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Unformatted text preview: 0 1+t+t
Solution:
1
( 1 + 2x)
Note that f 0 (x) =
= (1 + x + x2 ) 1 and f 00 (x) =
and the only potential
1 + x + x2
(1 + x + x2 )2
00 (x) > 0 if x < 1/2 and f 00 (x) < 0 if x > 1/2 and so f (x) is
point of inﬂection is x = 1/2 and f
concave up on ( 1, 1/2) and concave down on ( 1/2, +1).
p = d
dx cos2 (y ) p Concavity Test ^ _

1/2 6. The velocity function (in meters per second) of a particle in motion is given by v (t) = 3t 5, ﬁnd
(a) the displacement and (b) the total distance traveled by the particle in the ﬁrst 3 seconds.
Solution:
Z
(a) Here, 3 v (t) dt =
0 Z 3 3t 5 dt =
0 3 3t 2
2 5t =
0 27
15 =
2 3/2. So the displacement is 1.5 meters. (b) Note, total distance is given by,
Z 3 v (t) dt = 0 Z 3
0  3t 5 dt =
= Z 5/3 (5
0 5t 3t 2
2 3t) dt +
!
5 /3 Z + 0 3 (3t
5/3 3t2
2 5) dt
!
3 5t 5/3 = 41
6 So the total distance travelled is 41/6 meters.
Z2
7. (a) Find an approximation to the integral
(2 x2 ) dx using a Riemann sum with a regular partition
0...
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 Fall '07
 SEGUR,HARV

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