Explain why or why not Determine whether the following statements are true and give an explanation or counterexample. a. A region R is revolved about the y -axis to generate a solid S . To find the volume of S , you could use either the disk/washer method and integrate with respect to y or the shell method and integrate with respect to x . b. Given only the velocity of an object moving on a line, it is possible to find its displacement, but not its position. c. If water flows into a tank at a constant rate (for example, 6 gal/min), the volume of water in the tank increases according to a linear function of time. d. The variable y = t + 1 doubles in value whenever t increases by 1 unit. e. The function y = Ae 0.1 t increases by 10% when t increases by 1 unit. f. ln xy = (ln x )(ln y ). g. sinh ( ln x ) = x 2 − 1 2 x .

Question

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Answer 1

Textbook Question

Chapter 6, Problem 1RE

Explain why or why not Determine whether the following statements are true and give an explanation or counterexample.

a. A region R is revolved about the y-axis to generate a solid S. To find the volume of S, you could use either the disk/washer method and integrate with respect to y or the shell method and integrate with respect to x.
b. Given only the velocity of an object moving on a line, it is possible to find its displacement, but not its position.
c. If water flows into a tank at a constant rate (for example, 6 gal/min), the volume of water in the tank increases according to a linear function of time.
d. The variable y = t + 1 doubles in value whenever t increases by 1 unit.
e. The function y = Ae^0.1t increases by 10% when t increases by 1 unit.
f. ln xy = (ln x)(ln y).
g. $sinh ( ln x ) = x 2 − 1 2 x$ .

(a)

Expert Solution

To determine

Whether the statement “A region R is revolved about the y axis to generate a solid S. To find the volume S, you could use either the disk/ washer method and integrate with respect to y or the shell method and integrate with respect to x.” is true or false.

Answer to Problem 1RE

The statement is true.

Explanation of Solution

The shell method is a method of finding volumes by decomposing a solid of revolution into cylindrical shells.

Suppose that a thin vertical strip is revolved about the y axis. An object of revolution (one that looks like a cylindrical shell or an empty tin can with the top and bottom removed) is obtained.

Then the resulting volume of the cylindrical shell is the surface area of the cylinder times the thickness of the cylindrical wall, which is shown below.

$V=2π∫abx[f(x)]dx$ , where x is the distance to the axis of revolution, $f(x)$ is the length and dx is the width.

Disk method models the resulting three dimensional shape as a stack of an infinite number of disks of varying radius and infinitesimal thickness.

This is in contrast to shell integration which integrates along the axis perpendicular to the axis of revolution.

$V=∫ab[R(x)]2dx$ , where $R(x)$ is the length and dx is the width.

Thus, the given statement is true.

(b)

Expert Solution

To determine

Whether the statement “Given only the velocity of an object moving on a line, it is possible to find its displacement, but not its position.” is true or false.

Answer to Problem 1RE

The statement is true.

Explanation of Solution

In order to find the displacement between two time periods, it is required to integrate the velocity between given time periods.

To evaluate the position at any time, that is either the initial position of the object at any given time or at least the position of the object at a particular time is required.

Thus, the given statement is true.

(c)

Expert Solution

To determine

Whether the statement is “If water flows into a tank at a constant rate (for example 6gal/min.), the volume of water in the tank increases according to a linear function of time” is true or false.

Answer to Problem 1RE

The statement is true.

Explanation of Solution

If $dVdt$ is constant, then V is linear function of time.

Here the rate of inflow is in units of volume per unit time.

Hence, the change in volume in the tank will also be linear.

Thus, the given statement is true.

d.

Expert Solution

To determine

To explain: Whether the statement “The variable $y=t+1$ doubles in value whenever t increases by 1 unit” is true or false.

Explanation of Solution

Suppose $t=2$ , the value of y becomes,

$y=2+1=3$

If t is increased by 1 unit, $t=3$ ,

$y=2+1=3$

Hence, the value of y will not be doubled whenever t increases by 1 unit.

Therefore, the given statement is false.

e.

Expert Solution

To determine

To explain: Whether the statement “The function $y=Ae0.1t$ increases by 10% when t increases by 1 unit” is true or false.

