004 Sample Final A, Problem 11

Find and simplify the difference quotient ${\frac {f(x+h)-f(x)}{h}}$ for $f(x)={\sqrt {x-3}}$

Foundations
1) $f(x+h)=?$
2) How do you eliminate the $h$ in the denominator?
Answer:
1) We have $f(x+h)={\sqrt {x+h-3}}$
2) The difference quotient is ${\frac {{\sqrt {x+h-3}}-{\sqrt {x-3}}}{h}}$. To eliminate the $h$ in the denominator,
you need to multiply the numerator and denominator by ${\sqrt {x+h-3}}+{\sqrt {x-3}}$ (the conjugate).

Solution:

Step 1:
The difference quotient we need to simply is Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \frac{f(x + h) - f(x)}{h}=\frac{\sqrt{x+h-3}-\sqrt{x-3}}{h}} .

Step 2:
Multiplying the numerator and denominator by Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \sqrt{x+h-3}+\sqrt{x-3}} , we get
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \frac{f(x + h) - f(x)}{h}=\frac{x+h-3-(x-3)}{h(\sqrt{x+h-3}+\sqrt{x-3})} }

Step 3:
Now, simplifying the numerator, we get
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \frac{f(x + h) - f(x)}{h}=\frac{h}{h(\sqrt{x+h-3}+\sqrt{x-3})} } . Now, we can cancel the Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle h} in the denominator.
Thus, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \frac{f(x + h) - f(x)}{h}=\frac{1}{(\sqrt{x+h-3}+\sqrt{x-3})} } .

Final Answer:
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \frac{f(x + h) - f(x)}{h}=\frac{1}{(\sqrt{x+h-3}+\sqrt{x-3})} }

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