Page 69 - Schaum's Outline of Theory and Problems of Advanced Calculus

60 FUNCTIONS, LIMITS, AND CONTINUITY [CHAP. 3

2
3.37. If f ðxÞ¼ 2x ,0 < x @ 2, find (a)the l.u.b. and (b)the g.l.b. of f ðxÞ.Determine whether f ðxÞ attains its
l.u.b. and g.l.b.
Ans.(a)8, (b)0

3.38. Construct a graph for each of the following functions.

f ðxÞ¼jxj; 3 @ x @ 3 x ½xŠ where ½xŠ¼ greatest integer @ x
x
ðaÞ ð f Þ
jxj
; 2 @ x @ 2 f ðxÞ¼ cosh x
x
ðbÞ f ðxÞ¼ 2 ðgÞ
8
0; x < 0
>
< sin x
1
ðcÞ f ðxÞ¼ 2 ; x ¼ 0 ðhÞ f ðxÞ¼ x
>
1; x > 0
:
x; 2 @ x @ 0 x

x; 0 @ x @ 2 ðx 1Þðx 2Þðx 3Þ
ðdÞ f ðxÞ¼ ðiÞ f ðxÞ¼
2
2
f ðxÞ¼ x sin 1=x; x 6¼ 0 sin x
ðeÞ ð jÞ f ðxÞ¼ 2
x
2
2
2
2
2
2
2
2
2
2
3.39. Construct graphs for (a) x =a þ y =b ¼ 1, (b) x =a y =b ¼ 1, (c) y ¼ 2px, and (d) y ¼ 2ax x ,
where a; b; p are given constants. In which cases when solved for y is there exactly one value of y assigned to
each value of x,thus making possible definitions of functions f , and enabling us to write y ¼ f ðxÞ?In which
cases must branches be defined?
3.40. (a)From the graph of y ¼ cos x construct the graph obtained by interchanging the variables, and from
which cos 1 x will result by choosing an appropriate branch. Indicate possible choices of a principal value
1
1
of cos 1 x.Using this choice, find cos ð1=2Þ cos ð 1=2Þ. Does the value of this depend on the choice?
Explain.
3.41. Work parts (a) and (b)ofProblem 40 for (a) y ¼ sec 1 x, (b) y ¼ cot 1 x.
3.42. Given the graph for y ¼ f ðxÞ,show how to obtain the graph for y ¼ f ðax þ bÞ, where a and b are given
constants. Illustrate the procedure by obtaining the graphs of
(a) y ¼ cos 3x; ðbÞ y ¼ sinð5x þ =3Þ; ðcÞ y ¼ tanð =6 2xÞ.
3.43. Construct graphs for (a) y ¼ e jxj , (b) y ¼ ln jxj, (c) y ¼ e jxj sin x.
3.44. Using the conventional principal values on Pages 44 and 45, evaluate:
1 p ffiffiffi 1 1
(a) sin ð 3=2Þ ( f ) sin x þ cos x; 1 @ x @ 1
1 1 1
(b)tan ð1Þ tan ð 1Þ (g) sin ðcos 2xÞ; 0 @ x @ =2
ffiffiffi
1
1
ffiffiffi
1
p
p
(c) cot ð1= 3Þ cot ð 1= 3Þ (h) sin ðcos 2xÞ; =2 @ x @ 3 =2
2
(d)cosh 1 p ffiffiffi (i) tanh ðcsch 1 3xÞ; x 6¼ 0
1 1 2
(e) e coth ð25=7Þ ( j) cosð2tan x Þ
1 x 4
Ans. (a) =3 (c) =3 (e) 3 (g) =2 2x (i) jxj
4 p ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ð jÞ 1 þ x 4
2
ffiffiffi x 9x þ 1
p
(b) =2 (d)lnð1 þ 2 Þ ( f ) =2 (h)2x 3 =2
1
3.45. Evaluate (a)cosf sinhðln 2Þg, (b)cosh fcothðln 3Þg.
p ffiffiffi
Ans.(a) 2=2; ðbÞ ln 2