Explanation of Solution

Suppose $t=2$ , then

$y1=Ae0.1×2=Ae0.2≈1.221A$

If t is increased by 1 unit, that is $t=3$ ,

$y2=Ae0.1×3=Ae0.3≈1.35A$

Hence, the percentage increase of both values is approximately 10.57%.

Therefore, the given statement is false.

f.

Expert Solution

To determine

To explain: Whether the statement $lnxy=(lnx)(lny)$ is true or false.

Explanation of Solution

Consider the equation $lnxy=(lnx)(lny)$ .

Suppose $x=2 and y=3$ .

Compute the value of $lnxy$ .

$ln(2×3)=ln6≈1.791$

Compute the value of $(lnx)(lny)$ .

$(lnx)(lny)=(ln2)(ln3)=0.693×1.09≈0.762$

Clearly, $ln6≠(ln2)(ln3)$ .

Therefore, the given statement is false.

g.

Expert Solution

To determine

To explain: Whether the statement $sinh(lnx)=x2−12x$ is true or false.

Explanation of Solution

Compute $sinh(lnx)$ as follows.

$sinh(lnx)=elnx−e−lnx2 [∵sinhx=ex−e−x2]=elnx−elnx−12 [∵lnxa=alnx]=x−1x2 [∵elnx=x]=x2−12x$

Hence, $sinh(lnx)=x2−12x$ .

Therefore, the given statement is true.

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Answer 2

Textbook Question

Chapter 6, Problem 1RE

Explain why or why not Determine whether the following statements are true and give an explanation or counterexample.

a. A region R is revolved about the y-axis to generate a solid S. To find the volume of S, you could use either the disk/washer method and integrate with respect to y or the shell method and integrate with respect to x.
b. Given only the velocity of an object moving on a line, it is possible to find its displacement, but not its position.
c. If water flows into a tank at a constant rate (for example, 6 gal/min), the volume of water in the tank increases according to a linear function of time.
d. The variable y = t + 1 doubles in value whenever t increases by 1 unit.
e. The function y = Ae^0.1t increases by 10% when t increases by 1 unit.
f. ln xy = (ln x)(ln y).
g. $sinh ( ln x ) = x 2 − 1 2 x$ .

(a)

Expert Solution

To determine

Whether the statement “A region R is revolved about the y axis to generate a solid S. To find the volume S, you could use either the disk/ washer method and integrate with respect to y or the shell method and integrate with respect to x.” is true or false.

Answer to Problem 1RE

The statement is true.

Explanation of Solution

The shell method is a method of finding volumes by decomposing a solid of revolution into cylindrical shells.

Suppose that a thin vertical strip is revolved about the y axis. An object of revolution (one that looks like a cylindrical shell or an empty tin can with the top and bottom removed) is obtained.

Then the resulting volume of the cylindrical shell is the surface area of the cylinder times the thickness of the cylindrical wall, which is shown below.

$V=2π∫abx[f(x)]dx$ , where x is the distance to the axis of revolution, $f(x)$ is the length and dx is the width.

Disk method models the resulting three dimensional shape as a stack of an infinite number of disks of varying radius and infinitesimal thickness.

This is in contrast to shell integration which integrates along the axis perpendicular to the axis of revolution.

$V=∫ab[R(x)]2dx$ , where $R(x)$ is the length and dx is the width.

Thus, the given statement is true.

(b)

Expert Solution

To determine

Whether the statement “Given only the velocity of an object moving on a line, it is possible to find its displacement, but not its position.” is true or false.

Answer to Problem 1RE

The statement is true.

Explanation of Solution

In order to find the displacement between two time periods, it is required to integrate the velocity between given time periods.

To evaluate the position at any time, that is either the initial position of the object at any given time or at least the position of the object at a particular time is required.

Thus, the given statement is true.

(c)

Expert Solution

To determine

Whether the statement is “If water flows into a tank at a constant rate (for example 6gal/min.), the volume of water in the tank increases according to a linear function of time” is true or false.

Answer to Problem 1RE

The statement is true.

Explanation of Solution

If $dVdt$ is constant, then V is linear function of time.

Here the rate of inflow is in units of volume per unit time.

