الاجوبة النموذجية لامتحان الشهر الاول - الفصل الاول

1. Find the domain for the function f(x)=?(x^2-x-20 ) ?1?
2.If y=sec^(-1)?2t and x=?(?4t?^2-1) find dy/dx
3. The half-life of radioactive radium is 1600 years. If y_0 is the initial amount of the element, then the amount y remaining after t years is given by y=y_0 e^kt.
If a sample initially contains 50 gm, find the constant k.
4.Evaluate the limit ?lim???(x?2)?? sin^2?(?x)/(e^(3x-6)-3x+5)?


1. Find the domain for the function f(x)=?(x^2+x-20 ) ?2?
2.If y=sec^(-1)?3t and x=?(?9t?^2-1) find dy/dx
3. The half-life of radioactive radium is 1600 years. If y_0 is the initial amount of the element, then the amount y remaining after t years is given by y=y_0 e^kt.
If a sample initially contains 60 gm, find the constant k.
4.Evaluate the limit ?lim???(x?2)?? sin^2?(?x)/(e^(2x-4)-2x+3)?


1. Find the domain for the function f(x)=?(x^2-x-12 ) ?3?
2.If y=csc^(-1)?2t and x=?(?4t?^2-1) find dy/dx
3. The half-life of radioactive radium is 1600 years. If y_0 is the initial amount of the element, then the amount y remaining after t years is given by y=y_0 e^kt.
If a sample initially contains 40 gm, find the constant k.
4.Evaluate the limit ?lim???(x?1)?? sin^2?(?x)/(e^(3x-3)-3x+2)?


1. Find the domain for the function f(x)=?(x^2+x-12 ) ?4?
2.If y=csc^(-1)?3t and x=?(?9t?^2-1) find dy/dx
3. The half-life of radioactive radium is 1600 years. If y_0 is the initial amount of the element, then the amount y remaining after t years is given by y=y_0 e^kt.
If a sample initially contains 70 gm, find the constant k.
4.Evaluate the limit ?lim???(x?2)?? sin^2?(?x)/(e^(2x-4)-2x+3)?
1. Find the domain for the function f(x)=?(x^2-x-20 ) ?1?
x^2-x-20?0
x^2-x-20=0
(x+4)(x-5)=0 ? x=-4 ,5
? (-??,-4] ,[-4,5] and [5,? ?)?
If we choose 0?[-4,5] ?-20?0 False ?
Thus the domain is D=? (-??,-4] ?[5,? ?)?

2.If y=sec^(-1)?2t and x=?(?4t?^2-1) find dy/dx
dy/dt=2/(2t?(?4t?^2-1))=1/(t?(?4t?^2-1)) and dx/dt=8t/(2?(?4t?^2-1))=4t/?(?4t?^2-1)
dy/dx=dy/dt×dt/dx=1/(t?(?4t?^2-1))×?(?4t?^2-1)/4t=( 1 )/( ?4t?^2 )
3. The half-life of radioactive radium is 1600 years. If y_0 is the initial amount of the element, then the amount y remaining after t years is given by y=y_0 e^kt.
If a sample initially contains 50 gm, find the constant k.
25=50e^(k×1600) ? e^1600k=0.5
1600k=ln?0.5
k=-(6931×?10?^(-4))/1600=-4.33×?10?^(-4)

4.Evaluate the limit ?lim???(x?2)?? sin^2?(?x)/(e^(3x-6)-3x+5)?
?lim???(x?2)?? sin^2?(?x)/(e^(3x-6)-3x+5)? =?lim???(x?2)?? (2? sin^?(?x) cos?(?x))/(3e^(3x-6)-3) ?
=?lim???(x?2)?? (? sin^?(2?x))/(3e^(3x-6)-3)?
=?lim? ??(x?2)?? (2?^2 cos?(2?x))/(9e^(3x-6) )?
=(2?^2 cos?(4?))/(9e^(6-6) )=(2?^2)/9

1. Find the domain for the function f(x)=?(x^2+x-20 ) ?2?
x^2+x-20?0
x^2+x-20=0
(x-4)(x+5)=0 ? x=4 ,-5
? (-??,-5] ,[-5,4] and [4,? ?)?
If we choose 0?[-5,4] ?-20?0 False ?
Thus the domain is D=? (-??,-5] ?[4,? ?)?

