what is the purpose of limits and derivatives in calculus?

Question

fanduekisses · Answer

Please correct me if I'm wrong...

fanduekisses · Answer

So derivatives are basically slopes right?

anonymous · Answer

derivative is the slope of the tangent line

fanduekisses · Answer

but are limits and derivatives related? does it mean that you can only find the slope of a curve at certain points? or at certain limits?

anonymous · Answer

the limit tells you what the function approaches but does not touch

anonymous · Answer

how are they related is a very good question lol... im only a calc 2 student. i know how to solve them and have a pretty basic idea of what they are but you're over my head at that point

amistre64 · Answer

the definition of a derivative is that it is a limit:$$f'(x)=\lim_{h\to0}\frac{f(x+h)-f(x)}{(x+h)-(x)}$$

fanduekisses · Answer

that's like y2-y1/x2-x1 right

amistre64 · Answer

yeah, its a slope formula, but the limit as one point approaches the the other

amistre64 · Answer

now a derivative is NOT a slope of a line.  The slope of the line that is tangent to a curve can be determined by a derivative.

anonymous · Answer

so the derivative is NOT the slope of the tangent line?

amistre64 · Answer

the slope of the tangent line to the curve can be calculated using derivatives.  a derivative is a rate of change.

amistre64 · Answer

im reading your definition as:  a dog is a german shepherd.  which is backwards since not all dogs are german shepherds

fanduekisses · Answer

so it can be said that the slope of a tangent line is the slope of a curve?

amistre64 · Answer

the slope of a tangent line to a curve can be calculated using derivatives, since the slope expresses the rate of change at an instant in time.

amistre64 · Answer

derivatives are defined by limits,  but not all limits are derivatives.

the relationship between them is expressed by the limit definition i posted above

fanduekisses · Answer

oohh I see, and so since we need two points for a tangent line , we are trying to get the two points as close as possible, but don't let them have the same value.

fanduekisses · Answer

just approaching

anonymous · Answer

you only need one point for a tangent line don't you?

amistre64 · Answer

we need 2 points for a secant line :)

as the points approach each other, the slope between them approaches the slope of the tangent at 1 point

amistre64 · Answer

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