Questions: Let the function f be defined as follows. f(x) = 6 sqrt(x-3) Answer the following. (a) For h != 0, find the difference quotient (f(x+h)-f(x)) / h. Simplify your answer so that the resulting expression is defined at h=0. (f(x+h)-f(x)) / h =

Let the function f be defined as follows.

f(x) = 6 sqrt(x-3)

Answer the following. (a) For h != 0, find the difference quotient (f(x+h)-f(x)) / h.

Simplify your answer so that the resulting expression is defined at h=0.

(f(x+h)-f(x)) / h =

Transcript text: Let the function $f$ be defined as follows. \[ f(x)=6 \sqrt{x-3} \] Answer the following. (a) For $h \neq 0$, find the difference quotient $\frac{f(x+h)-f(x)}{h}$. Simplify your answer so that the resulting expression is defined at $h=0$. \[ \frac{f(x+h)-f(x)}{h}= \]

Solution

Solution Steps

To find the difference quotient $\frac{f(x+h)-f(x)}{h}$ for the function $f(x) = 6 \sqrt{x-3}$, we need to:

Substitute $f(x)$ and $f(x+h)$ into the difference quotient formula.
Simplify the expression to make it defined at $h=0$.

Step 1: Define the Function and Difference Quotient

Given the function $ f(x) = 6 \sqrt{x-3} $, we need to find the difference quotient: \[ \frac{f(x+h) - f(x)}{h} \]

Step 2: Substitute $ f(x) $ and $ f(x+h) $

First, we substitute $ f(x) $ and $ f(x+h) $ into the difference quotient: \[ f(x) = 6 \sqrt{x-3} \] \[ f(x+h) = 6 \sqrt{x+h-3} \] \[ \frac{f(x+h) - f(x)}{h} = \frac{6 \sqrt{x+h-3} - 6 \sqrt{x-3}}{h} \]

Step 3: Simplify the Expression

We can factor out the common factor of 6 in the numerator: \[ \frac{6 \sqrt{x+h-3} - 6 \sqrt{x-3}}{h} = 6 \cdot \frac{\sqrt{x+h-3} - \sqrt{x-3}}{h} \]

Step 4: Rationalize the Numerator

To simplify further, we rationalize the numerator by multiplying and dividing by the conjugate: \[ 6 \cdot \frac{\sqrt{x+h-3} - \sqrt{x-3}}{h} \cdot \frac{\sqrt{x+h-3} + \sqrt{x-3}}{\sqrt{x+h-3} + \sqrt{x-3}} \] \[ = 6 \cdot \frac{(\sqrt{x+h-3})^2 - (\sqrt{x-3})^2}{h (\sqrt{x+h-3} + \sqrt{x-3})} \] \[ = 6 \cdot \frac{(x+h-3) - (x-3)}{h (\sqrt{x+h-3} + \sqrt{x-3})} \] \[ = 6 \cdot \frac{h}{h (\sqrt{x+h-3} + \sqrt{x-3})} \] \[ = 6 \cdot \frac{1}{\sqrt{x+h-3} + \sqrt{x-3}} \]