اجابة امتحان التفاضل والتكامل
للصف الثالث الثاوى2009 – لغات
1- (a) * tan x – 3 sin 2x + c
* ( 3x – 1 )5 + c
(b) F/(x) = 0 x = ± 1 C.point. ------- +++ -----
L.max. value( 5) , L.min. value ( -3)
F//(x) = 0 then x = 0 Inflection . point ( 0 , 1)
++++ -------
[ F(X) is incr. at ] 0 , ∞ [ , decr. At ] - ∞ , 0
2- (a) ∫ 3 +4x dx = ∫ 6 y dy then 3x +2x2 + c = 3y2 C =2
Equation of curve : 3x +2x2 + 2 = 3y2
(b) y / = 1-2x /2-2y at y = 0 x2 – x -2 = 0 then x = -1 , x = 2
At (-1 , 0) at ( 2 , 0 )
m = 3/2 m = -3/2
equation of normal:
3 y + 2 x + 2 = 0 3 y – 2 x + 4 = 0
3- (a) A = (4x/5 )(3x/5 )
dA/dt = 24/25 (x) dx/dt then -60 = 24/25 (x) (-2.5)
then x =25 cm dimensions are : 20 , 15 cm
Area of rectangle = 20 * 15 = 300 cm2
x2 – 4 x
(b)(i) F(x) =
4 x - x2
F/(x)+ = lim h2 -4h – 0 = - 4
h 0 h
F/(x)- = lim -h2 +4h – 0 = 4 f(x) not diff.
h 0 h
(ii) F/(x) = 0 then x = 2 C.P.
F( 1 ) = -3
F( 4 ) = 0
F( 1 ) = -4 absolute max. value ( 0 ) , absolute min. value ( - 4 )
4- (a) y (2x+y) = 3 by diff. W. R .to x
y/ (2x+y) + ( 2+ y/ ) y = 0 by diff. W. R .to
y// (2x+y) + ( 2+ y/ ) y/ + y// y + y/ ( 2 +y/ ) = 0
2y//y +2 y2/ + 4 y/ + 2 x y// = 0
2 y / + y// ( x + y ) + y2/ = 0
2 dy/dx + d2y/dx2 ( x+ y ) + (dy/dx )2 = 0
(b) let Ө angle between two sides a , b
A = 0.5 a b sin Ө by diff. W. R .to Ө
dA/d Ө = 0.5 a b cos Ө
cos Ө = 0 then Ө = 900
Then: the third side is a diameter of circle passing its vertices.
5- (a)* at x € ] 1 , 2 [ : it is cont.
at x € ] 2 , 5 [ : it is cont .
* discuss cont . at x =1 : F(1)+ = F(1) = 1 + 1 = 2 it is cont.
* discuss cont . at x =1: F(1)- = F(2) = 2 +1 = 3
F(1)+ = (2)2 + 1 = 5 it is not cont
F(x) is cont . at [ 1 , 5 [ - 2
(b) x y = a y/ = - a x -2
By diff. w.r.to y
dx/dy . y + x = 0
dx/dy = - x/y
d2x/dy2 . y + dx/dy + dx/dy = 0
d2x/dy2 = -2 dx/dy
y
x y = a x/ = - a y -2
By diff. w.r.to x
dy/dx . x + y = 0
dy/dx = - y/x
d2y/dx2 . x + dy/dx + dy/dx = 0
d2y/dx2 = -2 dy/dx
x
- 2 dy/dx . -2dx/dy > - y . –x
X y x y
4 dy/dx . dx/dy > x y 4 (- a x -2 ) (- a y -2 ) > x y
4 a2 > x y a2 > x3y3
x2y2 4
a > x3y3
4
مع دوام النجاج الباهر
أ / أحمد البدرى