Explain why or why not Determine whether the following statements are true and give an explanation or counterexample. a. The variable y = t + 1 doubles in value whenever t increases by 1 unit. b. The function y = Ae 0.1 t increases by 10% when t increases by 1 unit. c. ln xy = (ln x )(ln y ). d. sinh ( ln x ) = x 2 − 1 2 x .

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Answer 1

Textbook Question

Chapter 7, Problem 1RE

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

a. The variable y = t + 1 doubles in value whenever t increases by 1 unit.
b. The function y = Ae^0.1t increases by 10% when t increases by 1 unit.
c. ln xy = (ln x)(ln y).
d. $sinh ( ln x ) = x 2 − 1 2 x .$

a.

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.

b.

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.

c.

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.

d.

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 7, Problem 1RE

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

a. The variable y = t + 1 doubles in value whenever t increases by 1 unit.
b. The function y = Ae^0.1t increases by 10% when t increases by 1 unit.
c. ln xy = (ln x)(ln y).
d. $sinh ( ln x ) = x 2 − 1 2 x .$

a.

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.

b.

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.

c.

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.

d.

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 7, Problem 1RE

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

a. The variable y = t + 1 doubles in value whenever t increases by 1 unit.
b. The function y = Ae^0.1t increases by 10% when t increases by 1 unit.
c. ln xy = (ln x)(ln y).
d. $sinh ( ln x ) = x 2 − 1 2 x .$

a.

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.

b.

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.

c.

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.

d.

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 7, Problem 1RE

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

a. The variable y = t + 1 doubles in value whenever t increases by 1 unit.
b. The function y = Ae^0.1t increases by 10% when t increases by 1 unit.
c. ln xy = (ln x)(ln y).
d. $sinh ( ln x ) = x 2 − 1 2 x .$

a.

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.

b.

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.

c.

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.

d.

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

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.

b.

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.

c.

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.

d.

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

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 9

a.

Expert Solution

Answer 10

a.

Expert Solution

Answer 11

Expert Solution

Answer 12

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 13

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 14

b.

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 15

b.

Expert Solution

Answer 16

b.

Expert Solution

Answer 17

Expert Solution

Answer 18

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 19

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 20

c.

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 21

c.

Expert Solution

Answer 22

c.

Expert Solution

Answer 23

Expert Solution

Answer 24

To determine

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

Answer 25

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 26

d.

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 27

d.

Expert Solution

Answer 28

d.

Expert Solution

Answer 29

Expert Solution

Answer 30

To determine

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

Answer 31

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 32

Ch. 7.1 - What is the domain of ln |x|?Ch. 7.1 - Simplify e ln 2x, ln (e2x), e2 ln x, and ln (2ex)Ch. 7.1 - What is the slope of the curve y = ex at x= ln 2?...Ch. 7.1 - Verify that the derivative and integral results...Ch. 7.1 - Prob. 1E Ch. 7.1 - Prob. 2E Ch. 7.1 - Evaluate 4xdx.Ch. 7.1 - What is the inverse function of ln x, and what are...Ch. 7.1 - Express 3x, x, and xsin x using the base e.Ch. 7.1 - Evaluate ddx(3x).

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