2.If y=sec^(-1)?3t and x=?(?9t?^2-1) find dy/dx
dy/dt=3/(3t?(?9t?^2-1))=1/(t?(?9t?^2-1)) and dx/dt=18t/(2?(?9t?^2-1))=9t/?(?9t?^2-1)
dy/dx=dy/dt×dt/dx=1/(t?(?9t?^2-1))×?(?9t?^2-1)/9t=( 1 )/( ?9t?^2 )

3. The half-life of radioactive radium is 1600 years. If y_0 is the initial amount of the element, then the amount y remaining after t years is given by y=y_0 e^kt.
If a sample initially contains 60 gm, find the constant k.
30=60e^(k×1600) ? e^1600k=0.5
1600k=ln?0.5
k=-(6931×?10?^(-4))/1600=-4.33×?10?^(-4)
4.Evaluate the limit ?lim???(x?2)?? sin^2?(?x)/(e^(2x-4)-2x+3)?
?lim???(x?2)?? sin^2?(?x)/(e^(2x-4)-2x+3)?=?lim???(x?2)?? (2? sin^?(?x) cos?(?x))/(2e^(2x-4)-2) ?
=?lim???(x?2)?? (? sin^?(2?x))/(2e^(2x-4)-2)?
=?lim? ??(x?2)?? (2?^2 cos?(2?x))/(4e^(2x-4) )?
=(2?^2 cos?(4?))/(4e^(4-4) )=(2?^2)/4=( ?^2)/2

1. Find the domain for the function f(x)=?(x^2-x-12 ) ?3?
x^2-x-12?0
x^2-x-12=0
(x-4)(x+3)=0 ? x=-3 ,4
? (-??,-3] ,[-3,4] and [4,? ?)?
If we choose 0?[-3,4] ?-12?0 False ?
Thus the domain is D=? (-??,-3] ?[4,? ?)?

2.If y=csc^(-1)?2t and x=?(?4t?^2-1) find dy/dx
dy/dt=(-2)/(2t?(?4t?^2-1))=(-1)/(t?(?4t?^2-1)) and dx/dt=8t/(2?(?4t?^2-1))=4t/?(?4t?^2-1)
dy/dx=dy/dt×dt/dx=(-1)/(t?(?4t?^2-1))×?(?4t?^2-1)/4t=(-1 )/( ?4t?^2 )
3. The half-life of radioactive radium is 1600 years. If y_0 is the initial amount of the element, then the amount y remaining after t years is given by y=y_0 e^kt.
If a sample initially contains 40 gm, find the constant k.
20=40e^(k×1600) ? e^1600k=0.5
1600k=ln?0.5
k=-(6931×?10?^(-4))/1600=-4.33×?10?^(-4)

4.Evaluate the limit ?lim???(x?1)?? sin^2?(?x)/(e^(3x-3)-3x+2)?
?lim???(x?1)?? sin^2?(?x)/(e^(3x-3)-3x+2)?=?lim???(x?1)?? (2? sin^?(?x) cos?(?x))/(?3e?^(3x-3)-3) ?
=?lim???(x?1)?? (? sin^?(2?x))/(?3e?^(3x-3)-3)?
=?lim? ??(x?1)?? (2?^2 cos?(2?x))/?9e?^(3x-3) ?
=(?2??^2 cos?(2?))/(9e^(3-3) ) =(2?^2)/9


1. Find the domain for the function f(x)=?(x^2+x-12 ) ?4?
x^2+x-12?0
x^2+x-12=0
(x+4)(x-3)=0 ? x=-3 ,4
? (-??,-4] ,[-4,3] and [3,? ?)?
If we choose 0?[-4,3] ?-12?0 False ?
Thus the domain is D=? (-??,-4] ?[3,? ?)?

2.If y=csc^(-1)?3t and x=?(?9t?^2-1) find dy/dx
dy/dt=(-3)/(3t?(?9t?^2-1))=(-1)/(t?(?9t?^2-1)) and dx/dt=18t/(2?(?9t?^2-1))=9t/?(?4t?^2-1)
dy/dx=dy/dt×dt/dx=(-1)/(t?(?9t?^2-1))×?(?9t?^2-1)/9t=(-1 )/( ?9t?^2 )
3. The half-life of radioactive radium is 1600 years. If y_0 is the initial amount of the element, then the amount y remaining after t years is given by y=y_0 e^kt.
If a sample initially contains 70 gm, find the constant k.
35=70e^(k×1600) ? e^1600k=0.5
1600k=ln?0.5
k=-(6931×?10?^(-4))/1600=-4.33×?10?^(-4)
4.Evaluate the limit ?lim???(x?2)?? sin^2?(?x)/(e^(2x-4)-2x+3)?
=?lim???(x?2)?? (2? sin^?(?x) cos?(?x))/(2e^(2x-4)-2) ?
=?lim???(x?2)?? (? sin^?(2?x))/(2e^(2x-4)-2)?
=?lim? ??(x?2)?? (2?^2 cos?(2?x))/(4e^(2x-4) )?
=(?^2 cos?(4?))/?2e?^(4-4) =?^2/2