Réponses

2013-03-04T14:42:34+01:00

Ex1
A) 1) a.
EF = BC - 2 * (tan(alpha)* AB/2 = 1 - 2 * (tan(alpha)* 1/2 = 1 - (tan(alpha)
EB² = (AB/2)² + ((tan(alpha)* AB/2)² = 1/4 + tan²(alpha)/4 = (1+tan²(alpha))/4
EB = 1/2cos(alpha)
         b.
f(alpha) = EF + 4 EB = 1 - tan(alpha) + 2/cos(alpha) = 1 + (2 - sin(alpha))/cos(alpha)
     2) a. f'(alpha) = (-cos²(alpha) + sin(alpha)(2-sin(alpha)))/cos²(alpha)
                        = ((sin²(alpha)-1) + sin(alpha)(2-sin(alpha)))/cos²(alpha)
                        = (sin²(alpha)-1 + 2sin(alpha) - sin²(alpha))/cos²(alpha)
                        = (2sin(alpha) - 1)/cos²(alpha)
         b.
f'(alpha) = 0 donne 2sin(alpha) = 1 alors alpha = pi/6 puisque 0 < alpha < pi/4
         c.
f(pi/6) = 1 + (2 - 1/2)/(V3/2) = 1 + V3

 

B) 1) 

BE = 1/2cos(alpha) et 0 < alpha < pi/4

V2/2 < cos(alpha) < 1

V2 < 2cos(alpha) < 2

1/2 < 1/2cos(alpha) < 1/V2 = V2/2

      2)

EF = BC - 2 * V(EB² - 1/2²) = 1 - 2V(x² - 1/4)

g(x) = EF + 4 EB = 4x + 1 - 2V(x² - 1/4)

      3) a.

g'(x) = 4 - 2x/V(x² - 1/4)

           b.