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anonymous
 4 years 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?
anonymous
 4 years 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?

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anonymous
 4 years ago
Best ResponseYou've already chosen the best response.0A 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.

anonymous
 4 years ago
Best ResponseYou've already chosen the best response.0Use 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: \[m=\frac{ f(2+h)f(2) }{ h }=\frac{ 0 }{ h }=0\]

anonymous
 4 years ago
Best ResponseYou've already chosen the best response.0Which only happnes if the line is horizontal

anonymous
 4 years ago
Best ResponseYou've already chosen the best response.0you 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

anonymous
 4 years ago
Best ResponseYou've already chosen the best response.0The only way the change in y is zero is if y=constant at that instant :)

anonymous
 4 years ago
Best ResponseYou've already chosen the best response.0Thus, line is horizontal
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