Reasoning (1 Viewer)

leehuan · Apr 10, 2016

Posting this in the 2U section as it's still within the scope of the course (even though it's just my homework).

$$The function $f$ is defined by $f(x)={x}^{2}-4x+3$ for $0 < x \le 3$, $f(-x)=-f(x)$ for all $x$ and $f(x+6)=f(x)$ for all $x$.$

I have the graph in front of me so I can pretty much infer the answer from just that. But say I wanted to do the following question without it.

$\\$Does $f^{\prime} (3)$ exist? If so, state its value.$$

Would the following method work:

$Solution: Verify that $\lim_{x\rightarrow {3}^{-}}{f^{\prime}(x)}=\lim_{x\rightarrow {3}^{+}}{f^{\prime}(x)}\quad (=2)\\ \because f$ is continuous at 3.$
_____________________________

Another question. Is this acceptable to find the limit as x approaches 3 from above (right)

$\lim_{x\rightarrow {3}^{+}}{f(x)}=\lim_{x\rightarrow {-3}^{+}}{f(x+6)}=\lim_{x\rightarrow {-3}^{+}}{({(x+6)}^{2}-4(x+6)+3)}=0$

InteGrand · Apr 10, 2016

leehuan said:
Posting this in the 2U section as it's still within the scope of the course (even though it's just my homework).

$$The function $f$ is defined by $f(x)={x}^{2}-4x+3$ for $0 < x \le 3$, $f(-x)=-f(x)$ for all $x$ and $f(x+6)=f(x)$ for all $x$.$

I have the graph in front of me so I can pretty much infer the answer from just that. But say I wanted to do the following question without it.

$\\$Does $f^{\prime} (3)$ exist? If so, state its value.$$

Would the following method work:

$Solution: Verify that $\lim_{x\rightarrow {3}^{-}}{f^{\prime}(x)}=\lim_{x\rightarrow {3}^{+}}{f^{\prime}(x)}\quad (=2)\\ \because f$ is continuous at 3.$
_____________________________

Another question. Is this acceptable to find the limit as x approaches 3 from above (right)

$\lim_{x\rightarrow {3}^{+}}{f(x)}=\lim_{x\rightarrow {-3}^{+}}{f(x+6)}=\lim_{x\rightarrow {-3}^{+}}{({(x+6)}^{2}-4(x+6)+3)}=0$

I can't see the graph (actually, realised now it was for the function whose formula was given), but if the slope (namely [f(x) – f(3)]/(x – 3) ) approaches 2 from either side and f is continuous there (In fact, if [f(x) – f(3)]/(x – 3) approaches 2 (or some other finite number in general) from either side, then this actually implies that f is continuous there), then yes, f'(3) = 2. What you said is correct too, due to the result proved here: http://math.stackexchange.com/quest...nction-has-a-limit-must-it-agree-with-that-li.

And yes, the answer is 0 for the other limit Q.

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Reasoning (1 Viewer)

leehuan

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InteGrand

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