CHAPTER 2 ADDITIONAL AND ADVANCED EXERCISES
3.1 THE DERIVATIVE OF A FUNCTION
1. Step 1: f(x)œ 4 x and f(x# h)œ 4 (x h)#
Step 2: f(x h)h f(x) œc4 (x h)h#d a4 x#b œ a4 x# 2xhhh#b 4 x# œ 2xhhh# œ h( 2xh h)
2xœ h
Step 3: f (x)w œ lim ( 2x h)œ 2x; f (w$ œ) 6, f (0)w œ0, f (1)w œ 2
hÄ !
2. F(x)œ(x1)#1 and F(xh)œ(x h 1)# " Ê F (x)w œ lim
hÄ !
c(x h 1) 1d c(x 1) 1d h
# #
lim lim lim (2x h 2)
œ œ œ
hÄ ! hÄ ! hÄ !
ax 2xh h 2x 2h 1 1b ax 2x 1 1b
h h
2xh h 2h
# # # #
2(x 1); F ( 1) 4, F (0) 2, F (2) 2 œ w œ w œ w œ
3. Step 1: g(t)œt"# and g(th)œ (t"h)#
Step 2: g(t h)h g(t) œ #" h "# œ h œ t (tth) t h2thh œ(t2thh) t hh
# #
(t h) t # #
t (t h) (t h) t
Œ † # a# #b
# # # #
#
† †
œ h( 2t(th) t h# #h)œ (t2th) t# #h
Step 3: g (t)w œ lim œ œ ; g ( 1)w œ2, g (2)w œ ", gw 3 œ
hÄ !
2t h 2t 2 2
(t h) t# # t t# #† t$ 4 ŠÈ ‹ 3 3
È
4. k(z)œ 1 z#z and k(zh)œ 12(z(zh)h) Ê k (z)w œ lim h
hÄ !
Š" # (z(zh)h)" #zz‹
lim lim lim lim
œ œ œ œ
hÄ ! hÄ ! hÄ ! hÄ !
(1 z h)z ( z)(z h)
(z h)zh 2(z h)zh 2(z h)zh (z h)z
z z zh z h z zh h
"
# # # # "
; k ( ) , k (1) , k 2
œ "2z# w" œ "# w œ "# wŠÈ ‹œ "4
5. Step 1: p( )) œÈ3 and p() )h)œÈ3()h)
Step 2: p( h)h p( ) 3( hh) 3 3 3hh 3 3 3h 3 (3 3h) 3
3 3h 3 h 3 3h 3
) ) ) ) ) ) ) ) ) )
) ) ) )
œ È È œŠÈ È ‹ ŠÈ È ‹ œ
ŠÈ È ‹ ŠÈ È ‹
† œ 3h œ 3
hŠÈ3)3hÈ3)‹ È3)3hÈ3)
Step 3: p ( )w lim ; p (1)w , p (3)w , pw
"
# #
) œ œ œ œ œ œ
hÄ !
3 3 3 3 2 3
3 3h 3 3 3 2 3 2 3 3 2
È ) È ) È ) È ) È) È ˆ ‰ È
6. r(s)œÈ2s1 and r(sh)œÈ2(s Êh) 1 r (s)w œ lim
hÄ !
È2s 2h 1 È2s 1 h
lim lim
œ œ
hÄ ! hÄ !
ŠÈ È ‹ ŠÈ È ‹
ŠÈ È ‹ ŠÈ È ‹
2s h 1 2s 1 2s 2h 1 2s 1
h 2s 2h 1 2s 1 h 2s 2h 1 2s 1
(2s 2h 1) (2s 1)
†
lim lim
œ œ œ œ
hÄ ! hÄ !
2h 2 2 2
hŠÈ2s2h 1 È2s1‹ È2s2h 1 È2s1 È2s 1 È2s1 2È2s1
; r (0) 1, r (1) , r
œ È2s"1 w œ w œ È"3 w "# œÈ"
ˆ ‰ 2
7. yœf(x)œ2x and f(x$ h)œ2(xh) $ Ê dydxœ lim 2(xh)h2x œ lim 2 x 3x h3xhh h 2x
hÄ ! hÄ !
$ $ a$ # # $b $
lim lim lim 6x 6xh 2h 6x
œ œ œ œ
hÄ ! hÄ ! hÄ !
