Assume F'(x)= f(x) and the Fundamental Theorem of Calculus applies in each case. For each problem complete the statement with if the value is a slope, point

Question

Assume F'(x)= f(x) and the Fundamental Theorem of Calculus applies in each case. For each problem complete the statement with if the value is a slope, point, length, or area.

I watched some videos but I am really having trouble grasping the FTC. I don't understand the what F'(x) is in relation to f(x). My notes say its the anti derivative, but I don't know what that means exactly.

 

Problem three I am having trouble on. It says:

On a graph of f(x), ∫A to B  f(x)dx represents

and on a graph of F(x), ∫A to B  f(x)dx represents

So for the first blank in the sentence I was thinking maybe the area between the curve and the x-axis and the second I don't know, a length?

 

The fifth question says

On a graph of f(x), (∫A to B  f(x)dx)/b-a represents
and on a graph of F(x), (∫A to B  f(x)dx)/b-a represents

 

For the first blank on this one I was thinking the average function value? but I really don't know how to fill these two blanks. Please help me!

 

Answer 1

So the FTC quite simply says that derivative and integral (or the anti-derivative) are like inverses, they cancel each other out.  When I have a function f(x) if I take the anti-derivative (or indefinite integral) we usually denote that F(x) if I take the derivative of F(x), I'm back to my original function, or in math notation F'(x)=f(x).

 

The notation is a little hard to do right here and I'm not entirely sure I understand the way the questions are phrased, but

 

The other thing of the FTC you should sort of think about is ∫A to B of f(x)dx = F(B)-F(A) where F(x) is the anti-derivative of f(x).  That will help you think through the part b of both questions below

 

3a) Would be an area, it would be the area under the curve of f(x) between a and b

3b) Would be a "length" it is F(b)-F(a) or how much your F(x) increases from a to b

 

5a) Would be a "point" its the average value of the function, that is a "point" on the curve

5b) Would be a "slope" if you thought about it literally it would be (F(b)-F(a)) / (b-a) which is the definition of the slope between those two points on the curve