Hence, the change in volume in the tank will also be linear.

Thus, the given statement is true.

d.

Expert Solution

To determine

To explain: Whether the statement “The variable $y=t+1$ doubles in value whenever t increases by 1 unit” is true or false.

Explanation of Solution

Suppose $t=2$ , the value of y becomes,

$y=2+1=3$

If t is increased by 1 unit, $t=3$ ,

$y=2+1=3$

Hence, the value of y will not be doubled whenever t increases by 1 unit.

Therefore, the given statement is false.

e.

Expert Solution

To determine

To explain: Whether the statement “The function $y=Ae0.1t$ increases by 10% when t increases by 1 unit” is true or false.

Explanation of Solution

Suppose $t=2$ , then

$y1=Ae0.1×2=Ae0.2≈1.221A$

If t is increased by 1 unit, that is $t=3$ ,

$y2=Ae0.1×3=Ae0.3≈1.35A$

Hence, the percentage increase of both values is approximately 10.57%.

Therefore, the given statement is false.

f.

Expert Solution

To determine

To explain: Whether the statement $lnxy=(lnx)(lny)$ is true or false.

Explanation of Solution

Consider the equation $lnxy=(lnx)(lny)$ .

Suppose $x=2 and y=3$ .

Compute the value of $lnxy$ .

$ln(2×3)=ln6≈1.791$

Compute the value of $(lnx)(lny)$ .

$(lnx)(lny)=(ln2)(ln3)=0.693×1.09≈0.762$

Clearly, $ln6≠(ln2)(ln3)$ .

Therefore, the given statement is false.

g.

Expert Solution

To determine

To explain: Whether the statement $sinh(lnx)=x2−12x$ is true or false.

Explanation of Solution

Compute $sinh(lnx)$ as follows.

$sinh(lnx)=elnx−e−lnx2 [∵sinhx=ex−e−x2]=elnx−elnx−12 [∵lnxa=alnx]=x−1x2 [∵elnx=x]=x2−12x$

Hence, $sinh(lnx)=x2−12x$ .

Therefore, the given statement is true.

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Answer 3

Textbook Question

Chapter 6, Problem 1RE

Explain why or why not Determine whether the following statements are true and give an explanation or counterexample.

a. A region R is revolved about the y-axis to generate a solid S. To find the volume of S, you could use either the disk/washer method and integrate with respect to y or the shell method and integrate with respect to x.
b. Given only the velocity of an object moving on a line, it is possible to find its displacement, but not its position.
c. If water flows into a tank at a constant rate (for example, 6 gal/min), the volume of water in the tank increases according to a linear function of time.
d. The variable y = t + 1 doubles in value whenever t increases by 1 unit.
e. The function y = Ae^0.1t increases by 10% when t increases by 1 unit.
f. ln xy = (ln x)(ln y).
g. $sinh ( ln x ) = x 2 − 1 2 x$ .

(a)

Expert Solution

To determine

Whether the statement “A region R is revolved about the y axis to generate a solid S. To find the volume S, you could use either the disk/ washer method and integrate with respect to y or the shell method and integrate with respect to x.” is true or false.

Answer to Problem 1RE

The statement is true.

Explanation of Solution

The shell method is a method of finding volumes by decomposing a solid of revolution into cylindrical shells.

Suppose that a thin vertical strip is revolved about the y axis. An object of revolution (one that looks like a cylindrical shell or an empty tin can with the top and bottom removed) is obtained.

Then the resulting volume of the cylindrical shell is the surface area of the cylinder times the thickness of the cylindrical wall, which is shown below.

$V=2π∫abx[f(x)]dx$ , where x is the distance to the axis of revolution, $f(x)$ is the length and dx is the width.

Disk method models the resulting three dimensional shape as a stack of an infinite number of disks of varying radius and infinitesimal thickness.

This is in contrast to shell integration which integrates along the axis perpendicular to the axis of revolution.

$V=∫ab[R(x)]2dx$ , where $R(x)$ is the length and dx is the width.

Thus, the given statement is true.

(b)

Expert Solution

To determine

Whether the statement “Given only the velocity of an object moving on a line, it is possible to find its displacement, but not its position.” is true or false.

