V. Jaquez
Calculus Part 2 Question
3/25/2015
Question: Let f Let f be be the function given by f(x !
4ince the li5it of the slo,e is 6. 7he gra,h of
(x2"#$(x"% for all x & '% an )hose erivative is
f(x never reaches or goes above 6 (5 8 6 but as
given by f’(x) by f’(x) ! (x2"*x"#$(x"%2 for all x & '%.
it a,,roaches infinity it co5es closer an closer
a +n the the vie)in vie)ing g )ino) )ino) ,rov ,rovie ie belo) belo)-setch the gra,h of f(x
to reaching it.
9eing able to analyze a function an its Gr
erivative an no)ing its li5its- o5ainan range are very i5,ortant for ,assing the P Calculus test. +n this question )e )ill learn ho) to setch a gra,h- fin the range of f using its erivative as 1ustification- fin the li5its of the erivative- an ex,lain ho)
aph of f(x) = (x2+8)/(x+3)
b /in the range of f of f . 0se f’ 0se f’ (x) to (x) to 1ustify your ans)er.
7o solve ,art a )e can change the
y ≥ -4 and y ≤ -8
c /in
lim
)ino) size to exactly )hat )e nee f ( x ) '
because the question tells us )hat it )ants.
❑
x →± ∞
2
7hen )e can fin all the i5,ortant ,oints that give the gra,h its curvature such as
2
x + 6 x + 8 x + 6 x + 8 = ( x + 3 )2 x2 + 6 x + 9
extre5e values or asy5,totes. 7his is very
x x
lim
f ' ( ( x x )
it affects the original gra,h.
5uch the easiest ,art of the ,roble5 because
2 2
1
=
❑
it requires 5ore )or ra)ing then it oes thining.
=1
x →± ∞
;o) )e )ill 5ove on to ,art 9
x,lain )h )hat your an ans)er to to 3 tells
involving the erivative is given by
about the gra,h of f(x.
f’(x) f’ (x) = (x2+6x+8)/(x+3)2 for all x& ('%. +t
V. Jaquez
Calculus Part 2 Question
is i5,ortant for us to no) )hat ty,e of
3/25/2015
iscontinuity this is to unerstan its relationshi, to the original
function. 4ince )e no) it is not efine at ('% )e tae the x an y values to the left an right of it to iscover ho) far the gra,h goes. -'#? an fro5 the right @'=- >. Looing at the erivative the function )e see ho) it starts ecreasing after it ,asses the critical ,oints an a,,roaches '> at ('%- this is an i5,ortant observation for later. 7o ans)er Part C an A- you nee to tae the li5it of the erivative by either exa5ining the gra,h on the calculator or solving it algebraicallyB )e )ill solve it using the latter 5etho in this case. 9y istributing the eno5inator of f’(x) )e ,ut the to, an botto5 of the fraction in the
x
sa5e ter5s- an )e can use the leaing ter5s to fin the li5it. 4ince
x
2 2
1
=
- )e foun the
li5it. Part A )ants to no) ho) the li5it of the slo,e relates to f, )e alreay ans)ere this ho)ever. cross the entire function the slo,e never excees 6- so )e no) )here its increasingan )here it is ecreasing to)ar infinity. +n conclusion- )e no) ho) to ,ro,erly inter,ret a function an analyze the infor5ation that the erivative gives.
