4.5.1 Compute the following definite integrals and compare with
your answer from the earlier problem. ... Compare with Section
4.4, Exercises 9 and 17.
Get solution
4.5.2
Compute the following definite integrals and compare with your answer
from the earlier problem. ... Compare with Section 4.4, Exercises
10 and 18.
Get solution
4.5.3
Compute the following definite integrals and compare with your answer
from the earlier problem. ... Compare with Section 4.4, Exercises
11 and 19.
Get solution
4.5.4
Compute the following definite integrals and compare with your answer
from the earlier problem. ... Compare with Section 4.4, Exercises
12 and 20.
Get solution
4.5.5 Compute the following definite integrals. ...
Get solution
4.5.6 Compute the following definite integrals. ...
Get solution
4.5.7 Compute the following definite integrals. ...
Get solution
4.5.8 Compute the following definite integrals. ...
Get solution
4.5.9 Compute the following definite integrals. ...
Get solution
4.5.10 Compute the following definite integrals. ...
Get solution
4.5.11 Compute the following definite integrals. ...
Get solution
4.5.12 Compute the following definite integrals. ...
Get solution
4.5.13 Compute the following definite integrals. ...
Get solution
4.5.14 Compute the following definite integrals. ...
Get solution
4.5.15 Compute the following definite integrals. ...
Get solution
4.5.16 Compute the following definite integrals. ...
Get solution
4.5.17 Compute the following definite integrals. ...
Get solution
4.5.18 Compute the following definite integrals. ...
Get solution
4.5.19 Compute the definite integrals of the following functions from t =1 to t =2, from t =2 to t =3, and finally from t =1 to t = 3 to check the summation property of definite integrals. ...
Get solution
4.5.20 Compute the definite integrals of the following functions from t =1 to t =2, from t =2 to t =3, and finally from t =1 to t = 3 to check the summation property of definite integrals. ...
Get solution
4.5.21 Compute the definite integrals of the following functions from t =1 to t =2, from t =2 to t =3, and finally from t =1 to t = 3 to check the summation property of definite integrals. ...
Get solution
4.5.22 Compute the definite integrals of the following functions from t =1 to t =2, from t =2 to t =3, and finally from t =1 to t = 3 to check the summation property of definite integrals. ...
Get solution
4.5.23 Compute the definite integrals of the following functions from t =1 to t =2, from t =2 to t =3, and finally from t =1 to t = 3 to check the summation property of definite integrals. ...
Get solution
4.5.24 Compute the definite integrals of the following functions from t =1 to t =2, from t =2 to t =3, and finally from t =1 to t = 3 to check the summation property of definite integrals. ...
Get solution
4.5.25 Another
way to write the Fundamental Theorem of Calculus (sometimes called the
First Fundamental Theorem of Calculus) relates the definite integral and
the derivative. It states that if we treat the definite integral ...as a function of x, then ... for any value of a. Check this in the following cases by computing the definite integral and then taking its derivative. ...
Get solution
4.5.26 Another
way to write the Fundamental Theorem of Calculus (sometimes called the
First Fundamental Theorem of Calculus) relates the definite integral and
the derivative. It states that if we treat the definite integral ...as a function of x, then ... for any value of a. Check this in the following cases by computing the definite integral and then taking its derivative. ...
Get solution
4.5.27 Another
way to write the Fundamental Theorem of Calculus (sometimes called the
First Fundamental Theorem of Calculus) relates the definite integral and
the derivative. It states that if we treat the definite integral ...as a function of x, then ... for any value of a. Check this in the following cases by computing the definite integral and then taking its derivative. ...
Get solution
4.5.28 Another
way to write the Fundamental Theorem of Calculus (sometimes called the
First Fundamental Theorem of Calculus) relates the definite integral and
the derivative. It states that if we treat the definite integral ...as a function of x, then ... for any value of a. Check this in the following cases by computing the definite integral and then taking its derivative. ... with a = 0. Taking the derivative with the chain rule returns the original function.
Get solution
4.5.29 Find
the change in the state variable between the given times first by
solving the differential equation with the given initial conditions and
then by using the definite integral. The change of position by a rock between times t = 1 and t = 5 with position following the differential equation ...= −.8t − 5.0 and initial condition p(0) = 200.
Get solution
4.5.30 Find
the change in the state variable between the given times first by
solving the differential equation with the given initial conditions and
then by using the definite integral. The amount a fish grows between ages t = 1 and t = 5 if it follows the differential equation ...with initial condition L(0) = 5.0.
Get solution
4.5.31 Find
the change in the state variable between the given times first by
solving the differential equation with the given initial conditions and
then by using the definite integral. The amount a fish grows between ages t = 0.5 and t = 1.5 if it follows the differential equation ...with initial condition L(0) = 5.0.
