Chapter 2.4, Problem 46E

(a)

To determine

To find: The difference quotient $\frac{f\left(1+h\right)-f\left(1\right)}{h}$ for $f\left(x\right)=2^x$.

Answer to Problem 46E

The difference quotient $\frac{f\left(1+h\right)-f\left(1\right)}{h}$ for $f\left(x\right)=2^x$ is equal to $\frac{2\left(2^h-1\right)}{h}$.

Explanation of Solution

Given information: The function is $f\left(x\right)=2^x$.

Calculation:

Substitute $1+h$ for $x$ in the function.

  $\begin{array}{c}f\left(x\right)=2^x\\ f\left(1+h\right)=2^{1+h}\\ f\left(1+h\right)=2^1\cdot 2^h\\ f\left(1+h\right)=2\cdot 2^h\end{array}$

Substitute $1$ for $x$ in the function.

  $\begin{array}{c}f\left(x\right)=2^x\\ f\left(1\right)=2^1\\ f\left(1\right)=2\end{array}$

Substitute $2\cdot 2^h$ for $f\left(1+h\right)$ and $2$ for $f\left(1\right)$ in the difference quotient.

  $\begin{array}{c}\frac{f\left(1+h\right)-f\left(1\right)}{h}=\frac{2\cdot 2^h-2}{h}\\ =\frac{2\left(2^h-1\right)}{h}\end{array}$

Therefore, the difference quotient $\frac{f\left(1+h\right)-f\left(1\right)}{h}$ for $f\left(x\right)=2^x$ is $\frac{2\left(2^h-1\right)}{h}$.

(b)

To determine

To find: The limit of difference quotient in part (a) as $h\to 0$ with the help of graph and table.

Answer to Problem 46E

The limit of difference quotient in part (a) as $h\to 0$ is equal to $2.718$.

Explanation of Solution

Given information: The function is $f\left(x\right)=2^x$.

Calculation:

As calculated in part(a), the difference quotient function is equal to $\frac{2\left(2^h-1\right)}{h}$.

Check the limit of difference quotient as $h\to 0$ graphically.

To graph a function $y=\frac{2\left(2^h-1\right)}{h}$ , follow the steps using graphing calculator.

First press “ON” button on graphical calculator, press $\boxed{\text{Y}=}$ key and enter right hand side of the equation $y=\frac{2\left(2^h-1\right)}{h}$ after the symbol ${\text{Y}}_1$ . Enter the keystrokes $\boxed{2}\boxed{(}\boxed{(}\boxed{(}\boxed{2}\boxed{^}\boxed{\text{X}}\boxed{)}\boxed{-}\boxed{1}\boxed{)}\boxed{÷}\boxed{\text{X}}\boxed{)}$.

The display will show the equation,

  ${\text{Y}}_{\text{1}}=2\left(\left(\left(2^\text{X}\right)-1\right)/\text{X}\right)$

Now, press the $\boxed{\text{GRAPH}}$ key and $\boxed{\text{TRACE}}$ key to produce the graph of given function in standard window as shown below.

  

Figure (1)

As observed from graph, the function $y=\frac{2\left(2^h-1\right)}{h}$ tends to $1.386$ as $h$ tends to $0$.

Now check the limit by table.

To make the table of function $y=\frac{2\left(2^h-1\right)}{h}$ , follow the steps using graphing calculator.

First press “ON” button on graphical calculator, press $\boxed{\text{Y}=}$ key and enter right hand side of the equation $y=\frac{2\left(2^h-1\right)}{h}$ after the symbol ${\text{Y}}_1$ . Enter the keystrokes $\boxed{2}\boxed{(}\boxed{(}\boxed{(}\boxed{2}\boxed{^}\boxed{\text{X}}\boxed{)}\boxed{-}\boxed{1}\boxed{)}\boxed{÷}\boxed{\text{X}}\boxed{)}$.

First set the Table setup, Enter the keystrokes $\boxed{2\text{nd}}\boxed{\text{WINDOW}}$ , Then set to independent variable to ask.

Now draw the table, Enter the keystrokes $\boxed{2\text{nd}}\boxed{\text{GRAPH}}$ , Enter the values of $\text{X}$ as $-0.1$ , $-0.01$ , $-0.001$ , $0$ , $0.1$ , $0.01$ and $0.001$ . The table is shown in window.

  

Figure (2)

As observed from table the value of $\text{Y}$ approaches $1.386$ as $h$ approaches to $0$.

Therefore, the limit of difference quotient in part (a) as $h\to 0$ is equal to $1.386$.

(c)

To determine

To find: The difference between the value $\ln 4$ and the estimate in part (b).

Answer to Problem 46E

The difference between the value $\ln 4$ and the estimate in part (b) is equal to $0.0002$.

Explanation of Solution

Given information: The function is $f\left(x\right)=2^x$.

Calculation:

As calculated in part (b), the limit of difference quotient function as $h\to 0$ is equal to $2.718$.

Check the difference between $e$ and estimate.

  $\begin{array}{c}\ln 4-1.386=1.3862-1.386\\ =0.0002\end{array}$

Therefore, the difference between the value $\ln 4$ and the estimate in part (b) is equal to $0.0002$.

(d)

To determine

To check: The graph of $f\left(x\right)=2^x$ has a tangent at $x=1$ or not and also the slope of the tangent if exists.

Answer to Problem 46E

Yes, the graph of $f\left(x\right)=2^x$ has a tangent at $x=1$ and the slope of the tangent at $x=1$ is equal to $1.386$ approximately.

Explanation of Solution

Given information: The function is $f\left(x\right)=2^x$.

Calculation:

The formula for the slope of the function $f\left(x\right)$ at $x=1$ is equal to $\underset{h\to 0}{\lim }\frac{f\left(1+h\right)-f\left(1\right)}{h}$.

As calculated in part (b), the limit of difference quotient function as $h\to 0$ is equal to $1.386$ . So, the slope of the tangent to the graph of $f\left(x\right)=2^x$ at $x=1$ is equal to $1.386$.

Therefore, the graph of $f\left(x\right)=2^x$ has a tangent at $x=1$ and the slope of the tangent at $x=1$ is equal to $1.386$ approximately.