Answer to Problem 1RE

The statement is true.

Explanation of Solution

In order to find the displacement between two time periods, it is required to integrate the velocity between given time periods.

To evaluate the position at any time, that is either the initial position of the object at any given time or at least the position of the object at a particular time is required.

Thus, the given statement is true.

(c)

Expert Solution

To determine

Whether the statement is “If water flows into a tank at a constant rate (for example 6gal/min.), the volume of water in the tank increases according to a linear function of time” is true or false.

Answer to Problem 1RE

The statement is true.

Explanation of Solution

If $dVdt$ is constant, then V is linear function of time.

Here the rate of inflow is in units of volume per unit time.

Hence, the change in volume in the tank will also be linear.

Thus, the given statement is true.

d.

Expert Solution

To determine

To explain: Whether the statement “The variable $y=t+1$ doubles in value whenever t increases by 1 unit” is true or false.

Explanation of Solution

Suppose $t=2$ , the value of y becomes,

$y=2+1=3$

If t is increased by 1 unit, $t=3$ ,

$y=2+1=3$

Hence, the value of y will not be doubled whenever t increases by 1 unit.

Therefore, the given statement is false.

e.

Expert Solution

To determine

To explain: Whether the statement “The function $y=Ae0.1t$ increases by 10% when t increases by 1 unit” is true or false.

Explanation of Solution

Suppose $t=2$ , then

$y1=Ae0.1×2=Ae0.2≈1.221A$

If t is increased by 1 unit, that is $t=3$ ,

$y2=Ae0.1×3=Ae0.3≈1.35A$

Hence, the percentage increase of both values is approximately 10.57%.

Therefore, the given statement is false.

f.

Expert Solution

To determine

To explain: Whether the statement $lnxy=(lnx)(lny)$ is true or false.

Explanation of Solution

Consider the equation $lnxy=(lnx)(lny)$ .

Suppose $x=2 and y=3$ .

Compute the value of $lnxy$ .

$ln(2×3)=ln6≈1.791$

Compute the value of $(lnx)(lny)$ .

$(lnx)(lny)=(ln2)(ln3)=0.693×1.09≈0.762$

Clearly, $ln6≠(ln2)(ln3)$ .

Therefore, the given statement is false.

g.

Expert Solution

To determine

To explain: Whether the statement $sinh(lnx)=x2−12x$ is true or false.

Explanation of Solution

Compute $sinh(lnx)$ as follows.

$sinh(lnx)=elnx−e−lnx2 [∵sinhx=ex−e−x2]=elnx−elnx−12 [∵lnxa=alnx]=x−1x2 [∵elnx=x]=x2−12x$

Hence, $sinh(lnx)=x2−12x$ .

Therefore, the given statement is true.

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Answer 4

Textbook Question

Chapter 6, Problem 1RE

Explain why or why not Determine whether the following statements are true and give an explanation or counterexample.

a. A region R is revolved about the y-axis to generate a solid S. To find the volume of S, you could use either the disk/washer method and integrate with respect to y or the shell method and integrate with respect to x.
b. Given only the velocity of an object moving on a line, it is possible to find its displacement, but not its position.
c. If water flows into a tank at a constant rate (for example, 6 gal/min), the volume of water in the tank increases according to a linear function of time.
d. The variable y = t + 1 doubles in value whenever t increases by 1 unit.
e. The function y = Ae^0.1t increases by 10% when t increases by 1 unit.
f. ln xy = (ln x)(ln y).
g. $sinh ( ln x ) = x 2 − 1 2 x$ .

(a)

Expert Solution

To determine

Whether the statement “A region R is revolved about the y axis to generate a solid S. To find the volume S, you could use either the disk/ washer method and integrate with respect to y or the shell method and integrate with respect to x.” is true or false.

Answer to Problem 1RE

The statement is true.

Explanation of Solution

The shell method is a method of finding volumes by decomposing a solid of revolution into cylindrical shells.

Suppose that a thin vertical strip is revolved about the y axis. An object of revolution (one that looks like a cylindrical shell or an empty tin can with the top and bottom removed) is obtained.

