Review&for&the&Final&Exam&
No&Calculator&Allowed&
1.&Find&the&limit:&lim$$$$$
%→'
% ( )*
% ( )+%)'
&&
&
&
&
&
&
&
&
&
&
&
&
2.&The&position&of&a&pig&is&given&by&the&equation&, = . ' − 6. + + 9.&&
where&s&is&in&feet&and&t&is&in&seconds.&Include&units&on&all&answers.&
A.!What&is&its&average&velocity&for&0 ≤ . ≤ 5?&
&
&
&
&
B.!What&is&its&instantaneous&velocity&at&t&=&2?&
&
&
&
C.!What&is&its&acceleration&at&t&=&4?&
&
$
3.&Find&the&derivative:&&
A.&6 = (8 9 − 38 + + 5)' &
&
&
&
&
&
&
&
&
&
&
$
$
$
(
B.&6 = 2x 8 + + 1 + ? '% &
&
&
&
&
&
&
&
&
&
&
&
&
$
B
C.&6 = 8.@A 8 + ,CA)D (28)&
&
&
&
& &
&
&
&
&
&
&
&
&
&
&
D.&E 8 = sinh$(cosh 8 )&
&
&
&
&
&
&
&
&
&
&
&
&
&
4.&Find&the&equation&of&the&tangent&line&to&6 = 1 + 4,CA8&&at&the&
point&(0,1).&
&
&
&
&
&
&
&
&
&
&
&
&
LM
5.&Use&implicit&differentiation&to&find& :&&8 + + 486 + 6 + = 13&
L%
&
&
&
&
&
&
&
&
&
&
&
&
&
6.&Consider&y&=&x3&–&6x2&–&15x&+&4&
A.!Using&the&derivative&and&a&sign&graph&determine&the&intervals&
where&y&is&increasing&and&decreasing.&
&
&
&
&
&
&
&
B.!State&any&local&maximum&or&minimum&points.&
&
&
&
&
C.!Find&the&second&derivative&of&y.&
&
&
D.!Determine&the&intervals&were&y&is&concave&up&or&concave&
down&using&the&second&derivative&and&a&sign&graph.&
&
&
&
&
&
&
&
&
&
7.&Evaluate&the&integrals:&&
+
A.& D 88 ' + 38 + O8&
&
&
&
&
&
&
&
&
&
&
&
$
D
B.& Q sin$(3P.) $O.&
&
&
&
&
&
&
&
&
&
&
&
&
&
C.& sin 8 cos$(cos 8 )$O8&
&
&
&
&
&
&
&
&
&
&
$
$
$
D.&
&
&
&
&
&
&
&
&
&
&
&
&
$
R S
%
O8&
TUV$(WX %)
E.&
O8&
%
&
&
&
&
&
&
&
&
&
&
&
&
$
F.& 7? Z% O8&
&
&
&
&
&
&
&
&
&
&
&
&
&
G.&
&
&
&
&
&
&
&
&
&
&
&
&
&
sec 48 tan 48 O8&
H.&
&
&
&
&
&
&
&
&
&
&
&
&
&
%
^% ( _'
$O8&
Calculator&Allowed&
1.&Use&logarithmic&differentiation&to&find&the&derivative&of&
&6 = (cos 8)% &
&
&
&
&
&
&
&
&
&
&
&
&
&
&
&
&
&
$
$
$
$
$
$
$
$
2.&A&pig&colony&starts&with&500&pigs&and&grows&at&a&rate&
proportional&to&its&size.&After&3&hours&there&are&8000&pigs.&Find&
an&exponential&function&for&the&number&of&pigs&after&t&hours&
and&find&the&rate&of&growth&after&t&hours.&
&
&
&
&
&
&
&
&
&
&
&
&
$
$
$
$
$
$
$
$
$
$
$
&
3.&A&pig&farmer&with&2400&feet&of&fencing&wants&to&enclose&a&
rectangular&area&that&borders&a&straight&river.&He&needs&no&fence&
along&the&river.&&The&farmer&wants&to&maximize&the&area&of&the&
pens&so&his&pigs&have&plenty&of&waddling&space.&What&are&the&
dimensions&of&the&rectangular&area&that&has&the&largest&area?&
&
&
&
&
&
&
&
&
&
&
&
&
&
&
&
&
&
$
$
$
$
$
$
$
4.&A&cat&moves&along&a&line&so&that&its&velocity&at&time&t&is&
&` . = . + − .&&(measured&in&meters&per&second).&Find&the&
displacement&and&the&distance&traveled&during&the&time&interval&
[0,5].&You&must&show&the&integral(s)&used&to&answer&the&question.&
&
&
&
&
&
&
&
$
$
$
$
$
$
$
$
$
$
$
$
$
$
$
$
5.&The&volume&of&a&cube&is&increasing&at&a&rate&of&10&cm3/min.&How&
fast&is&the&surface&area&increasing&when&the&length&of&an&edge&is&30&
cm?&
&
&
&
&
&
&
&
&
&
&
&
&
&
&
&
&
&
&
$
$
$
$
$
$
$
$
6.&Predict the limit by finding at least four values of the function
for x near a.
lim
x→1
&
x −1
x 2 −1
&
&
&
&
&
$
$
$
$
$
7.&Find 6′(1) given the graph of y below.
&
&