الاجوبة النموذجية لاسئلة الامتحان الشهري الاول شعبة B

Questions Form No.(5)
(1) Solve ( x-2)/(x+4)<0 . (2) Find the domain of f(x)=?( x^2-5x+4 ).
(3) If y=?(t )-t and x=?(t )+t find dy/dx
(4) Find y^ for y=2x sin^(-1)?2x+?(1-4x^2 )
(5) Evaluate lim?(x??)???(4x^2+3)/(3x+1)? (6) Evaluate lim?(x?0)??((e^3x-1)^2 cos?x)/x^2 ?
Typical answers(5)
(1) x=-4,2
(-?,-4) ,(-4,2) , (4,?)
0?(-4,2)? (-2)/4<0 ?-4<0 T
S=(-4,2)
(2) x^2-5x+4 ?0
(x-4)(x-1)=0 ?x=1,4
(-?,? 1]?,[1,4],[4,? ?)?
0? (-?,? 1]? ? 4>0 T
?( D=(-?,? 1]??[4,? ?)? )
(3) dy/dt=1/(2?(t ))-1=(1-2?(t ))/(2?(t ))
dx/dt=1/(2?(t ))+1=(1+2?(t ))/(2?(t ))
dy/dx= dy/dt×dt/dx=(1-2?(t ))/(2?(t )) ×(2?(t ))/(1+2?(t ))=(1-2?(t ))/(1+2?(t ))
(4) y^ =2x×1/?(1-4x^2 )×2+2 sin^(-1)?2x+(-8x)/(2?(1-4x^2 ))=2 sin^(-1)?2x
(5) lim?(x??)???(4x^2+3)/(3x+1)?=lim?(x??)???(4+3?x^2 )/(3+1?x)? =( 2 )/( 3 )
(6) lim?(x?0)??((e^3x-1)^2 cos?x)/x^2 ?=(lim?(x?0)??(e^3x-1)/x? )^2×lim?(x?0) cos?x=(lim?(x?0)???3e?^3x/1? )^2×1=9
Questions Form No.(6)
(1) Solve x^2-5x+4<0 . (2) Find the domain of f(x)=?(( x+2)/(x-4) ).
(3) If y=t+?(t ) and x=t-?(t ) find dy/dx
(4) Find y^ for y=3x sin^(-1)?3x+?(1-9x^2 )
(5) Evaluate lim?(x??)???(9x^2+1)/(2x+3)? (6) Evaluate lim?(x?0)??((e^3x-1)^2 cos?2x)/x^2 ?
Typical answers(6)
(1) (x-4)(x-1)=0 ?x=1,4
(-?,? 1)?,(1,4),(4,?)
0? (-?,? 1)? ? 4<0 F
?(S=(1,4) )
(2) ( x+2)/(x-4)?0 ?x=-2,4
(-??,? -2],[-2?,? 4),(4,?)
0? [-2?,? 4) ?( 2)/(-4)?0 F
D= (-?,? 1)??(4,?)
(3) dy/dt=1+1/(2?(t ))=(2?(t )+1)/(2?(t ))
dx/dt=1-1/(2?(t )) =(2?(t )-1)/(2?(t ))
dy/dx= dy/dt×dt/dx=(2?(t )+1)/(2?(t )) ×(2?(t ))/(2?(t )-1)=(2?(t )+1)/(2?(t )-1)
(4) y^ =3x×1/?(1-9x^2 )×3+3 sin^(-1)?3x+(-18x)/(2?(1-9x^2 ))=3 sin^(-1)?3x
(5) lim?(x??)???(9x^2+1)/(2x+3)?=lim?(x??)???(9+1?x^2 )/(2+3?x)? =( 3 )/( 2 )
(6) lim?(x?0)??((e^3x-1)^2 cos?2x)/x^2 ?=(lim?(x?0)??(e^3x-1)/x? )^2×lim?(x?0) cos?2x=(lim?(x?0)???3e?^3x/1? )^2×1=9
Questions Form No.(7)
(1) Solve (x+1)/(x-3)<0 . (2) Find the domain of f(x)=?( x^2-3x+2 ).
