Question: Find and simplify the difference quotient 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}}
for f(x) = 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{2}{3x+1}}
| Foundations
|
| 1) f(x + h) = ?
|
| 2) How do you eliminate the 'h' in the denominator?
|
| Answer:
|
| 1)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 + h) = \frac{2}{3(x + h) + 1}}
.
|
| 2) The numerator of the difference quotient 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{2}{3(x + h) + 1} - \frac{2}{3x + 1}}
so the first step is to simplify this expression. This then allows us to eliminate the 'h' in the denominator.
|
Solution:
| Step 1:
|
| The difference quotient that we want to simplify 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} = \left(\frac{2}{3(x + h) + 1} - \frac{2}{3x + 1}\right) \div h}
|
| Step 2:
|
| Now we simplify the numerator:
|
- 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{\frac{f(x + h) - f(x)}{h}} &=& \displaystyle{\left(\frac{2}{3(x + h) + 1} - \frac{2}{3x + 1}\right) \div h}\\ & & \\ &=& \displaystyle{\frac{2(3x + 1) -2(3(x + h) + 1)}{h(3(x + h) + 1)(3x + 1))}} \end{array}}
|
| Arithmetic:
|
| Now we simplify the numerator:
|
- Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle {\begin{array}{rcl}\displaystyle {\frac {2(3x+1)-2(3(x+h)+1)}{h(3(x+h)+1)(3x+1))}}&=&\displaystyle {\frac {6x+2-6x-6h-2}{h(3(x+h)+1)(3x+1))}}\\&&\\&=&\displaystyle {\frac {-6}{(3(x+h)+1)(3x+1))}}\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{-6}{(3(x + h) + 1)(3x + 1))}}
|
Return to Sample Exam