# Can you check the follow statements about derivatives?

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#f'(c)# 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'(p)(x-p)f(p)# ?

An estimate of #f'(c)# 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_(xrarr0)(f(x+h)-f(x))/h# **What happens if the limit does not exist**??

*instantaneous* rate of change of *tangent* to the graph of

An estimate of *average* rate of change of **What happens if the limit does not exist**??

##### 1 Answer

*instantaneous* rate of change of

TRUE

It is the graph of the line *tangent* to the graph of

FALSE (I am assuming that "it" refers to "*slope* of the tangent line at

You can find this by the equation

FALSE (I am assuming that "this" refers to "the equation of the tangent line to the graph of

You can find the equation of the tangent line at

An estimate of *average* rate of change of

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

?what does "it" refer to?

You can find this by the limit of the difference quotient as

What does "this" refer to?

The derivative of

The derivative of

**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.