Then the resulting volume of the cylindrical shell is the surface area of the cylinder times the thickness of the cylindrical wall, which is shown below.

$V=2π∫abx[f(x)]dx$ , where x is the distance to the axis of revolution, $f(x)$ is the length and dx is the width.

Disk method models the resulting three dimensional shape as a stack of an infinite number of disks of varying radius and infinitesimal thickness.

This is in contrast to shell integration which integrates along the axis perpendicular to the axis of revolution.

$V=∫ab[R(x)]2dx$ , where $R(x)$ is the length and dx is the width.

Thus, the given statement is true.

(b)

Expert Solution

To determine

Whether the statement “Given only the velocity of an object moving on a line, it is possible to find its displacement, but not its position.” is true or false.

Answer to Problem 1RE

The statement is true.

Explanation of Solution

In order to find the displacement between two time periods, it is required to integrate the velocity between given time periods.

To evaluate the position at any time, that is either the initial position of the object at any given time or at least the position of the object at a particular time is required.

Thus, the given statement is true.

(c)

Expert Solution

To determine

Whether the statement is “If water flows into a tank at a constant rate (for example 6gal/min.), the volume of water in the tank increases according to a linear function of time” is true or false.

Answer to Problem 1RE

The statement is true.

Explanation of Solution

If $dVdt$ is constant, then V is linear function of time.

Here the rate of inflow is in units of volume per unit time.

Hence, the change in volume in the tank will also be linear.

Thus, the given statement is true.

d.

Expert Solution

To determine

To explain: Whether the statement “The variable $y=t+1$ doubles in value whenever t increases by 1 unit” is true or false.

Explanation of Solution

Suppose $t=2$ , the value of y becomes,

$y=2+1=3$

If t is increased by 1 unit, $t=3$ ,

$y=2+1=3$

Hence, the value of y will not be doubled whenever t increases by 1 unit.

Therefore, the given statement is false.

e.

Expert Solution

To determine

To explain: Whether the statement “The function $y=Ae0.1t$ increases by 10% when t increases by 1 unit” is true or false.

Explanation of Solution

Suppose $t=2$ , then

$y1=Ae0.1×2=Ae0.2≈1.221A$

If t is increased by 1 unit, that is $t=3$ ,

$y2=Ae0.1×3=Ae0.3≈1.35A$

Hence, the percentage increase of both values is approximately 10.57%.

Therefore, the given statement is false.

f.

Expert Solution

To determine

To explain: Whether the statement $lnxy=(lnx)(lny)$ is true or false.

Explanation of Solution

Consider the equation $lnxy=(lnx)(lny)$ .

Suppose $x=2 and y=3$ .

Compute the value of $lnxy$ .

$ln(2×3)=ln6≈1.791$

Compute the value of $(lnx)(lny)$ .

$(lnx)(lny)=(ln2)(ln3)=0.693×1.09≈0.762$

Clearly, $ln6≠(ln2)(ln3)$ .

Therefore, the given statement is false.

g.

Expert Solution

To determine

To explain: Whether the statement $sinh(lnx)=x2−12x$ is true or false.

Explanation of Solution

Compute $sinh(lnx)$ as follows.

$sinh(lnx)=elnx−e−lnx2 [∵sinhx=ex−e−x2]=elnx−elnx−12 [∵lnxa=alnx]=x−1x2 [∵elnx=x]=x2−12x$

Hence, $sinh(lnx)=x2−12x$ .

Therefore, the given statement is true.

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Answer 5

Textbook Question

Answer 6

Textbook Question

Answer 7

(a)

Expert Solution

To determine

Whether the statement “A region R is revolved about the y axis to generate a solid S. To find the volume S, you could use either the disk/ washer method and integrate with respect to y or the shell method and integrate with respect to x.” is true or false.

Answer to Problem 1RE

The statement is true.

Explanation of Solution

The shell method is a method of finding volumes by decomposing a solid of revolution into cylindrical shells.

Suppose that a thin vertical strip is revolved about the y axis. An object of revolution (one that looks like a cylindrical shell or an empty tin can with the top and bottom removed) is obtained.

