Here's (c)
u(x,t) = cos(x + ct )
ut (x, t) = - sin(x ct) (c) = - (c)sin (x + ct)
So
utt = -(c)cos(x + ct)(c) = -(c2) cos (x + ct)
And
ux (x, t) = -sin(x + ct)(1)= -sin(x + ct)
So
uxx = -cos( x + t)(1) = -cos( x + ct)
So
(c2) uxx = (c2)[-cos( x + ct) ] = -(c2) cos (x + ct)
And ......we have shown that.....
utt = (c2) uxx
∫√(4 - x^2) dx
Let x = 2sinΘ dx = 2cosΘ dΘ ..... when x =0, Θ = 0 and when x = 2, Θ = pi/2 ... so we have
∫√(4 - 4sin^2 Θ) 2cosΘ dΘ =
2∫√[(4(1 - sin^2 Θ) ] cos Θ dΘ =
2 ∫2 √[ 1 - sin^2 Θ) ] cos Θ dΘ =
4 ∫ cos Θ * cos Θ dΘ =
4 ∫ cos ^2 Θ dΘ and using cos^2 Θ = (1/2)(1 + cos2 Θ) , we have
2 ∫ [1 + cos2 Θ] dΘ =
2Θ + sin 2Θ ....and substituting in the limits of integration, we have
2 [ pi/2 - 0 ] + [sin 2 (pi/2) - sin 2(0)] =
pi + [ sin (pi) - sin(0)] =
[pi ] + [0] =
pi = about 3.1416
Here's (b)
q = [P1P2 + 2P1 ] / [P1P2 - 2P2 ].......we can use the quotient rule here.....
∂q/∂P1 = [ [ (P2 + 2)(P1P2 - 2P2) ] - [ (P1P2 + 2P1) (P2) ] ] / [P1P2 - 2P2 ]2 =
[ P1P22 + 2P1P2 - 2P22 - 4P2 - P1P22 - 2P1P2 ] / [P1P2 - 2P2 ]2 =
- [ 2P22 + 4P2 ] / [P1P2 - 2P2 ]2 =
-2P2 [ P2 + 2 ] / [P1P2 - 2P2 ]2
-2P2 [ P2 + 2 ] / [P2 (P1 - 2)]2 =
-2 [ P2 + 2 ] / [P2 (P1 - 2)2 ]
Here's (c)
u(x,t) = cos(x + ct )
ut (x, t) = - sin(x ct) (c) = - (c)sin (x + ct)
So
utt = -(c)cos(x + ct)(c) = -(c2) cos (x + ct)
And
ux (x, t) = -sin(x + ct)(1)= -sin(x + ct)
So
uxx = -cos( x + t)(1) = -cos( x + ct)
So
(c2) uxx = (c2)[-cos( x + ct) ] = -(c2) cos (x + ct)
And ......we have shown that.....
utt = (c2) uxx