Chapter 3.3, Problem 8QR

(a)

To determine

To calculate: The value of the function.

Answer to Problem 8QR

The required value of the function $f\left(10\right)$ is 7.

Explanation of Solution

Given information:

The given function $f\left(x\right)=7$ for all real number $x$.

Calculation:

The given function is $f\left(x\right)=7$.

It is given that the value of the function $f\left(x\right)$ is 7 for all value of $x$.

Thus, for $x=10$.

  $f\left(10\right)=7$

Hence, the required value of the function $f\left(10\right)$ is 7.

(b)

To determine

To calculate: The value of the function.

Answer to Problem 8QR

The required value of the function $f\left(0\right)$ is 7.

Explanation of Solution

Given information:

The given function $f\left(x\right)=7$ for all real number $x$.

Calculation:

The given function is $f\left(x\right)=7$

It is given that the value of the function $f\left(x\right)$ is 7 for all value of $x$.

Thus, for $x=0$.

  $f\left(0\right)=7$

Hence, the required value of the function $f\left(0\right)$ is 7.

(c)

To determine

To calculate: The value of the function.

Answer to Problem 8QR

The required value of the function $f\left(x+h\right)$ is 7.

Explanation of Solution

Given information:

The given function $f\left(x\right)=7$ for all real number $x$.

Calculation:

The given function is $f\left(x\right)=7$

It is given that the value of the function $f\left(x\right)$ is 7 for all value of $x$.

Thus, for $x=x+h$.

  $f\left(x+h\right)=7$

Hence, the required value of the function $f\left(x+h\right)$ is 7.

(d)

To determine

To calculate: The value of the function.

Answer to Problem 8QR

The required value of the function is 0.

Explanation of Solution

Given information:

The given function: $f\left(x\right)=7$ for all real number $x$.

The function to determine: $\underset{x\to 0}{\lim }\frac{f\left(x\right)-f\left(a\right)}{x-a}$

Formula Used:

Differential formula:

  $f\text{'}\left(x\right)=\underset{h\to 0}{\lim }\frac{f\left(x+h\right)-f\left(x\right)}{h}$

Here $f\left(x\right)$ is the function.

Calculation:

The given function is $\underset{x\to 0}{\lim }\frac{f\left(x\right)-f\left(a\right)}{x-a}$.

From the differential formula rewrite the given function as

  $\underset{x\to 0}{\lim }\frac{f\left(x\right)-f\left(a\right)}{x-a}=f\text{'}\left(x\right)$

It is given that the value of the function $f\left(x\right)$ is 7 for all value of $x$.

Thus, $f\left(x\right)=7$ , $f\left(a\right)=7$

Substitute these value in the above function.

  $\begin{array}{c}f\text{'}\left(x\right)=\underset{x\to 0}{\lim }\frac{f\left(x\right)-f\left(a\right)}{x-a}\\ =\underset{x\to 0}{\lim }\frac{7-7}{x-a}\\ =0\end{array}$

Hence, the required value of the function is 0.