Then the resulting volume of the cylindrical shell is the surface area of the cylinder times the thickness of the cylindrical wall, which is shown below.

$V=2π∫abx[f(x)]dx$ , where x is the distance to the axis of revolution, $f(x)$ is the length and dx is the width.

Disk method models the resulting three dimensional shape as a stack of an infinite number of disks of varying radius and infinitesimal thickness.

This is in contrast to shell integration which integrates along the axis perpendicular to the axis of revolution.

$V=∫ab[R(x)]2dx$ , where $R(x)$ is the length and dx is the width.

Thus, the given statement is true.

(b)

Expert Solution

To determine

Whether the statement “Given only the velocity of an object moving on a line, it is possible to find its displacement, but not its position.” is true or false.

Answer to Problem 1RE

The statement is true.

Explanation of Solution

In order to find the displacement between two time periods, it is required to integrate the velocity between given time periods.

To evaluate the position at any time, that is either the initial position of the object at any given time or at least the position of the object at a particular time is required.

Thus, the given statement is true.

(c)

Expert Solution

To determine

Whether the statement is “If water flows into a tank at a constant rate (for example 6gal/min.), the volume of water in the tank increases according to a linear function of time” is true or false.

Answer to Problem 1RE

The statement is true.

Explanation of Solution

If $dVdt$ is constant, then V is linear function of time.

Here the rate of inflow is in units of volume per unit time.

Hence, the change in volume in the tank will also be linear.

Thus, the given statement is true.

d.

Expert Solution

To determine

To explain: Whether the statement “The variable $y=t+1$ doubles in value whenever t increases by 1 unit” is true or false.

Explanation of Solution

Suppose $t=2$ , the value of y becomes,

$y=2+1=3$

If t is increased by 1 unit, $t=3$ ,

$y=2+1=3$

Hence, the value of y will not be doubled whenever t increases by 1 unit.

Therefore, the given statement is false.

e.

Expert Solution

To determine

To explain: Whether the statement “The function $y=Ae0.1t$ increases by 10% when t increases by 1 unit” is true or false.

Explanation of Solution

Suppose $t=2$ , then

$y1=Ae0.1×2=Ae0.2≈1.221A$

If t is increased by 1 unit, that is $t=3$ ,

$y2=Ae0.1×3=Ae0.3≈1.35A$

Hence, the percentage increase of both values is approximately 10.57%.

Therefore, the given statement is false.

f.

Expert Solution

To determine

To explain: Whether the statement $lnxy=(lnx)(lny)$ is true or false.

Explanation of Solution

Consider the equation $lnxy=(lnx)(lny)$ .

Suppose $x=2 and y=3$ .

Compute the value of $lnxy$ .

$ln(2×3)=ln6≈1.791$

Compute the value of $(lnx)(lny)$ .

$(lnx)(lny)=(ln2)(ln3)=0.693×1.09≈0.762$

Clearly, $ln6≠(ln2)(ln3)$ .

Therefore, the given statement is false.

g.

Expert Solution

To determine

To explain: Whether the statement $sinh(lnx)=x2−12x$ is true or false.

Explanation of Solution

Compute $sinh(lnx)$ as follows.

$sinh(lnx)=elnx−e−lnx2 [∵sinhx=ex−e−x2]=elnx−elnx−12 [∵lnxa=alnx]=x−1x2 [∵elnx=x]=x2−12x$

Hence, $sinh(lnx)=x2−12x$ .

Therefore, the given statement is true.

Answer 8

(a)

Expert Solution

To determine

Whether the statement “A region R is revolved about the y axis to generate a solid S. To find the volume S, you could use either the disk/ washer method and integrate with respect to y or the shell method and integrate with respect to x.” is true or false.

Answer to Problem 1RE

The statement is true.

Explanation of Solution

The shell method is a method of finding volumes by decomposing a solid of revolution into cylindrical shells.

Suppose that a thin vertical strip is revolved about the y axis. An object of revolution (one that looks like a cylindrical shell or an empty tin can with the top and bottom removed) is obtained.

Then the resulting volume of the cylindrical shell is the surface area of the cylinder times the thickness of the cylindrical wall, which is shown below.

