009A Sample Final 1, Problem 2

Consider the following piecewise defined function:

Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle f(x)=\left\{{\begin{array}{lr}x+5&{\text{if }}x<3\\4{\sqrt {x+1}}&{\text{if }}x\geq 3\end{array}}\right.}

a) Show that $f(x)$ is continuous at Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle x=3.}

b) Using the limit definition of the derivative, and computing the limits from both sides, show that $f(x)$ is differentiable at Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle x=3.}

Foundations:
Recall:
1. $f(x)$  is continuous at $x=a$  if $\lim _{x\rightarrow a^{+}}f(x)=\lim _{x\rightarrow a^{-}}f(x)=f(a).$
2. The definition of derivative for $f(x)$  is  $f'(x)=\lim _{h\rightarrow 0}{\frac {f(x+h)-f(x)}{h}}.$

Solution:

(a)

Step 1:
We first calculate Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle \lim _{x\rightarrow 3^{+}}f(x).} We have
Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle {\begin{array}{rcl}\displaystyle {\lim _{x\rightarrow 3^{+}}f(x)}&=&\displaystyle {\lim _{x\rightarrow 3^{+}}4{\sqrt {x+1}}}\\&&\\&=&\displaystyle {4{\sqrt {3+1}}}\\&&\\&=&\displaystyle {8.}\end{array}}}

Step 2:
Now, we calculate Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle \lim _{x\rightarrow 3^{-}}f(x).} We have
Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle {\begin{array}{rcl}\displaystyle {\lim _{x\rightarrow 3^{-}}f(x)}&=&\displaystyle {\lim _{x\rightarrow 3^{-}}x+5}\\&&\\&=&\displaystyle {3+5}\\&&\\&=&\displaystyle {8.}\end{array}}}

Step 3:
Now, we calculate Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle f(3).} We have
Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle f(3)=4{\sqrt {3+1}}\,=\,8.}
Since Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle \lim _{x\rightarrow 3^{+}}f(x)=\lim _{x\rightarrow 3^{-}}f(x)=f(3),~f(x)}   is continuous.

(b)

Step 1:
We need to use the limit definition of derivative and calculate the limit from both sides.
So, we have
Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle {\begin{array}{rcl}\displaystyle {\lim _{h\rightarrow 0^{-}}{\frac {f(3+h)-f(3)}{h}}}&=&\displaystyle {\lim _{h\rightarrow 0^{-}}{\frac {(3+h)+5-8}{h}}}\\&&\\&=&\displaystyle {\lim _{h\rightarrow 0^{-}}{\frac {h}{h}}}\\&&\\&=&\displaystyle {\lim _{h\rightarrow 0^{-}}1}\\&&\\&=&\displaystyle {1.}\end{array}}}

Step 2:
Now, we have
Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle {\begin{array}{rcl}\displaystyle {\lim _{h\rightarrow 0^{+}}{\frac {f(3+h)-f(3)}{h}}}&=&\displaystyle {\lim _{h\rightarrow 0^{+}}{\frac {4{\sqrt {3+h+1}}-8}{h}}}\\&&\\&=&\displaystyle {\lim _{h\rightarrow 0^{+}}{\frac {4({\sqrt {4+h}}-{\sqrt {4}})}{h}}}\\&&\\&=&\displaystyle {\lim _{h\rightarrow 0^{+}}{\frac {4({\sqrt {4+h}}-{\sqrt {4}})({\sqrt {4+h}}+{\sqrt {4}})}{h({\sqrt {4+h}}+{\sqrt {4}})}}}\\&&\\&=&\displaystyle {\lim _{h\rightarrow 0^{+}}{\frac {4(4+h-4)}{h({\sqrt {4+h}}+{\sqrt {4}})}}}\\&&\\&=&\displaystyle {\lim _{h\rightarrow 0^{+}}{\frac {4h}{h({\sqrt {4+h}}+{\sqrt {4}})}}}\\&&\\&=&\displaystyle {\lim _{h\rightarrow 0^{+}}{\frac {4}{({\sqrt {4+h}}+{\sqrt {4}})}}}\\&&\\&=&\displaystyle {\frac {4}{2{\sqrt {4}}}}\\&&\\&=&\displaystyle {1.}\\\end{array}}}

Step 3:
Since Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle \lim _{h\rightarrow 0^{-}}{\frac {f(3+h)-f(3)}{h}}=\lim _{h\rightarrow 0^{+}}{\frac {f(3+h)-f(3)}{h}},}
$f(x)$  is differentiable at Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle x=3.}

Final Answer:
(a) Since $\lim _{x\rightarrow 3^{+}}f(x)=\lim _{x\rightarrow 3^{-}}f(x)=f(3),~f(x)$  is continuous.
(b) Since Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle \lim _{h\rightarrow 0^{-}}{\frac {f(3+h)-f(3)}{h}}=\lim _{h\rightarrow 0^{+}}{\frac {f(3+h)-f(3)}{h}},}
$f(x)$  is differentiable 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 x=3.}

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