Thomas Calculus 13th [Solutions]

Section 4.4 Concavity and Curve Sketching 267
67.
y
2
sec x, y ( )
/2 /2
rising on ,
2 2
, never falling
no local extrema;
y 2(sec x)(sec x)(tan x) 2
2 (sec x) (tan x),
y ( | ) concave up on
/2 0 /2
on , 0 ,0 is a point of inflection.
2
0, , concave down
2
68.
y tan x, y ( | ) rising on 0, , falling on
2
/2 0 /2
,0 no local maximum, a local minimum at
2
2
x 0; y sec x, y ( )
/2 /2
concave up on , no points of inflection
2 2
69.
y cot 2
, y ( | ) rising on (0, ), falling on
0 2
( , 2 ) a local maximum at , no
2
local minimum; y 1 csc , y ( ) never concave
2 2
0 2
up, concave down on (0, 2 ) no points of inflection
70.
2
y csc 2
, y ( ) rising on (0, 2 ) , never falling
0 2
no local extrema;
y 2 csc csc cot 1
2 2 2 2
csc 2
2 cot 2
, y ( | )
0 2
concave up on ( , 2 ), concave down on (0, )
a point of inflection at
71.
2
y tan 1 (tan 1)(tan 1),
y ( | | ) rising on
/2 /4 /4 /2
, , falling on
4 2
local minimum at
, a local maximum at
4 4
4 ; 2
y 2 tan sec ,
y ( | ) concave up on
/2 0 /2
on , 0 a point of inflection at 0
2
, and
2 4
4 , a
0, , concave down
2
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