Assignment 6

course MTH 174

n?????????????assignment #006

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Physics II

11-25-2007

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20:05:18

Query problem 7.5.13 (3d edition #10) graph concave DOWN and decreasing (note changes indicated by CAPS)

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RIGHT(n), TRAP(n), Exact value, MID(n), LEFT(n).

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20:07:34

list the approximations and their rules in order, from least to greatest

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RIGHT(n), TRAP(n), Exact value, MID(n), LEFT(n).

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20:07:54

between which approximations does the actual integral lie?

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Between TRAP(n) and MID(n)

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20:10:03

Explain your reasoning

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RESPONSE -->

Since the graph is concave down and decreasing,

LEFT will over-estimate the most

RIGHT will under-estimate the most

TRAP will under-estimate the actual, but will be closer than RIGHT

MID will over-estimate the actual, but will be closer than LEFT

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20:12:10

if you have not done so explain why when a function is concave down the trapezoidal rule UNDERestimates the integral

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MID:

A line(within a domain) tangent to a curve that is concave down will extend the trapezoid to some area above the curve.

TRAP:

A line connecting two points on a curve (secant) that is concave down, defining the top of the trapezoid, will lie in an area below the curve.

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20:12:22

if you have not done so explain why when a function is concave down the midpoint rule OVERrestimates the integral

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20:12:30

Query NOTE: this problem has been left out of the new edition of the text, which is a real shame; you can skip on to the next problem (was problem 7.5.18) graph positive, decreasing, concave upward over interval 0 < x < h

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20:12:32

why is the area of the trapezoid h (L1 + L2) / 2?

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20:12:33

Describe how you sketched the area E = h * f(0)

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20:12:35

Describe how you sketched the area F = h * f(h)

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20:12:37

Describe how you sketched the area R = h*f(h/2)

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20:12:40

Describe how you sketched the area C = h * [ f(0) + f(h) ] / 2

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20:12:43

Describe how you sketched the area N = h/2 * [ f(0) + f(h/2) ] / 2 + h/2 * [ f(h/2) } f(h) ] / 2

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20:12:46

why is C = ( E + F ) / 2?

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20:12:47

Why is N = ( R + C ) / 2?

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20:12:49

Is E or F the better approximation to the area?

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20:12:51

Is R or C the better approximation to the area?

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21:01:09

query problem 7.5.24 show trap(n) = left(n) + 1/2 ( f(b) - f(a) ) `dx

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RESPONSE -->

The trapezoid rule is:

TRAP(n) = (LEFT(n) + RIGHT(n)) / 2

Since,

| LEFT(n) - RIGHT(n) | = ( f(b) - f(a) ) `dx

Then,

1/2 | LEFT(n) - RIGHT(n) | = 1/2 ( f(b) - f(a) ) `dx

It follows that,

TRAP(n) = LEFT(n) + 1/2 ( f(b) - f(a) ) `dx

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21:05:55

Explain why the equation must hold.

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RESPONSE -->

If f is increasing over the interval [a,b],

1/2 ( f(b) - f(a) ) `dx = the area of a triangle that when added to LEFT(n) = the trapezoid under that region.

The trapezoid rule is:

TRAP(n) = (LEFT(n) + RIGHT(n)) / 2

Since,

| LEFT(n) - RIGHT(n) | = ( f(b) - f(a) ) `dx

Then,

1/2 | LEFT(n) - RIGHT(n) | = 1/2 ( f(b) - f(a) ) `dx

It follows that,

TRAP(n) = LEFT(n) + 1/2 ( f(b) - f(a) ) `dx

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21:10:05

In terms of a graph describe how trap(n) differs from left(n) and what this difference has to do with f(b) - f(a).

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RESPONSE -->

If f is increasing over the interval [a,b],

1/2 ( f(b) - f(a) ) `dx = the area of a triangle that when added to LEFT(n) equals the trapezoid under that region.

( f(b) - f(a) ) `dx = the area of a rectangle that equals the difference between LEFT(n) and RIGHT(n). The difference between the trapezoid and LEFT(n) = half of the rectangle.

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Once more, excellent work.