الاجوبة النموذجية لامتحان الدور الثاني -فصل اول / مرحلة اولى

Q1:Answer ?Two of the following questions ?20 marks?
A. Solve the inequality ?2x?^2+4x?x^2+32

B.Find the domain for the function f(x)=?((x-4)/(x+1) )
C. Placed a metal bar, at a temperature of 100°F in a room with constant temperature of 0°F. After 20 min the temperature of the bar is 50°F. Find a function of the temperature at any time

Q2: A. If y=t-( 1 )/( t ) and x=t+( 1 )/( t ) ,find dy/dx ?10 marks?

B. Find y^ for y=?(?4x?^2-1)-sec^(-1)?2x ?10 marks?

Q3: Evaluate Two of the following limits ?20 marks?
A. lim?(x?0)??(e^x-x-1)/(x sin?x )? B. lim?(x?1/2)??((1-4x^2 ) sin^(-1)?x)/cos?(?x) ? C. lim?(x??)??(x^2+2x-3)^2/(2x+1)^4 ?

Q4: Evaluate Two of the following integrals ?20 marks?
A. ???(x-1)^2/(x^3+x) dx? B.???x^3 e^(-3x) ? dx C.??(?2x?^2+3x+2)/(x+1)^3 dx

Q5:A. The radioactive element radium-226 has a half-life of 1620 years. If a sample initially contains 120 gm, find a function of the amount at any time. ?10 marks?
B.Solve for x: 3 cosh?x- 2 sinh??x ?=3 ?10 marks?





Ass. Lecturer Fouad H. A. Alsharify Prof. Dr. Raheem G. Kadhim
Instructor Head of department
??? With our best wishes ???

Q1: A.x^2-4x-32?0 ? (x+4)(x-8)=0 ? x=-4 ,8
? (-??,-4] ,[-4,8] and [8,? ?)?
0?[-4,8] ? 0-4×0-32?0 ? -32?0 True ?
S=[-4,8]
B. (x-4)/(x+1)?0 such that x?-5
x-4=0 ? x=4 and is x+1=0 ? x=-1
(-?,-1) ,? (-1?,4] and [4,? ?)?
0?? (-1?,4] ? (x-4)/(x+1)?0 ? (0-4)/(0+1)?0 ?-4?0 False ?
Thus the domain is D=(-?,-1)?[4,? ?)?

C. T= T_s+ce^(-kt) & (T_s=0) ? T= ce^(-kt)
t=0 ,T=100°F ? 100=ce^(-k×0) ? c=100 ?T=100e^(-kt)
To find k we have t=20 min , T=50°F ? 50=100e^(-k×20)
e^(-20k)=0.5 ? -20k=ln?(0.5)? ? k=0.0347
T=100e^(-0.0347 t)
Q2: A. dy/dt=1+1/t^2 =(t^2+1)/t^2 and dx/dt=1-1/t^2 =(t^2-1)/t^2
dy/dx=dy/dt×dt/dx=(t^2+1)/t^2 ×t^2/(t^2-1)=(t^2+1)/(t^2-1)
B. y^ =8x/(2?(?4x?^2-1))-2/(2x?(?4x?^2-1))=(?4x?^2-1)/(x?(?4x?^2-1))=?(?4x?^2-1)/x


Q3 A. lim?(x?0)??(e^x-x-1)/(x sin?x )?=lim?(x?0)??(e^x-1)/(x cos?x+sin?x )? =lim?(x?0)??e^x/(x sin?x+cos?x+cos?x )?=( 1 )/( 2 )
B. lim?(x?1/2)??((1-4x^2 ) sin^(-1)?x)/cos?(?x) ?=lim?(x?1/2)?sin^(-1)?x ×lim?(x?1/2)??(1-4x^2)/cos?(?x) ?=sin^(-1)?(1/2) lim?(x?1/2)??(-8x)/(-? sin?(?x) )?
=?/( 6 )×4/(? sin?(?/2) )=?/( 6 )×( 4 )/?=( 2 )/( 3 )
C. lim?(x??)??(x^2+2x-3)^2/(2x+1)^4 ?=(lim?(x??)??(x^2+2x-3)/(2x+1)^2 ? )^2=(lim?(x??)??(x^2+2x-3)/(?4x?^2+4x+1)? )^2
=(lim?(x??)??(1+(2?x)-(3?x^2 ))/(4+(4?x)+(1?x^2 ) )? )^2=(( 1 )/4)^2=( 1 )/16

Q4 A.???(x-1)^2/(x^3+x) dx=? ???(x^2-2x+1)/x(x^2+1) dx?=???(x^2+1)/x(x^2+1) dx?-???2x/x(x^2+1) dx?
=???( 1 )/x dx?-???2/((x^2+1) ) dx?=ln?|x|-2 tan^(-1)?x+c

C. (?2x?^2+3x+2)/(x+1)^3 =A/(x+1)^3 +B/(x+1)^2 +C/((x+1) )
?2x?^2+3x+2=A+B(x+1)+C(x+1)^2?2-3+2=A ? A=1
4x+3=B+2C(x+1)?-4+3=B ? B=-1 and 4=2C ? C=2
??(?2x?^2+3x+2)/(x+1)^3 dx=??(1/(x+1)^3 -1/(x+1)^2 +2/((x+1) )) dx
=(-1)/?2(x+1)?^2 +1/((x+2) )+2 ln?|x+2|+c
Q5:A. A=A_0 e^kt ? 60=120 e^(k×1620) ? e^1620k=0.5?1620k=ln?(0.5)
k=ln?(0.5)/1620=(-0.6931)/1620=-0.00028?A=120e^(-0.00028t)

B. 3 cosh?x- 2 sinh??x ?=3? (3e^x+3e^(-x)-2e^x+2e^(-x))/2=3
e^x+5e^(-x)=6 ? e^2x-6e^x+5=0
(e^x-1)(e^x-5)=0 ? e^x=1 or e^x=5
x=ln?1=0 or x=ln?5=1.609