Thomas Calculus 13th [Solutions]

280 Chapter 4 Applications of Derivatives
127. The graph of f falls where f 0, rises where f 0,
and has horizontal tangents where f 0. It has
local minima at points where f changes from
negative to positive and local maxima where f
changes from positive to negative. The graph of f is
concave down where f 0 and concave up where
f 0. It has an inflection point each time f
changes sign, provided a tangent line exists there.
128. The graph f is concave down where f 0, and
concave up where f 0. It has an inflection point
each time f changes sign, provided a tangent line
exists there.
4.5 INDETERMINATE FORMS AND LHÔPITALS RULE
1. l Hôpital: lim x 2 1 1
2
4 2 x 2
x 2 x x 4
or lim x 2 lim x 2 lim 1 1
2
x 2 x 4 x 2
( x 2)( x 2)
x 2
x 2 4
2. l Hôpital:
lim sin 5x
5cos5x
1
5 x 0
x 0
x
or lim sin 5x
5 lim sin 5x
0
x
5 0
5x
5 1 5
x
x
3. l Hôpital:
2
lim 5x 3x lim 10x
3 10 5
2
7 1 14 lim
x x x
x
x
14 7
or
lim 2
5x
3x
lim 5
x 5
x 7x
1
7
2 1
x 7
x
2
3
4. l Hôpital:
3 2
lim x 1 lim 3x
3
3 2
x 1 4x x 3 x 112x
1 11
or
2
3 ( x 1) x x 1
2
lim x 1 lim lim x x 1 3
3 2
2
x 1 4x x 3 x 1 ( x 1) 4x 4x
3 x 1 4x 4x
3 11
5. l Hôpital: lim 1 cos x lim sin x cos 1
2
0 0
2 lim x
x x x
x
x 0
2 2
2
lim sin x lim sin x sin x 1 1
2
x 0 x (1 cos x)
x 0
x x 1 cos x 2
1 cos x (1 cos x)
1 cos x
0 x x 0 x 1 cos x
or lim lim
2 2
x
6. l Hôpital:
2
lim 2x 3x lim 4x
3 lim 4
3 2
1 3 1 6x
0
x x x x x x
or
2 3
2
x x
x x
2
3
1
1 1
x 1 x
x
2
x
3
lim 2 3 lim 0
1
0
x x
7. lim x 2 lim 1 1
2
x 2 x 4 x 2
2x
4
8.
lim x
2 25 2
5
5 lim x
x
5
1
10
x
x
9.
3 2 3( 3) 4
lim t 4t 15 lim 3t
4 23
2
t 3 t t 12 t 3
2t
1 2( 3) 1 7
2
10.
3 2
lim 3t
3 9 9
3
lim t
2
t 1 4t t 3 t 1 12t
1 11
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