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A basic calculus question? 
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#1
Jul2910, 11:44 AM

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Okay, I am new to calculus so can you guys just explain something to me.
What is exactly are you doing to a function when you differentiate or take the derivative of it? I thought that when you do that, you make an instantaneous solution to where something is at all times with respect to the other axis. Is that correct? Also, whenever you see calculus in advanced physics, I don't understand. You can only use indefinite integrals and derivatives when you have a valid function right? So all these advanced physics problems need functions to go along with them right? Thank you guys! 


#2
Jul2910, 12:19 PM

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Try to formulate what the following quantity is, for nonzero h: [tex]\frac{f(x+h)f(x)}{h}[/tex] Hint: Draw an arbitray graph, pick two points on the xaxis ("x" and "x+h"), and think of h=(x+h)x 


#3
Jul2910, 12:23 PM

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#4
Jul2910, 12:24 PM

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A basic calculus question?
The derivative of a function is another function that gives the slope of the tangent line to the graph of the first function.



#5
Jul2910, 12:27 PM

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#6
Jul2910, 12:48 PM

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a) First off, what do you call a straight line going through two points on the graph? b) What, EXACTLY, is the QUANTITY you calculate? A line is not a "quantity" in this sense! 


#7
Jul2910, 12:50 PM

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b) You are calculating the area under the curve???? 


#8
Jul2910, 12:53 PM

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If not, what geometric interpretation does this ratio have? 


#9
Jul2910, 12:58 PM

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#10
Jul2910, 01:05 PM

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Now I get some of your problems. No, the calculated number will not be "tangent" to the curve here, but it WILL give you the tangent value associated with the angle formed by the secant line and the horizontal line formed at the point (x,f(x)). You have a natural triangle here in the (x,y)plane, with corners: (x,f(x)), (x+h,f(x)) and (x+h,f(x+h)) We usually call that ratio the slope of the line. Have you heard that expression? 


#11
Jul2910, 01:12 PM

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#12
Jul2910, 01:25 PM

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No, let us forget about that angle for now. Okay?
1. The most important to remember about that slope, is that it tells you how steep the secant line is between the two points on the graph. Agreed? 2. Now, a tangent line at some point on the graph is that line which a) intersects with graph there, and b) is equally steep as the curve itself there. 3. When you perform the limiting operation (letting h be smaller and smaller) in order to calculate the derivative, you are simply calculating the slopes of successive secant lines and, in the end, when your two points overlap (h=0), sit back with the slope of the tangent line at (x,f(x)), i.e, the slope of the curve itself 4. So, the derivative at some point tells you how steep the curve is at that point. 


#13
Jul2910, 01:30 PM

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