6x h 6xh 2h
h h
h 6x 6xh 2h
# # $ a # #b # # #
a b
8. rœ Ês#$ 1 drds œ lim 1h 1 œ#" lim (sh)$ h2 s$2
hÄ ! hÄ !
” • ’ “
c d c d
(s $h) s
# #
$
lim lim lim 3s 3sh h s
œ "# œ "# œ"# # # œ # #
hÄ ! hÄ ! hÄ !
s 3s h 3sh h 2 s 2 3
h h
h 3s 3sh h
$ # # $ $ c # #d
a b
9. sœr(t)œ 2t 1t and r(th)œ 2(t h) 1t h Ê dsdt œ lim h
hÄ !
Š2(tb btbh)h 1‹ ˆ2tbt1‰
lim lim
œ œ
hÄ ! hÄ !
Š(tbh)(2t(2tbb c2h1)b1)(2tt(2tbb2h1)b1)‹
h (2t 2h 1)(2t 1)h
(th)(2t 1) t(2t2h1)
lim lim lim
œ œ œ
hÄ ! hÄ ! hÄ !
2t t 2ht h 2t 2ht t h
(2t 2h 1)(2t 1)h (2t 2h 1)(2t 1)h (2t 2h 1)(2t 1)
# # "
œ (2t1)(2t" 1) œ (2t"1)#
10. dvdt œ lim (t h) h t œ lim h h œ lim h
hÄ ! hÄ ! hÄ !
’ " “ ˆ ‰ " Š ‹
t h t h (t h)t
h(t h)t t (t h)
" "
t t
lim lim 1
œ œ œ œ
hÄ ! hÄ !
ht h t h t ht 1 t 1
h(t h)t (t h)t t t
# # # #
# #
"
11. pœf(q)œ Èq"1 and f(qh)œÈ(q "h) 1 Ê dpdqœ lim Š h‹Š ‹
hÄ !
" "
È(q h) 1 Èq 1
lim lim
œ œ
hÄ ! hÄ !
Œ È È
È È
È È
Èq 1 qÈh 1
q h 1 q 1
b c b b
b b b
h
q 1 q h 1
h q h 1 q1
lim lim
œ œ
hÄ ! hÄ !
ˆÈ È ‰ ˆÈ È ‰
È È ˆÈ È ‰ È È ˆÈ È ‰
q 1 q h 1 q 1 q h 1
h q h 1 q 1 q 1 q h 1 h q h 1 q 1 q 1 q h 1
(q 1) (q h 1)
†
lim lim
œ œ
hÄ ! hÄ ! "
h
hÈq h 1Èq 1ˆÈq 1 Èq h 1‰ Èq h 1Èq 1ˆÈq 1 Èq h 1‰
œ Èq1Èq1ˆ"Èq 1 Èq1‰œ 2(q1)"Èq1
12. dwdz lim h lim 3w 2 3w 3h 2
h 3w 3h 2 3w 2
œ œ
hÄ ! hÄ !
Š ‹ È È
È È
" "
È3(w h) 2È3w2
lim œhÄ !
ŠÈ È ‹ ŠÈ È ‹
È È ŠÈ È ‹
3w 2 3w 3h 2 3w 2 3w 3h 2
h 3w 3h 2 3w 2 3w 2 3w 3h 2
†
lim œhÄ !
(3w 2) (3w 3h 2)
h 3w 3h 2 3w 2 3w 2 3w 3h 2
È È ŠÈ È ‹
lim
œ œ
hÄ !
3 3
3w 3h 2 3w 2 3w 2 3w 3h 2 3w 2 3w 2 3w 2 3w 2
È È ŠÈ È ‹ È È ŠÈ È ‹
œ 2(3w2)3È3w2
13. f(x)œ x 9x and f(xh)œ(x h) (x9h) Ê f(x h)h f(x)œ ’(x h) (xb9hh)“’x9x“ œ x(xh)#9xx(xx (x#h)hh) 9(xh) œx$2x h# xh#x(x9xh)hx$ x h# 9x9h œx h#x(xxh#h)h9h
; f (x) lim 1 ; m f ( 3) 0
œ h(xx(x#xhh)h9) œx#x(xxhh)9 œ x#x(xxhh)9 œx#x#9 œ x9# œ œ
w w
hÄ !
