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Area and fundamental theorem

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http://img521.imageshack.us/img521/4549/bbav9.png [Broken]

The problem states that it wants the upper and lower estimate of total distance. Therefore, I used rectangles to solve for it. However, let's say I'm working on upper limits. For my initial rectangles, I use the right endpoints, but then it begins to slope downward, so at that point, if I use right endpoints, I am below the graph. Do I continue using right endpoints, or do I switch to left endpoints to have my rectangles stay over the graph?

http://img521.imageshack.us/img521/22/bb2by3.png [Broken] <--I forgot to put on there that this curve = f(x)

This problem states... for each quantity in questions a-d, copy the diagram and show the quantity on the diagram. Explain what the expression means in terms of the graph. Note that F'(x) = f(x).

For example if the question showed the quantity square root of (b-a)^2 + (f(b)-f(a))^2 then you would explain that this is the distance on PQ and you would draw a line segment PQ on the diagram.

a) f(b)-f(a)

b) f(b)-f(a)/b-a

c) F(b) - F(a)

d) F(b) - F(a)/b-a

I have no idea how to start this.. b and d both seem like slope equations to me, but I don't know the difference.
 
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Answers and Replies

  • #2
Gib Z
Homework Helper
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1st problem, switch. Or better, use midpoint rectangles, or simpsons rule.

2nd part,

a) part a perpendicular line, long enough to cross the segment from B(b) to q. The distance from the point of intersect, and q, is what you want.

b) average change, gradient, gradient of secant connting those 2 points.
c)Area between b and a. shade it in.
d) the average value of the function. Its the height of the rectable with base b to a, with the same area as part c.
 

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