Velocity is positive on (0,1.5)∪(6,7), negative on (1.5,2)∪(5,6), and zero on (2,5).
Acceleration is positive on (5,7), negative on (0,2), and zero on (2,5).
The object is speeding up on (6,7)∪(1.5,2) and slowing down on (0,1.5)∪(5,6).
R(x)=10x−0.001x2
R′(x)=10−0.002x
$6 per item, $0 per item
C′(x)=65
R(x)=143x−0.03x2,R′(x)=143−0.06x
R′(1000)=83,R′(4000)=−97. At a production level of 1000 cordless drills, revenue is increasing at a rate of $83 per drill; at a production level of 4000 cordless drills, revenue is decreasing at a rate of $97 per drill.
P(x)=−0.03x2+78x−75000,P′(x)=−0.06x+78
P′(1000)=18,P′(4000)=−162. At a production level of 1000 cordless drills, profit is increasing at a rate of $18 per drill; at a production level of 4000 cordless drills, profit is decreasing at a rate of $162 per drill
N′(t)=3000(−4t2+400(t2+100)2)
N′(0)=120,N′(10)=0,N′(20)=−14.4,N′(30)=−9.6
The bacteria population increases from time 0 to 10 hours; afterwards, the bacteria population decreases.
N″(0)=0,N″(10)=−6,N″(20)=0.384,N″(30)=0.432. The rate at which the bacteria is increasing is decreasing during the first 10 hours. Afterwards, the bacteria population is decreasing at a decreasing rate.