009A Sample Midterm 3, Problem 3 Detailed Solution
Let Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle y=3{\sqrt {2x+5}},x\geq 0.}
(a) Use the definition of the derivative to compute
(b) Find the equation of the tangent line to Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle y=3{\sqrt {2x+5}}} at 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 (2,9).}
| Background Information: |
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| Recall |
| 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:
(a)
| Step 1: |
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| 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 |
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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: |
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| Now, we multiply the numerator and denominator by the conjugate of the numerator. |
| Hence, 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{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}} |
(b)
| Step 1: |
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| We start by finding the slope of the tangent line to 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}} at 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 (2,9).} |
| Using the derivative calculated in part (a), the slope 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 \begin{array}{rcl} \displaystyle{m} & = & \displaystyle{f'(2)}\\ &&\\ & = & \displaystyle{\frac{3}{\sqrt{2(2)+5}}}\\ &&\\ & = & \displaystyle{\frac{3}{\sqrt{9}}}\\ &&\\ & = & \displaystyle{1.} \end{array}} |
| Step 2: |
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| Now, the tangent line to 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}} at 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 (2,9)} |
| has slope 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 m=1} and passes through the point 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 (2,9).} |
| Hence, the equation of this line is |
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| If we simplify, we get |
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| Final Answer: |
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| (a) 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}}} |
| (b) 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=x+7} |