14. k(x)œ # "x and k(xh)œ2 (x" h) Ê k (x)w œ lim k(x h)h k(x) œ lim h
hÄ ! hÄ !
Š# "x h# "x‹
lim lim lim ;
œ œ œ œ
hÄ ! hÄ ! hÄ !
( x) (2 x h)
h(2 x)(2 x h) h(2 x)(2 x h) (2 x)( x h) (2 x)
# h
"# "
#
k (2)w œ 16"
15. dsdt œ lim (t h) (t hh) t t œ lim t 3t h 3th h h t 2th h t t
hÄ ! hÄ !
c $ #da$#b a$ # # $ba# #b $ #
lim lim lim 3t 3th h 2t h
œ œ œ
hÄ ! hÄ ! hÄ !
3t h 3th h 2th h
h h
h 3t 3th h 2t h
# # $ # a # # b # #
a b
3tœ #2t; mœ dsdt¸œ"œ5
t
16. dydxœ lim (x h 1)h (x 1) œ lim (x 1) 3(x ) h 3(xh 1)h h (x 1)
hÄ ! hÄ !
"
$ $ $ # # $ $
lim 3(x 1) 3(x 1)h h 3(x 1) ; m 3
œ œ œ œ
hÄ !c # #d # dydx¹
x=#
17. f(x)œÈx82 and f(xh)œ È(x 8h) 2 Ê f(x h)h f(x)œ h
8 8
(x h) 2 x 2
È b c È c
œ 8 h xx 2h 2x xh 22 x 2 x h 2 œ
x 2 x h 2 h x h 2 x 2 x 2 x h 2
8[(x 2) (x h 2)]
ŠÈ È ‹ ŠÈ È ‹
È È ŠÈ È ‹ È È ŠÈ È ‹
†
f (x) lim
œ Ê œ
w
8h 8
hÈx h 2Èx 2ŠÈx 2 Èx h 2‹ hÄ ! Èx h 2Èx 2ŠÈx 2 Èx h 2‹
; m f (6) the equation of the tangent
œ œ œ œ œ Ê"
w
8 4 4 #
x 2 x 2 x 2 x 2 (x 2) x 2 4 4
È È ŠÈ È ‹ È È
line at (6 4) is yß œ 4 "#(x6)Ê œ y "#x $ % Ê œ y "#x (.
18. g (z)w lim lim
œ œ
hÄ ! hÄ !
ˆ È ‰ Š È ‹ ŠÈ È ‹ ŠÈ È ‹
ŠÈ È ‹
1 4 (z h) 1 4 z 4 z h 4 z 4 z h 4 z
h h † 4 z h 4 z
lim lim lim ;
œ œ œ œ
hÄ ! hÄ ! hÄ !
(4 z h) (4 z)
h 4 z h 4 z h 4 z h 4 z 4 z h 4 z
h
2 4 z
" "
ŠÈ È ‹ ŠÈ È ‹ ŠÈ È ‹ È
mœg (3)w œ 2È"43 œ Ê"# the equation of the tangent line at ($ß #) is w œ 2 "#(z3)
w z w z .
Ê œ # Ê œ "# $# "# (#
19. sœf(t)œ 1 3t and f(t# h)œ 1 3(th)# œ 1 3t#6th3h # Ê dsdt œ lim f(t h)h f(t)
hÄ !
lim lim ( 6t 3h) 6t 6
œ œ œ Ê œ
hÄ ! hÄ !
a1 3t 6th 3hb a1 3tb
h dt
# # # ds¸t="
20. yœf(x)œ " "x and f(xh)œ 1 x"h Ê dxdy œ lim f(x h)h f(x)œ lim 1 h 1
hÄ ! hÄ !
Š x"h‹ Š "x‹
lim lim lim
œ œ œ œ Ê œ
hÄ ! hÄ ! hÄ !
" "
xx h
" " "
h x(x h)h x(x h) x dx 3
h dy
# ¹
x=È3
21. rœf( )) œ È42) and f()h)œÈ4 2() h) Ê drd) œ lim f() h)h f( )) œ lim h
hÄ ! hÄ !
2 2
4 h 4
È c c) È c)
lim lim
œ œ
hÄ ! hÄ !