$V=2π∫abx[f(x)]dx$ , where x is the distance to the axis of revolution, $f(x)$ is the length and dx is the width.

Disk method models the resulting three dimensional shape as a stack of an infinite number of disks of varying radius and infinitesimal thickness.

This is in contrast to shell integration which integrates along the axis perpendicular to the axis of revolution.

$V=∫ab[R(x)]2dx$ , where $R(x)$ is the length and dx is the width.

Thus, the given statement is true.

Answer 9

(a)

Expert Solution

Answer 10

(a)

Expert Solution

Answer 11

Expert Solution

Answer 12

To determine

Whether the statement “A region R is revolved about the y axis to generate a solid S. To find the volume S, you could use either the disk/ washer method and integrate with respect to y or the shell method and integrate with respect to x.” is true or false.

Answer 13

Answer to Problem 1RE

The statement is true.

Answer 14

Explanation of Solution

The shell method is a method of finding volumes by decomposing a solid of revolution into cylindrical shells.

Suppose that a thin vertical strip is revolved about the y axis. An object of revolution (one that looks like a cylindrical shell or an empty tin can with the top and bottom removed) is obtained.

Then the resulting volume of the cylindrical shell is the surface area of the cylinder times the thickness of the cylindrical wall, which is shown below.

$V=2π∫abx[f(x)]dx$ , where x is the distance to the axis of revolution, $f(x)$ is the length and dx is the width.

Disk method models the resulting three dimensional shape as a stack of an infinite number of disks of varying radius and infinitesimal thickness.

This is in contrast to shell integration which integrates along the axis perpendicular to the axis of revolution.

$V=∫ab[R(x)]2dx$ , where $R(x)$ is the length and dx is the width.

Thus, the given statement is true.

Answer 15

(b)

Expert Solution

To determine

Whether the statement “Given only the velocity of an object moving on a line, it is possible to find its displacement, but not its position.” is true or false.

Answer to Problem 1RE

The statement is true.

Explanation of Solution

In order to find the displacement between two time periods, it is required to integrate the velocity between given time periods.

To evaluate the position at any time, that is either the initial position of the object at any given time or at least the position of the object at a particular time is required.

Thus, the given statement is true.

Answer 16

(b)

Expert Solution

Answer 17

(b)

Expert Solution

Answer 18

Expert Solution

Answer 19

To determine

Whether the statement “Given only the velocity of an object moving on a line, it is possible to find its displacement, but not its position.” is true or false.

Answer 20

Answer to Problem 1RE

The statement is true.

Answer 21

Explanation of Solution

In order to find the displacement between two time periods, it is required to integrate the velocity between given time periods.

To evaluate the position at any time, that is either the initial position of the object at any given time or at least the position of the object at a particular time is required.

Thus, the given statement is true.

Answer 22

(c)

Expert Solution

To determine

Whether the statement is “If water flows into a tank at a constant rate (for example 6gal/min.), the volume of water in the tank increases according to a linear function of time” is true or false.

Answer to Problem 1RE

The statement is true.

Explanation of Solution

If $dVdt$ is constant, then V is linear function of time.

Here the rate of inflow is in units of volume per unit time.

Hence, the change in volume in the tank will also be linear.

Thus, the given statement is true.

Answer 23

(c)

Expert Solution

Answer 24

(c)

Expert Solution

Answer 25

Expert Solution

Answer 26

To determine

Whether the statement is “If water flows into a tank at a constant rate (for example 6gal/min.), the volume of water in the tank increases according to a linear function of time” is true or false.

Answer 27

Answer to Problem 1RE

The statement is true.

Answer 28

Explanation of Solution

If $dVdt$ is constant, then V is linear function of time.

Here the rate of inflow is in units of volume per unit time.

Hence, the change in volume in the tank will also be linear.

Thus, the given statement is true.

Answer 29

d.

Expert Solution

To determine

To explain: Whether the statement “The variable $y=t+1$ doubles in value whenever t increases by 1 unit” is true or false.

Explanation of Solution

Suppose $t=2$ , the value of y becomes,

$y=2+1=3$

If t is increased by 1 unit, $t=3$ ,

$y=2+1=3$

Hence, the value of y will not be doubled whenever t increases by 1 unit.

