justiﬁed. Use the format given on UWACE
under Content, in the ﬁle Assignment Format for Math 135 and Math 137. TEXT PROBLEMS:
Section 2.5 — 3, 8, 9, 13, 18, 26, 38, -,-, 55
Section 2.7 - 4, I, 16, 19, M, 49
Section 2.8 — 23,-, -, 33 Section 3.3 — 41,-, -, 46,-,
Page 171— 3, EXTRA PROBLEMS: E4.1 Use continuity theorems and known properties of basic functions: a) to evaluate the limit, if appropriate (if not, explain why): (i) lim 6731—] (ii) lim arcsin(:r + sin 7m)
b) to ﬁnd the interval(s) on which each function is continuous.
a: — a:
' —— ii 111 arctan x2 — 1
(1) “#9,, (> ( ( > )) E42 Use the Squeeze Theorem and known properties of basic functions to provide the proofs
of the following: 1
cos (_) - :1: __ - 3 ' 7T _ - i :rzarctanCr) _
a) Inn —— — 0 b) Inna: sm(—) — 0 c) lime — 1‘"°° 3: 55-‘0 (L’ 3—0 d) Suppose h(a:) satisﬁes 3:4 — 3:2 g Mac) S 3:132 for —2 S a; g 2. What limits of h(x)
could you determine using the Squeeze theorem? Illustrate your explanation with a
sketch. ' 1 E43 a) Determine all points of discontinuity for each of the given functions.
a: 3:2 — 2
i a: = ln( ) ii a: =
()f() $_1 ()9() $4_4
b) When a function has a removable discontinuity at a: = a (see page 120 of your
text), we can deﬁne its continuous extension by simply deﬁning f (a) appropriately.
Do this for any of the functions in part a) which have removeable discontinuities. E44 A 50 cm long rod lies along the x-axis for 0 g a: g 50. The rod has circular cross-
sections with diameters which vary with x. The total mass of the portion of the rod
on [0,x] is given by 6020
NICE) = $+10 g. a) Sketch the graph of M (:c) (rewrite M(:1:), and hence Show that M (at) —> 60 as :c —> 00, and note that its graph is basically a shifted, and scaled version of
1 y = 2: +10
rapidly. ). Hence describe Where the mass of the rod is increasing most b) The average linear density of the rod on a < :1: g b is deﬁned by M(b) — M(a) b _ a g/cm. ﬂ: Find b on each of the intervals [10,11], [10,14], [10,20], and [10, 50]. c) The exact linear density is of the rod at the point :r is deﬁned by 006) = M'Cv) Use the deﬁnition of derivative to ﬁnd IVI’(:E). d) Which of the averages in b) would you expect to best approximate [14’ (10)? Verify
your prediction and explain your reasoning. E45 A desired bend in the direction of a set of railway tracks is sometimes achieved by
means of a segment of a cubic curve, y = (1:133 + (2:62 + etc + d, linking two linear pieces
of track in such a way that the connected tracks have no ’Sharp corners’. If the initial
track direction is y = a: + 2 for a: S 0, and the ﬁnal direction is y = 2 — at, for :E 2 2, d
ﬁnd the required values for a, b, c, d. (Recall that £0133) = 3x2 and diﬁrz) = 2x.)
as Illustrate your answer with a sketch.
CHALLENGE: (Optional: Submit to your instructor in class Friday, if requested.) Show that if f is continuous and 0 g f (:r) S 1 for a: E [0, 1], then there exists 0 in [0, 1]
such that f(c) = c. [HINTz Apply the IVT to g(:r) = f(:z:) — a: on [0, 1].] ...
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- Fall '08