2 4 2 4 h 2 4 h
h 4 4 h h 4 4 h
2 2 4 h
2 4 4 h
È È È È
È È È È
Š È È ‹
Š È È ‹
# %
% #
) ) ) )
) ) ) )
) )
) )
†
lim lim
œ œ
hÄ ! hÄ !
4( ) 4( h)
2h 4 4 h 4 4 h 4 4 h 4 h
% % 2
%
) )
) ) ) ) ) ) ) )
È È ŠÈ È ‹ È È ŠÈ È ‹
œ 2 œ Ê dr œ
(4 ) 2 4 (4 ) 4 d 8
" "
œ!
) ŠÈ )‹ )È ) ) )¸
22. wœf(z)œ z Èz and f(zh)œ (z h) Èz Êh dw œ lim
dz h
f(z h) f(z) hÄ !
lim lim lim 1
œ œ œ
hÄ ! hÄ ! hÄ !
Š È ‹ ˆ È ‰ È È È È ŠÈ È ‹
ŠÈ È ‹
z h z h z z z h z
h h h
h z h z z h z
z h z
– † —
1 lim 1 lim
œ œ œ " Ê œ
hÄ ! hÄ !
(z h) z
h z h z z h z 2 z
dw 5
dz 4
" "
ŠÈ È‹ È È È ¸
z 4œ
23. f xwa bœz limÄxf za bz xf xa b œz limÄxz #"zxx #" œz limÄxazax # #x zbab #azbax #bb œz limÄxazx zbax #zbax #b œz limÄxaz #"bax #b œ ax" #b#
24. f xw lim lim lim " " lim
" "
Ò " " ÓÒ "
a bœzÄxf z f xa bz xa b œzÄxaz""b#z xax""b# œzÄxazaxx zbab# b aa#zx b#b# œzÄxax b az b ax b az " Ób " "
az x zba b a#x b#
lim lim
œzÄxazaxx zbaz xba "b a#zx "2bb# œzÄxaz" "axb a#xz "2bb# œ" # #axa "x b%b œ # "axa "x b%b œax# "b$
25. g xwa bœz limÄxg z g xa bz xa b œz limÄxzc "zz xx "x œz limÄxaz xza " x zbab "x zabax " "bb œz limÄxazx zbaz "xbax "b œz limÄxaz ""bax "b œ ax" "b#
26. g xwa bœz limÄxg z g xa bz xa b œz limÄxˆ" Èzz ‰ ˆ " x Èx‰ œz limÄxÈz zÈxx†ÈÈzzÈÈxx œz limÄxaz xbˆz ÈzxÈx‰ œzlimÄxÈz"Èx œ#È"x
27. Note that as x increases, the slope of the tangent line to the curve is first negative, then zero (when xœ0), then positive Ê the slope is always increasing which matches (b).
28. Note that the slope of the tangent line is never negative. For x negative, f (x) is positive but decreasing as x#w increases. When xœ0, the slope of the tangent line to x is 0. For x0, f (x) is positive and increasing. This#w graph matches (a).
29. f (x) is an oscillating function like the cosine. Everywhere that the graph of f has a horizontal tangent we$ $ expect f to be zero, and (d) matches this condition.$w
30. The graph matches with (c).
31. (a) f is not defined at xw œ0, 1, 4. At these points, the left-hand and right-hand derivatives do not agree.
For example, lim slope of line joining ( 0) and ( ) but lim slope of
xÄ !c xÄ !b
f(x) f(0) f(x) f(0)
x 0 x 0
œ %ß !ß # œ #" œ
line joining (0 2) and (ß "ß œ 2) 4. Since these values are not equal, f (0)w œ lim does not exist.
xÄ !
f(x) f(0) x 0
(b)
32. (a) (b) Shift the graph in (a) down 3 units
33.