Get solution
4.5.32 Find
the change in the state variable between the given times first by
solving the differential equation with the given initial conditions and
then by using the definite integral. The number of new AIDS cases between 1985 and 1987 if the number of AIDS cases follows ...with initial condition A(0)= 13, 400 and t measured in years since 1981.
Get solution
4.5.33 Find
the change in the state variable between the given times first by
solving the differential equation with the given initial conditions and
then by using the definite integral. The amount of chemical produced between times t = 5 and t = 10 if the amount P follows ...with initial condition P(0.0)= 2.0, and t is measured in minutes and P in moles.
Get solution
4.5.34 Find
the change in the state variable between the given times first by
solving the differential equation with the given initial conditions and
then by using the definite integral. The amount of chemical produced between times t = 5 and t = 10 if the amount P follows ...with initial condition P(0.0) = 2.0, and t is measured in minutes and P in moles.
Get solution
4.5.35 Check
the summation property for the solutions of the differential equations
by showing that change in value of the whole interval is equal to the
change during the first half of the interval plus the change during the
second half of the interval. The position of a rock obeys the differential equation ...=−9.8t − 5.0 with initial condition p(0)=200 (as in Exercise 29.) how that the distance moved between times t = 1 and t = 5 is equal to the sum of the distance moved between t = 1 and t = and the distance moved between t = 3 and t =5.
Get solution
4.5.36 Check
the summation property for the solutions of the differential equations
by showing that change in value of the whole interval is equal to the
change during the first half of the interval plus the change during the
second half of the interval. The growth of a fish obeys the differential equation ...=6.48e−0.09t with initial condition L(0)=5.0 (as in Exercise 30.) how that the growth between times t = 1 and t =5 is equal to the sum of the growth between t = 1 and t = 3 and the growth between t = 3 and t = 5.
Get solution
4.5.37 Check
the summation property for the solutions of the differential equations
by showing that change in value of the whole interval is equal to the
change during the first half of the interval plus the change during the
second half of the interval. The growth of a fish obeys the differential equation ...with initial condition L(0)=5.0 (as in Exercise 31.) Show that the growth between times t = 0.5 and t = 1.5 is equal to the sum of the growth between t = 0.5 and t = 1.0 and the growth between t = 1.0 and t = 1.5.
Get solution
4.5.39 Tw
o rockets are shot from the ground. Each has a different upward
acceleration, and a different amount of fuel. After the fuel runs out,
each rocket falls with an acceleration of −9.8 .... For each rocket,
a. Write down and solve differential equations describing the velocity and position of the rocket while it still has fuel.
b. Find the velocity and height of the rocket when it runs out of fuel. c.
Write down and solve differential equations describing the velocity and
position of the rocket after it has run out of fuel. What is the
initial condition for each?
d. Find the maximum height reached by the rocket. Does it rise more with or without fuel. Why?
e. Find the velocity when it hits the ground. The upward acceleration is 12.0 ...and it has 10 seconds worth of fuel.
Get solution
4.5.39 Tw
o rockets are shot from the ground. Each has a different upward
acceleration, and a different amount of fuel. After the fuel runs out,
each rocket falls with an acceleration of −9.8 .... For each rocket,
a. Write down and solve differential equations describing the velocity and position of the rocket while it still has fuel.
b. Find the velocity and height of the rocket when it runs out of fuel. c.
Write down and solve differential equations describing the velocity and
position of the rocket after it has run out of fuel. What is the
initial condition for each?
d. Find the maximum height reached by the rocket. Does it rise more with or without fuel. Why?
e. Find the velocity when it hits the ground. The upward acceleration is 12.0 ...and it has 10 seconds worth of fuel.
Get solution
4.5.40 Tw
o rockets are shot from the ground. Each has a different upward
acceleration, and a different amount of fuel. After the fuel runs out,
each rocket falls with an acceleration of −9.8 .... For each rocket,
a. Write down and solve differential equations describing the velocity and position of the rocket while it still has fuel.
b. Find the velocity and height of the rocket when it runs out of fuel. c.
Write down and solve differential equations describing the velocity and
position of the rocket after it has run out of fuel. What is the
initial condition for each?
d. Find the maximum height reached by the rocket. Does it rise more with or without fuel. Why?
e. Find the velocity when it hits the ground. The upward acceleration is 2.0 ...and it has 60 seconds worth of fuel.
Get solution
4.5.41
Toward the end of the universe, acceleration due to gravity begins to
break down. Suppose that ... where time is measured in seconds
after the beginning of the end. An object begins falling from 10.0 m
above the ground.
a. Find the velocity at time t.
b. Find the position at time t.
c. Graph acceleration, velocity, and position on the same graph. Which of these measurements are integrals of each other?
d. When will this object hit the ground?
Get solution
