Chapter 3, Problem 3.17P

(a)

Interpretation Introduction

Interpretation:

The given calculation has to be verified.

Concept Introduction:

Uncertainty for powers and roots:

For the function $y=x^a$ , the relative uncertainty in $y(\%e_y)$ is $a$ times the relative uncertainty in $x(\%e_x)$

$y=x^a\Rightarrow \%e_y=a(\%e_x)$

To Verify: The given calculation

(b)

Interpretation Introduction

Interpretation:

The given calculation has to be verified.

Concept Introduction:

Exponents and logarithm:

Consider, $y=\log x$ .

Here, the absolute uncertainty in $y(e_y)$ is proportional to the relative uncertainty in x, which is $e_x/x$

$\begin{array}{l}\text{Uncertainty for logarithm:}\\ y=1\text{og}x\\ \Rightarrow e_y=\frac{1}{\text{ln 10 }x}\frac{e_x}{x}\\ \approx 0.43429\frac{e_x}{x}\end{array}$

To Verify: The given calculation.

(c)

Interpretation Introduction

Interpretation:

The given calculation has to be verified.

Concept Introduction:

Exponents and logarithm:

Consider, $y=\text{antilog }x\Rightarrow y={10}^x$ .

Here, the relative uncertainty in y is proportional to the absolute uncertainty in x.

$\begin{array}{l}{\text{Uncertainty for 10}}^{x:}\\ y={10}^x\\ \Rightarrow \frac{e_y}{y}=(\text{ln 10)}{e}_x\\ \approx 2.3026e_x\end{array}$

To Verify: The given calculation

(d)

Interpretation Introduction

Interpretation:

The given calculation has to be verified.

Concept Introduction:

Exponents and logarithm:

Consider, $y=\text{ln }x$ .

Here, the absolute uncertainty in y is equal to the relative uncertainty in x.

$\begin{array}{l}\text{Uncertainty for natural logarithm:}\\ y=\text{ln }x\\ \Rightarrow e_y=\frac{e_x}{x}\end{array}$

To Verify: The given calculation

(e)

Interpretation Introduction

Interpretation:

The given calculation has to be verified.

Concept Introduction:

Uncertainty for powers and roots:

For the function $y=x^a$ , the relative uncertainty in $y(\%e_y)$ is $a$ times the relative uncertainty in $x(\%e_x)$

$y=x^a\Rightarrow \%e_y=a(\%e_x)$

To Verify: The given calculation