34. (a) (b) The fastest is between the 20 and 30 days;th th
slowest is between the 40 and 50 days.th th
35. Left-hand derivative: For h0, f(0h)œf(h)œh (using y# œx curve) # Ê lim
hÄ !c
f(0 h) f(0) h
lim lim h 0;
œ œ œ
hÄ !c hÄ !c
h 0
h
#
Right-hand derivative: For h0, f(0h)œf(h)œh (using yœx curve) Ê lim
hÄ !b
f(0 h) f(0) h
lim lim 1 1;
œ œ œ
hÄ !b hÄ !b h 0
h
Then lim lim the derivative f (0) does not exist.
hÄ !c hÄ !b
f(0 h) f(0) f(0 h) f(0)
h Á h Ê w
36. Left-hand derivative: When h ! , 1 h 1 Ê f(1h)œ2 Ê lim œ lim
hÄ !c hÄ !c
f(1 h) f(1)
h h
22
lim 0 0;
œ œ
hÄ !c
Right-hand derivative: When h ! , 1 h 1 Ê f(1h)œ2(1h)œ 2 2h Ê lim
hÄ !b
f(1 h) f(1) h
lim lim lim 2 2;
œ œ œ œ
hÄ !b hÄ !b hÄ !b
(2 2h) 2
h h
2h
Then lim lim the derivative f (1) does not exist.
hÄ !c hÄ !b
f(1 h) f(1) f(1 h) f(1)
h Á h Ê w
37. Left-hand derivative: When h0, 1 h 1 Ê f(1h)œÈ1 Êh lim
hÄ !c
f( h) f(1) h
"
lim lim lim lim ;
œ œ œ œ œ
hÄ !c hÄ !c hÄ !c hÄ !c
È ŠÈ ‹ ŠÈ ‹
ŠÈ ‹ ŠÈ ‹ È
1 h
h h
1 h 1 h
1 h 1 h 1 h
(1 h)
1 h 1 " " "
"
" " "
† #
Right-hand derivative: When h0, 1 h 1 Ê f(1h)œ2(1 œh) 1 2h Ê1 lim
hÄ !b
f( h) f(1) h
"
lim lim 2 2;
œ œ œ
hÄ !b hÄ !b
(2h 1) h "
Then lim lim the derivative f (1) does not exist.
hÄ !c hÄ !b
f(1 h) f(1) f(1 h) f(1)
h Á h Ê w
38. Left-hand derivative: lim lim lim 1 1;
hÄ !c hÄ !c hÄ !c
f(1 h) f( ) (1 h)
h h
" œ "œ œ
Right-hand derivative: lim lim lim
hÄ !b hÄ !b hÄ !b
f(1 h) f( )
h h h
" "
œ Š1"h ‹ œ Š11(1hh)‹
lim lim 1;
œ œ œ
hÄ !b hÄ !b "
h
h(1 h) 1 h
Then lim lim the derivative f (1) does not exist.
hÄ !c hÄ !b
f(1 h) f(1) f(1 h) f(1)
h Á h Ê w
39. (a) The function is differentiable on its domain $ Ÿ Ÿx 2 (it is smooth) (b) none
(c) none
40. (a) The function is differentiable on its domain # Ÿ Ÿx 3 (it is smooth) (b) none
(c) none
41. (a) The function is differentiable on $ Ÿ x 0 and ! Ÿx 3 (b) none
(c) The function is neither continuous nor differentiable at xœ0 since lim f(x)Á lim f(x)
xÄ !c xÄ !b
42. (a) f is differentiable on # Ÿ " x 1, x 0, 0 x 2, and 2 Ÿx 3
(b) f is continuous but not differentiable at xœ 1: lim f(x)œ0 exists but there is a corner at xœ 1 since
xÄ 1
lim 3 and lim 3 f ( 1) does not exist
hÄ !c hÄ !b
f( 1 h) f( ) f( h) f( 1)
h h
" œ " œ Ê w
(c) f is neither continuous nor differentiable at xœ0 and xœ2:
at xœ0, lim f(x)œ3 but lim f(x)œ0 Ê lim f(x) does not exist;
xÄ !c xÄ !b xÄ0
at xœ2, lim f(x) exists but lim f(x)xÄ # xÄ # Áf(2) 43. (a) f is differentiable on " Ÿ x 0 and 0 Ÿx 2
(b) f is continuous but not differentiable at xœ0: lim f(x)œ0 exists but there is a cusp at xœ0, so
xÄ !
f (0)w œ lim does not exist
hÄ !
