CALCULUS I
SOLt./TltJNS
PRACTICEPROBLEMS
CHAPTER 2
'I
~
1. The graph of a function
f is drawn on the right.
Find the following:
..1= (a) z~cc
fun f(x); (b) z-+-2
Jim f(x);
0.: (c) funf(x);
z~O
40 =(e)
(d) z~r
Jim f(x);
Jimf(x);
(f)
-3
;::..t..
!
-2- -J.
z~2
v
1\
.,~
Jimf(x);=.d./I,e,
z~r
J..
:: ~
/
,
-f. -l.
.
3
.'
"
.
(h) What are the horizontal and vertical asymptotes? 1/: ~::::./ , L 1-==;L
.
~~~
~&~
(i) Where isf discontinuous? G) Where isf not differentiable?'
)
(g) fun f(x).
:::
_cc
/f..::; -,) ) :2-
1- =
I
-;).,-/' ~-J 0
~
}
2. Sketch the graph of an example of a single function which satisfies all of the following
conditions:
4
-
~
.
r----
(a)limf(x)=-2; (b)1imf(x)=l; (c)f(O)=-1; (d)limf(x)=cx:>;-- - :3. ,--z~+
z~-
:C~r
(e) :c~2+
funf(x) =-00; (f) z-+-cc
Jimf(x)=3;
(g)Z~<D
Jimf(x) =4 .
.
2
3. Do not use your calculator. Let f(x)
.
= x-2 x
x+l'.
~
Find the fOllO ':
(b). :C~r
llin-j(x);
(c) z~l+
~f~X);
(f) z~";
Jim.f(x);
(g) :c-+-c:o
lim f(x).
(h) Sketch the graph of y
= 1.
I
7,I
.
(a) ;r~r
fun/ex);
.
.
)( (d/~J(x);
z-+-r
.
.
(e) funf(x);;
.
.
=I(x).
~
-r
.
=L
4. Use the Intermediate Value Theorem to show that there
eqUation
2. is a solution .:tothe
~
. e-z1= x sozmewhere
in th~interval (0, 1).
.
U -iUi/:: e: 't.- 1L ~.h
.
~
.
:rvrfJ~ 'f Aba
.
I
:tku.- ":
"-
IH...~o! 11.
~
e
S. For the graph on the right, arrangethe
following numbers in increasing order:
f'(l); f'(2); f'(3);
e.-'t- 'f. ~
~
l(g)
e;'¥--1- = Q.
=
eO:/
(O"jl)~
~
I,(r)
£(fy~~
=.
I
e. :~z <0
e =e.
f(4)- f(2).
2
I
i{ljL~) (~(1)(t{3)
z.;
:;;
4
~
,
,
6. 'If you take a 5 year $20,000 loaD.to buy a car, the total cost (C) of the loan is a
functionof (r), theannualinterestratethatyouarecharged.I.e., C = I(r) .
-I
(a) 'Whatdoes 1(6) =26,000 mean? Whatarethe units? (a) a. ''I..~
k Do.
-;:t;:t;£.4ii:I~,~
(b) What does 1'(6)=1200 mean? V/batarethe units? Lb)~~QJ.Lc£1k<-~4
J.wat
~
(c) Us~ (a)and-(b}w approximate'-f(6:25):- 'Mc1d";(""
(d) Whatis f(O)?
(a) jfk ~,,-j;. .iff; + 1200(;'5'}
=
f!-) lee) = 1&2~ ~()() d:iDU1 ~ 0/
Ua.
-
~~~
7. The graph of a,function
0
~£.
t,~ ~ poo~
0"
J.+
f
I
is drawn on the right. Draw
the graph of I'
x
.
-
= I(x)
8. (a) If the tangent line to the curve y
(b) Make a careful sketch of the graph for I(x)
~
r(x).
...2-
at the point ( 3 ,2)
also goes through the
= cos(x). Belowit sketcha graphfor
y~
Guessa fonnuJafur r(x).
,/'
9. (a) Label, on the graph to the right,
~
~~""'("J=
-1,
bl I
~~
.r
-~C)(J
~t(Y.)~~
.f
'/
f(x); f(x + h); f(x + h) - f(x); h .
(b) Whatis f(x+h)-f(x)?
h
(c) Give the definition for f' (x) .
(d) Use your definition in (c) to
find f'(x)
~,
tw;~ j ~&) ~~~~;) = 2..
point ( 4 , 4), find 1(3) and 1'(3).
y
X~
.
--
,'J
-f-")
if f(x) = .!.+x.
x
.1
)(
1.+"-..
10. The table below gives C(t), the total value of U.S. currency in circulation (in
billionsof dollars) as a function oftime t (years).
(a) Find the average rate of change of the currency over the time interval [1980 , 2000].
(b) Givean estimatefor the valueof C'(1990).
f / )
A;'-/, j ~~,
:=
leA ewe-I~
t
1995
2 00,
C(t)
409
568
I
.
=
::1
f, /,
,
.
(~
.~
!HV r, ~f) 95'"()O
1)-'
,~,g-I ~
.
~Oo()-/'?'l1J
.
~~t
~
()
~
~
~
AC-
4r
=.1.il:: :J19~i
~
:u
'~
;£'910
0
!ttgSj ,cr;b-l
tf.()9-J'ifl
]
==
[ 1f(9,j--IIf'iS-
~~'S-Ui
r