AP Calculus AB Unit 2 Differential Calculus Applications Drill Solutions questions with accurate answers

Unit,2—Differential,,
Calculus,Applications,
Drill,Solutions
SOLUTIONS,TO,UNIT,2,DRILL

1. y,−,4,=,−(x,−,2)


,,, R,emember,that,the,equation,of,a,line,through,a,point,(x,y1,1,,),with,slope,m,is,y−,=y1,m,x(,−x

1).,,We,find,the,y-

coordinate,by,plugging,x,=,2,into,the,equation,y,=,8x,,,and,we,find,the,slope,by,plugg

ing,x,=,2,into,the,derivative,of,the,equation.


, First,,we,find,the,y-
,

coordinate,,y1 :,y,=,8,2(,),=,4.,This,means,that,the,line,passes,through,the,
,




point
(2,4,,),.

dy 4 dy
=,4 =1.,
,



, , Next,,we,take,the,derivative:,=
.,Now,,we,can,find,the,slope,,m:
dx 8x dx,x=2 8,2(,)

,Cracking,the,AP,Calculus,AB,Exam


However,,this,is,the,slope,of,the,tangent,line.,The,normal,line,is,perpendicular,to,the,ta
ngent,line,,so,its,slope,will,be,the,negative,reciprocal,of,the,tangent,line’s,slope.,In,thi

s,case,,the,slope,of,the,normal,line,is,

, =,−1.,Finally,,we,plug,in,the,point,(2,4, ),and,the,slope,m=−1,to,get,the,equation,of,th
,




e,normal,line:,y,−,4,=,−(x,−,2).,

2. y,=−3,4x,+

, ,

Remember,that,the,equation,of,a,line,through,a,point,(x,y1 1,,),with,slope,m,is,y−,=
,




y1 m x(,−x1).,,We,find,the,slope,by,plugging,x,=,0,into,the,derivative,of,the,equation,y,=,
,
,




−4,3x−x2 .,First,,we,take,the,derivative:,
,




dy,
=,−3,2−,x,.,Now,,we,can,find,the,slope,,m:,dy =,−3,−,3(,)0,=,−3.,,dx dx,x=0

Finally,,we,plug,in,the,point,(0,4, ),and,the,slope,m,=,−3,to,get,the,equation,of,the,tang
,




ent,line:,y,−,=−,−4,3(x,0)or,y,=−3,4x,+,.

3. y,=0


,,, Remember,that,the,equation,of,a,line,through,a,point,,(x,y1,,1),with,slope,m,is,y−,=y1 m x(,−x1).,
,
,




We,find,the,y-coordinate,by,plugging,x,=,–

2,into,the,equation,y,=,(x2 +,4x,+,4)2 ,,and,we,find,the,slope,by,plugging,x,=,–
, ,




2,into,the,derivative,of,the,equation.

, , First,,we,find,the,y-coordinate,,y1 :,y,= −((,2)2 +4(−2)+4)2 =0,.,This,means,that,the,line,passes,
,
, , ,




through,the,point,(−2,0,,).



,2,|,Unit,2—Differential,Calculus,Applications,Drill,Solutions

, Cracking,the,AP,Calculus,AB,Exam



dxdy,=2(x2 +4x,+4 2)(,x,+4),.,Now,,we,can,find,the,slope,,,,,
, ,


Next,,we,take,the,derivative:,


m:,dxdy,x,2,=,−2((,2)2,+,−4(,2)+4,2)(,(−2)+4)=0.,Finally,,we,plug,in,the,point,(−2,0,,),
and,the,
=−,slope,m=0,to,get,the,equation,of,the,tangent,line:,y,−,0,=,
0(x,+,2)or,y,=0,.

4. c,
, , , ,The,Mean,Value,Theorem,says,that,if,f(x),is,continuous,on,the,interval,[a,,b],and,is,dif-



ferentiable,everywhere,on,the,interval,(a,,b),,then,there,exists,at,least,one,number,c,on,
the,interval,(a,,b),such,that, f,′(,)c,=,f b( )− f a( ) .,Here,,the,function,is,b,−,a
, , ,, , ,




f(x),=,x3,+,12x2,+,7x,and,the,interval,is,[–4,,4].,Thus,,the,Mean,Value,Theorem,says,

((,)4,3 +12,4(,)2 +,7,4(,))−,((−4)3 +12(−4)2 +,7(−4))
, , , ,



that, f,′(,)c,= ., This, simplifies, to,,
(4,4+,)

f′(c),=,23.,Next,,we,need,to,find,f′(c).,The,derivative,of,f(x),is,f′(x),=,3x2,+,24x,+,7,,so,,


f′(c),=,3c2,+,24c,+,7.,Now,,we,can,solve,for,c:,3c2,+,24c,+,7,=,23,and,c,=,


,, . Note,that,c, , is,in,the,interval,(–4,,4),,but,

, is,not,in,the,interval.,Thus,,the,answer,is,only,c,
,, . It’s,very,important,to,check,that,the,answers,you,get,for,c,fall,in,the,

given,interval,when,doing,Mean,Value,Theorem,problems.

5. c,=±
, ,
Rolle’s,Theorem,says,that,if,f(x),is,continuous,on,the,interval,[a,,b],and,is,differentiable,every-
Unit,2—Differential,Calculus,Applications,Drill,Solutions,|,3