f(0 h) f(0) h
(c) none
44. (a) f is differentiable on $ Ÿ x 2, 2 x 2, and 2 Ÿx 3
(b) f is continuous but not differentiable at xœ 2 and xœ2: there are corners at those points (c) none
45. (a) f (x)w œ lim œ lim œ lim œ lim ( 2x h)œ 2x
hÄ ! hÄ ! hÄ ! hÄ !
f(x h) f(x)
h h h
(x h)# a x#b x# 2xh h# x#
(b)
(c) ywœ 2x is positive for x0, y is zero when xw œ0, y is negative when xw 0
(d) yœ x is increasing for # _ x 0 and decreasing for ! _x ; the function is increasing on intervals where yw0 and decreasing on intervals where yw0
46. (a) f (x)w œ lim œ lim œ lim œ lim " œ "
hÄ ! hÄ ! hÄ ! hÄ !
f(x h) f(x) x (x h)
h h x(x h)h x(x h) x
Šxc"h x1‹
#
(b)
(c) y is positive for all xw Á0, y is never 0, y is never negativew w (d) yœ "x is increasing for _ x 0 and ! _x
47. (a) Using the alternate formula for calculating derivatives: f (x)w œz lim Äx f(z) z f(x)x œz lim Äx Šz3z xx3‹
$ $
lim lim lim x f (x) x
œzÄx 3(z z$xx)$ œzÄx (z x) z 3(z a#x) zx x #b œzÄx z#zx 3x# œ #Ê w œ # (b)
(c) y is positive for all xw Á0, and ywœ0 when xœ0; y is never negativew
(d) yœx3$ is increasing for all xÁ0 (the graph is horizontal at xœ0) because y is increasing where yw0; y is never decreasing
48. (a) Using the alternate form for calculating derivatives: f (x)w œz lim Äx f(z) z f(x)x œz lim Äx Œ z x
z x
4 4
% %
lim lim lim x f (x) x
œzÄx 4(z z% xx)% œzÄx (zx) za$4(z xz#x) x z x# $b œzÄx z$ xz#4 x z# x$ œ $ Ê w œ $ (b)
(c) y is positive for xw 0, y is zero for xw œ0, y is negative for xw 0 (d) yœx4% is increasing on 0 _x and decreasing on _ x 0
49. ywœx lim Äc f(x)xf(c)c œx lim Äc xx$cc$ œx lim Äc (x c) x ax#cxc c#b œx lim xÄca #xcc#bœ3c .#
The slope of the curve yœx at x$ œc is ywœ3c . Notice that 3c# # 0 for all c Ê yœx never has a negative$ slope.
50. Horizontal tangents occur where ywœ0. Thus, ywœ lim
hÄ !
2 x h 2 x
h
È È
lim lim lim .
œ œ œ œ
hÄ ! hÄ ! hÄ !
2 x h x x h x
h x h x h x h x
2((x h) x)) 2
x h x x
ŠÈ È ‹ ŠÈ È ‹
ŠÈ È ‹ ŠÈ È ‹ È È È"
†
Then ywœ0 when Èx" œ0 which is never true Ê the curve has no horizontal tangents.
51. ywœ lim œ lim
hÄ ! hÄ !
a2(x h) 13(x h) 5b a2x 13x 5b
h h
2x 4xh 2h 13x 13h 5 2x 13x 5
# # # # #
lim lim (4x 2h 13) 4x 13, slope at x. The slope is 1 when 4x 13
œ œ œ œ "
hÄ ! hÄ !
4xh 2h 13h h#
4x 12 x 3 y 2 3 13 3 5 16. Thus the tangent line is y 16 ( 1)(x 3) Ê œ Ê œ Ê œ † # † œ œ
y x and the point of tangency is (3 16).
Ê œ "$ ß
52. For the curve yœÈx, we have ywœ lim œ lim
hÄ ! hÄ !
ŠÈ È ‹ ŠÈ È‹
ŠÈ È‹ ŠÈ È ‹
x h x x h x
h x h x x h x h
(x h) x
†
lim . Suppose a is the point of tangency of such a line and ( ) is the point
œ œ +ß "ß !
hÄ ! Èx "h Èx #È"x ˆ È ‰
on the line where it crosses the x-axis. Then the slope of the line is aÈa( 1)0 œ aÈa1 which must also equal
; using the derivative formula at x a 2a a 1 a 1. Thus such a line does
" "
#
2 a a
a a 1
È È
œ Ê È œ Ê œ Ê œ
exist: its point of tangency is ("ß "), its slope is #Èa" œ "#; and an equation of the line is y œ1 "#(x1)
y x .
