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True or False and explain why?
For a differentiable function y=f(x), f'(2)=0 means that the tangent line to the graph of f at x=2 is horizontal.
I know it's true, but how do I explain why?
 one year ago
 one year ago
True or False and explain why? For a differentiable function y=f(x), f'(2)=0 means that the tangent line to the graph of f at x=2 is horizontal. I know it's true, but how do I explain why?
 one year ago
 one year ago

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SujayBest ResponseYou've already chosen the best response.0
A derivative gives the slope of a tangent point for any point where it applies. So since the slope at x=2 would be 0, that means it must be a horizontal line.
 one year ago

quantum77Best ResponseYou've already chosen the best response.1
Use the definition limit definition of a derivative: If thederivative is 0 then \[f \prime(2)=\lim_{h \rightarrow 0}=\frac{ f(2+h)f(2) }{ h }\] If this is zero than This implies that f(2+h)f(2)=0, so thinking about the defintion of the slope: \[m=\frac{ f(2+h)f(2) }{ h }=\frac{ 0 }{ h }=0\]
 one year ago

quantum77Best ResponseYou've already chosen the best response.1
Which only happnes if the line is horizontal
 one year ago

eseidlBest ResponseYou've already chosen the best response.0
you could prove this using the definition of the derivative:\[f'(x)=\lim_{h \rightarrow 0} \frac{f(x+h)f(x)}{h}\]Another way to say this is:\[f'(x)=\lim_{\Delta x \rightarrow 0}\frac{\Delta y}{\Delta x}=\frac{dy}{dx}\]Graphically, this is the slope of the tangent line at the point as @Sujay said.dw:1354847377868:dw
 one year ago

eseidlBest ResponseYou've already chosen the best response.0
The only way the change in y is zero is if y=constant at that instant :)
 one year ago

eseidlBest ResponseYou've already chosen the best response.0
Thus, line is horizontal
 one year ago
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