Calculus: Early Transcendentals
8th Edition
ISBN:9781285741550
Author:James Stewart
Publisher:James Stewart
Chapter1: Functions And Models
Section: Chapter Questions
Problem 1RCC: (a) What is a function? What are its domain and range? (b) What is the graph of a function? (c) How...
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![Net Change
Inverse Property:
If f(x) is integrated over [a,b] and f(x) = F (x) , then
S* f(x) dx = S f(x)dx = F' (x)dx = F(x) = F(b) - F(a)
Net Change Theorem
Applications: Examples follow to help you capture the concept.
V(t) = volume
V'(1) = rate of liquid flow
v'(t)dt = V(t2) - V(t1)
change in amount of water from [t1, t2]
N(t) population at time t
N (t) = rate of growth of population
SN'(t)dt = N(t2) – N(t1)
is net change in population or quantity from [t1, t2]
S(1) = position function of an object that moves along a straight line
V(t) = S'(1) = velocity of particle
Sv(t) = Ss'(t) dt = s(t2) – s(t1)
net change (displacement) in position of particle [t1,t2]
Problem [3]
(1) An object is moving with velocity v(t) = e + 4t -5. Find the displacement and the total distance travelled
from t[0,7]
Displacement: s(t)
Total Distance = S1 v(t)| dt
v(t)dt + - v(t) dt
=](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fec70a3af-7012-41b5-84d0-077491201fef%2Fb23a6153-51f4-45e3-a2fb-a20df1f144a0%2Fvzxejg_processed.png&w=3840&q=75)
Transcribed Image Text:Net Change
Inverse Property:
If f(x) is integrated over [a,b] and f(x) = F (x) , then
S* f(x) dx = S f(x)dx = F' (x)dx = F(x) = F(b) - F(a)
Net Change Theorem
Applications: Examples follow to help you capture the concept.
V(t) = volume
V'(1) = rate of liquid flow
v'(t)dt = V(t2) - V(t1)
change in amount of water from [t1, t2]
N(t) population at time t
N (t) = rate of growth of population
SN'(t)dt = N(t2) – N(t1)
is net change in population or quantity from [t1, t2]
S(1) = position function of an object that moves along a straight line
V(t) = S'(1) = velocity of particle
Sv(t) = Ss'(t) dt = s(t2) – s(t1)
net change (displacement) in position of particle [t1,t2]
Problem [3]
(1) An object is moving with velocity v(t) = e + 4t -5. Find the displacement and the total distance travelled
from t[0,7]
Displacement: s(t)
Total Distance = S1 v(t)| dt
v(t)dt + - v(t) dt
=
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