009A Sample Midterm 3, Problem 3

Use the definition of the derivative to compute   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{dy}{dx}}   for  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 y=3\sqrt{-2x+5}.}

Foundations:
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 f'(x)=\lim_{h\rightarrow 0}\frac{f(x+h)-f(x)}{h}}

Solution:

Step 1:
Let  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 f(x)=3\sqrt{-2x+5}.}
Using the limit definition of the derivative, we have
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 \begin{array}{rcl} \displaystyle{f'(x)} & = & \displaystyle{\lim_{h\rightarrow 0} \frac{f(x+h)-f(x)}{h}}\\ &&\\ & = & \displaystyle{\lim_{h\rightarrow 0} \frac{3\sqrt{-2(x+h)+5}-3\sqrt{-2x+5}}{h}}\\ &&\\ & = & \displaystyle{\lim_{h\rightarrow 0} \frac{3\sqrt{-2x+-2h+5}-3\sqrt{-2x+5}}{h}}\\ &&\\ & = & \displaystyle{3\lim_{h\rightarrow 0} \frac{\sqrt{-2x+-2h+5}-\sqrt{-2x+5}}{h}.} \end{array}}

Step 2:
Now, we multiply the numerator and denominator by the conjugate of the numerator.
Hence, we have
${\begin{array}{rcl}\displaystyle {f'(x)}&=&\displaystyle {3\lim _{h\rightarrow 0}{\frac {({\sqrt {-2x+-2h+5}}-{\sqrt {-2x+5}})}{h}}{\frac {({\sqrt {-2x+-2h+5}}+{\sqrt {-2x+5}})}{({\sqrt {-2x+-2h+5}}+{\sqrt {-2x+5}})}}}\\&&\\&=&\displaystyle {3\lim _{h\rightarrow 0}{\frac {(-2x+-2h+5)-(-2x+5)}{h({\sqrt {-2x+-2h+5}}+{\sqrt {-2x+5}})}}}\\&&\\&=&\displaystyle {3\lim _{h\rightarrow 0}{\frac {-2h}{h({\sqrt {-2x+-2h+5}}+{\sqrt {-2x+5}})}}}\\&&\\&=&\displaystyle {3\lim _{h\rightarrow 0}{\frac {-2}{{\sqrt {-2x+-2h+5}}+{\sqrt {-2x+5}}}}}\\&&\\&=&\displaystyle {3{\frac {-2}{{\sqrt {-2x+5}}+{\sqrt {-2x+5}}}}}\\&&\\&=&\displaystyle {-{\frac {3}{\sqrt {-2x+5}}}.}\end{array}}$

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{3}{\sqrt{-2x+5}}}

Return to Sample Exam