Therefore, the given statement is false.

Answer 30

d.

Expert Solution

Answer 31

d.

Expert Solution

Answer 32

Expert Solution

Answer 33

To determine

To explain: Whether the statement “The variable $y=t+1$ doubles in value whenever t increases by 1 unit” is true or false.

Answer 34

Explanation of Solution

Suppose $t=2$ , the value of y becomes,

$y=2+1=3$

If t is increased by 1 unit, $t=3$ ,

$y=2+1=3$

Hence, the value of y will not be doubled whenever t increases by 1 unit.

Therefore, the given statement is false.

Answer 35

e.

Expert Solution

To determine

To explain: Whether the statement “The function $y=Ae0.1t$ increases by 10% when t increases by 1 unit” is true or false.

Explanation of Solution

Suppose $t=2$ , then

$y1=Ae0.1×2=Ae0.2≈1.221A$

If t is increased by 1 unit, that is $t=3$ ,

$y2=Ae0.1×3=Ae0.3≈1.35A$

Hence, the percentage increase of both values is approximately 10.57%.

Therefore, the given statement is false.

Answer 36

e.

Expert Solution

Answer 37

e.

Expert Solution

Answer 38

Expert Solution

Answer 39

To determine

To explain: Whether the statement “The function $y=Ae0.1t$ increases by 10% when t increases by 1 unit” is true or false.

Answer 40

Explanation of Solution

Suppose $t=2$ , then

$y1=Ae0.1×2=Ae0.2≈1.221A$

If t is increased by 1 unit, that is $t=3$ ,

$y2=Ae0.1×3=Ae0.3≈1.35A$

Hence, the percentage increase of both values is approximately 10.57%.

Therefore, the given statement is false.

Answer 41

f.

Expert Solution

To determine

To explain: Whether the statement $lnxy=(lnx)(lny)$ is true or false.

Explanation of Solution

Consider the equation $lnxy=(lnx)(lny)$ .

Suppose $x=2 and y=3$ .

Compute the value of $lnxy$ .

$ln(2×3)=ln6≈1.791$

Compute the value of $(lnx)(lny)$ .

$(lnx)(lny)=(ln2)(ln3)=0.693×1.09≈0.762$

Clearly, $ln6≠(ln2)(ln3)$ .

Therefore, the given statement is false.

Answer 42

f.

Expert Solution

Answer 43

f.

Expert Solution

Answer 44

Expert Solution

Answer 45

To determine

To explain: Whether the statement $lnxy=(lnx)(lny)$ is true or false.

Answer 46

Explanation of Solution

Consider the equation $lnxy=(lnx)(lny)$ .

Suppose $x=2 and y=3$ .

Compute the value of $lnxy$ .

$ln(2×3)=ln6≈1.791$

Compute the value of $(lnx)(lny)$ .

$(lnx)(lny)=(ln2)(ln3)=0.693×1.09≈0.762$

Clearly, $ln6≠(ln2)(ln3)$ .

Therefore, the given statement is false.

Answer 47

g.

Expert Solution

To determine

To explain: Whether the statement $sinh(lnx)=x2−12x$ is true or false.

Explanation of Solution

Compute $sinh(lnx)$ as follows.

$sinh(lnx)=elnx−e−lnx2 [∵sinhx=ex−e−x2]=elnx−elnx−12 [∵lnxa=alnx]=x−1x2 [∵elnx=x]=x2−12x$

Hence, $sinh(lnx)=x2−12x$ .

Therefore, the given statement is true.

Answer 48

g.

Expert Solution

Answer 49

g.

Expert Solution

Answer 50

Expert Solution

Answer 51

To determine

To explain: Whether the statement $sinh(lnx)=x2−12x$ is true or false.

Answer 52

Explanation of Solution

Compute $sinh(lnx)$ as follows.

$sinh(lnx)=elnx−e−lnx2 [∵sinhx=ex−e−x2]=elnx−elnx−12 [∵lnxa=alnx]=x−1x2 [∵elnx=x]=x2−12x$

Hence, $sinh(lnx)=x2−12x$ .

Therefore, the given statement is true.

Answer 53

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