Can you check the follow statements about derivatives?

$f ' \left(c\right)$ is the instantaneous rate of change of $f$ at $x = c$? It is the graph of the line tangent to the graph of $f$ at $x = c$? You can find this by the equation $y = f ' \left(p\right) \left(x - p\right) f \left(p\right)$? An estimate of $f ' \left(c\right)$ is the average rate of change of $f$ at $x = c$? It is the graph of the secant line whose points are close to $x = c$? You can find this by the limit of the difference quotient as $x$ approaches $0$? ${\lim}_{x \rightarrow 0} \frac{f \left(x + h\right) - f \left(x\right)}{h}$ What happens if the limit does not exist??

Aug 22, 2017

$f ' \left(c\right)$ is the instantaneous rate of change of $f$ at $x = c$?
TRUE

It is the graph of the line tangent to the graph of $f$ at $x = c$?
FALSE (I am assuming that "it" refers to "$f ' \left(c\right)$.) The derivative at $x = c$ is the slope of the tangent line at $x = c$

You can find this by the equation $y = f ' \left(p\right) \left(x - p\right) f \left(p\right)$?
FALSE (I am assuming that "this" refers to "the equation of the tangent line to the graph of $f$ at $x = c$.)
You can find the equation of the tangent line at $x = c$ using $y = f ' \left(c\right) \left(x - c\right) + f \left(c\right)$. (The statement given is ambiguous as to "the equation", uses the letter $p$, and also is missing an addition sign.)

An estimate of $f ' \left(c\right)$ is the average rate of change of $f$ at $x = c$?
False? (I would say meaningless.) There is no such thing as an average rate of change at one point. An average rate of change always involves 2 points.

It is the graph of the secant line whose points are close to $x = c$?
?what does "it" refer to?

You can find this by the limit of the difference quotient as $x$ approaches $0$? ${\lim}_{x \rightarrow 0} \frac{f \left(x + h\right) - f \left(x\right)}{h}$
What does "this" refer to?
The derivative of $f$ at $x = c$ is ${\lim}_{h \rightarrow 0} \frac{f \left(c + h\right) - f \left(c\right)}{h}$
The derivative of $f$ at $x$ is ${\lim}_{h \rightarrow 0} \frac{f \left(x + h\right) - f \left(x\right)}{h}$

What happens if the limit does not exist??
Nothing. There is no slope, there may be no tangent line depending on why the limit fails to exist.

If the limit does not exist because the limit increases or decreases without bound, then there is a vertical tangent line.
If the limit faiis to exist for some other reason, then there is no tangent line.