Ê œ "# "#
53. No. Derivatives of functions have the intermediate value property. The function f(x)œ Ú Ûx satisfies f(0)œ0 and f(1)œ1 but does not take on the value anywhere in ["# !ß " Ê] f does not have the intermediate value property. Thus f cannot be the derivative of any function on [!ß " Ê] f cannot be the derivative of any function on (_ß _).
54. The graphs are the same. So we know that for f(x)œk kx , we have f (x)w œ k kxx .
55. Yes; the derivative of f is f so that f (x ) exists w w ! Ê f (x ) exists as well.w !
56. Yes; the derivative of 3g is 3g so that g (7) exists w w Ê 3g (7) exists as well.w
57. Yes, lim can exist but it need not equal zero. For example, let g(t) mt and h(t) t. Then g(0) h(0)
tÄ ! g(t)
h(t) œ œ œ
0, but lim lim lim m m, which need not be zero.
œ œ œ œ
tÄ ! tÄ ! tÄ !
g(t)
h(t) t
mt
58. (a) Suppose f(x)k kŸx for # " Ÿ Ÿx 1. Then f(0)k kŸ0 # Ê f(0)œ0. Then f (0)w œ lim
hÄ !
f(0 h) f(0) h
lim lim . For h 1, h f(h) h h h f (0) lim 0
œ œ Ÿ Ÿ Ÿ Ê Ÿ Ÿ Ê œ œ
hÄ ! hÄ ! hÄ !
f(h) 0 f(h) f(h) f(h)
h h k k # # h w h
by the Sandwich Theorem for limits.
(b) Note that for xÁ0, f(x)k kœ¸x sin # "x¸œk k kx# sin xkŸk kx# †1œx (since # " Ÿsin xŸ1). By part (a), f is differentiable at xœ0 and f (0)w œ0.
59. The graphs are shown below for hœ1, 0.5, 0.1. The function yœ2È"x is the derivative of the function yœÈx so that #È" œ lim È È . The graphs reveal that yœÈ È gets closer to yœ#È"
x x
x h x x h x
h h
hÄ !
as h gets smaller and smaller.
60. The graphs are shown below for hœ2, 1, 0.5. The function yœ3x is the derivative of the function y# œx so$ that 3x#œ lim . The graphs reveal that yœ gets closer to yœ3x as h#
hÄ !
(x h) x (x h) x
h h
$ $ $ $
gets smaller and smaller.
61. Weierstrass's nowhere differentiable continuous function.
62-67. Example CAS commands:
: Maple
f := x -> x^3 + x^2 - x;
x0 := 1;
plot( f(x), x=x0-5..x0+2, color=black, title="Section 3_1, #62(a)" );
q := unapply( (f(x+h)-f(x))/h, (x,h) ); # (b) L := limit( q(x,h), h=0 ); # (c) m := eval( L, x=x0 );
tan_line := f(x0) + m*(x-x0);
plot( [f(x),tan_line], x=x0-2..x0+3, color=black, linestyle=[1,7], title="Section 3.1 #62(d)", legend=["y=f(x)","Tangent line at x=1"] );
Xvals := sort( [ x0+2^(-k) $ k=0..5, x0-2^(-k) $ k=0..5 ] ): # (e) Yvals := map( f, Xvals ):
evalf[4](< convert(Xvals,Matrix) , convert(Yvals,Matrix) >);
plot( L, x=x0-5..x0+3, color=black, title="Section 3.1 #62(f)" );
: (functions and x0 may vary) (see section 2.5 re. RealOnly ):
Mathematica
<<Miscellaneous`RealOnly`
Clear[f, m, x, y, h]
x0= 1/4;
f[x_]:=x 2Cos[x]
Plot[f[x], {x, x0 3, x03}]
q[x_, h_]:=(f[x h] f[x])/h m[x_]:=Limit[q[x, h], h Ä 0]
ytan:=f[x0]m[x0] (xx0)
Plot[{f[x], ytan},{x, x0 3, x03}]
m[x01]//N
m[x01]//N
Plot[{f[x], m[x]},{x, x03, x03}]