(3) If y=?(u )-u and x=?(u )+u find dy/dx
(4) Find y^ for y=2x sin^(-1)?2x+?(1-4x^2 )
(5) Evaluate lim?(x??)??(3x+1)/?(4x^2+3)? (6) Evaluate lim?(x?0)??(e^3x-1)^2/(x^2 cos?x )?
Typical answers(7)
(1) x=-1,3
(-?,-1) ,(-1,3) , (3,?)
0?(-1,3)? 1/(-3)<0 T
S=(-1,3)
(2) x^2-3x+2?0
(x-1)(x-2)=0 ? x=1,2
? (-??,1],[1,2],[2,? ?)?
0? ? (-??,1]? 2>0 T
?( D=? (-??,1]?[2,? ?)? )
(3) dy/du=1/(2?(u ))-1=(1-2?(u ))/(2?(u ))
dx/du=1/(2?(u ))+1=(1+2?(u ))/(2?(u ))
dy/dx= dy/du×du/dx=(1-2?u)/(2?(u )) ×(2?(u ))/(1+2?(u ))=(1-2?(u ))/(1+2?(u ))
(4) y^ =2x×1/?(1-4x^2 )×2+2 sin^(-1)?2x+(-8x)/(2?(1-4x^2 ))=2 sin^(-1)?2x
(5) lim?(x??)??(3x+1)/?(4x^2+3)?=lim?(x??)??(3+1?x)/?(4+3?x^2 )?=( 3 )/( 2 )
(6) lim?(x?0)??(e^3x-1)^2/(x^2 cos?x )?=(lim?(x?0)??(e^3x-1)/x? )^2×lim?(x?0) 1/cos?x =(lim?(x?0)???3e?^3x/1? )^2×1=9

Questions Form No.(8)
(1)Solve x^2-3x+2<0 . (2) Find the domain of f(x)=?((x-1)/(x+3) ).
(3) If y=u-?(u ) and x=u+?(u ) find dy/dx
(4) Find y^ for y=3x sin^(-1)?3x+?(1-9x^2 )
(5) Evaluate lim?(x??)??(2x+1)/?(9x^2+2)? (6) Evaluate lim?(x?0)??(e^3x-1)^2/(x^2 cos?x )?
Typical answers(8)
(1) x^2-3x+2=0
(x-1)(x-2)=0 ? x=1,2
(-?,1),(1,2),(2,?)
0? ? (-??,1]? 2<0 F ? S=(1,2)
(2) (x-1)/(x+3)?0
x=-3,1 ?(-?,-3),(-3,? 1]?,[1,? ?)?
0? (-3,? 1]?? (-1)?3>0 F
D=(-?,-3)?[1,? ?)?
(3) dy/du=1-1/(2?(u ))=(2?(u )-1)/(2?(u ))
dx/du=1+1/(2?(u ))=(2?(u )+1)/(2?(u ))
dy/dx= dy/du×du/dx=(2?u-1)/(2?(u )) ×(2?(u ))/(2?(u )+1)=(2?(u )-1)/(2?(u )+1)
(4) y^ =3x×1/?(1-9x^2 )×3+3 sin^(-1)?3x+(-18x)/(2?(1-9x^2 ))=3 sin^(-1)?3x
(5) lim?(x??)??(2x+1)/?(9x^2+2)? =lim?(x??)??(2+1?x)/?(9+2?x^2 )?=( 2 )/( 3 )
(6) Evaluate lim?(x?0)??(e^3x-1)^2/(x^2 cos?x )?=(lim?(x?0)??(e^3x-1)/x? )^2×lim?(x?0) 1/cos?x =(lim?(x?0)???3e?^3x/1? )^2×1=9