Analyze the function. f(x)={12x+13ifx&lt;88x+9ifx≥8 Is f(x) diferential at x=8? Explain why or why not in your Discussion post.

Step 1 {12x+13ifx&lt;88x+9ifx≥8 f(x) is lontinaus at x→8, f(8)⇒17 In order to find out differentialyti of function Limit of the dyferent quotient f(8+h)−f(8)h From the LHS and RHS must both exuct and lee equal to each other In LHS limh→0−(8+h)2+13−17h⇒limh→0−h2+17−17h ⇒limh→0−[h2h]=12 In RHS limn→0+8(8+h)+9−17h⇒limn→0+[8(8+h)−8h] Given fos m is indeterminale form {00} Step 2 Apply L-Hopital is rule limh→0+[8×128+h−01]⇒828⇒12 Therefore lim(LHS)=lim(RHS)=12 Yes, function f(x) is differentialite at x=8

Answered: "Analyze the function. f(x)={12x+13ifx<88x+9ifx≥8 Is f(x) diferential at..."

auskunftsgkp

Answered question

2022-02-14

Analyze the function.
$f (x) = {\begin{array}{ccc} \frac{1}{2} x + 13 & if & x < 8 \\ \sqrt{8 x} + 9 & if & x \geq 8 \end{array}$
Is f(x) diferential at $x = 8 ?$ Explain why or why not in your Discussion post.

Answer & Explanation

liofila3w7

Beginner2022-02-15Added 13 answers

Step 1
${\begin{array}{ccc} \frac{1}{2} x + 13 & if & x < 8 \\ \sqrt{8 x} + 9 & if & x \geq 8 \end{array}$
f(x) is lontinaus at $x \to 8, f (8) \Rightarrow 17$
In order to find out differentialyti of function
Limit of the dyferent quotient
$\frac{f (8 + h) - f (8)}{h}$
From the LHS and RHS must both exuct and lee equal to each other
In LHS
$lim_{h \to 0^{-}} \frac{\frac{(8 + h)}{2} + 13 - 17}{h} \Rightarrow lim_{h \to 0^{-}} \frac{\frac{h}{2} + 17 - 17}{h}$
$\Rightarrow lim_{h \to 0^{-}} [\frac{h}{2 h}] = \frac{1}{2}$
In RHS
$lim_{n \to 0^{+}} \frac{\sqrt{8 (8 + h)} + 9 - 17}{h} \Rightarrow lim_{n \to 0^{+}} [\frac{\sqrt{8 (8 + h)} - 8}{h}]$
Given fos m is indeterminale form ${\frac{0}{0}}$
Step 2
Apply L-Hopital is rule
$lim_{h \to 0^{+}} [\frac{\frac{\sqrt{8} \times 1}{2 \sqrt{8 + h}} - 0}{1}] \Rightarrow \frac{\sqrt{8}}{2 \sqrt{8}} \Rightarrow \frac{1}{2}$
Therefore $lim (L H S) = lim (R H S) = \frac{1}{2}$
Yes, function f(x) is differentialite at $x = 8$

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