Assignment 6

course Mth 174

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Physics II

06-10-2009

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20:28:48

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

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20:30:17

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

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RIGHT less than or equal to TRAP less than or equal to exact value greater than or equal to MID greater than or equal to LEFT

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20:30:43

between which approximations does the actual integral lie?

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The actual integral lies between TRAP and MID

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20:35:04

Explain your reasoning

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I found this because the graph is decreasing, therefore RIGHT(n) is less than or equal to exact value which is less than or equal to LEFT(n).

The graph is concave down , so TRAP(n) is less than or equal to exact value is less than or equal to MID(n)

If the graph is decreasing then we know that the left end of an interval gives us a higher estimate than the right.

The trapezoidal estimate is the mean of the left and right estimates, so it will lie between the two. The trapezoidal estimate also corresponds to a straight-line estimate between the graph points at the left and right endpoing of each interval.

Since the graph is concave down, any straight line between two graph points will lie below the graph. In particular the value of the function at the midpoint will lie above the straight line. The trapezoidal estimate corresponds to the straight-line, estimate so in this case the midpoint estimate exceeds the trapezoidal estimate.

The straight-line trapezoidal area is also clearly less than the area beneath the curve, so the trapezoidal estimate is lower than the actual value of the integral.

The midpoint rule uses the function value at the midpoint for the altitude of the rectangle. Thus we imagine a rectangle whose top is a horizontal line segment through the midpoint value of the graph. You should sketch this.

If the function is concave downward, the part of the actual graph that lies between the endpoints and above the midpoint-value rectangle (this segment lies to the left of the midpoint) represents graph area under the actual graph which is left out of the midpoint approximation.  The area under the horizontal segment at the midpoint which lies above the graph (this are lies to the right of the midpoint) represents area included in the midpoint approximation but which is not part of the area under the curve.

Since the function is concave down, the average altitude of the ‘left-out’ area to the right is greater than that of the ‘wrongly-excluded’ area to the left. Since the widths of the two regions are equal, the wrongly excluded area must be less than the wrongly-included area and the midpoint estimate must therefore be high. We conclude that the actual area is less than the midpoint area.

Since the trapezoidal approximation is less than the actual area, we have our final ordering:

RIGHT < TRAP < exact < MID < LEFT.

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20:36:18

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

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This occurs because when the function is concave down the gragh curves under the trapezoid, so the total area is not calculated, therefore is underestimated.

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20:38:55

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

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This occurs because the graph goes through the midpoint of the trapezoid, but when calculated the area exceeds the graph on the ends of the section.

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20:39:39

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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20:39:40

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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20:39:48

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

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20:39:50

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

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20:39:52

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

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20:39:54

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

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20:39:57

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

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20:40:00

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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20:40:03

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

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20:40:05

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

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20:40:09

Is E or F the better approximation to the area?

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20:40:12

Is R or C the better approximation to the area?

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20:42:32

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

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20:45:52

Explain why the equation must hold.

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Integral of TRAP from b to a= LEFT(n) + 1/2(f(b)-f(a))

This equation must hold because TRAP=[ LEFT(n) + RIGHT(n) ]/ 2

1/2(f(b)-f(a)) must = RIGHT(n)

GEOMETRIC SOLUTION:

First suppose n = 1. You have only one trapezoid. In this case left(n) represents the altitude of a rectangle constructed on the left-hand side of the trapezoid, so left(n) * `dx represents the area of this rectangle.

| f(b) - f(a) | is the altitude of the triangle formed by the top part of the trapezoid. Assuming b > a then if the slope of the top of the trapezoid is positive the area of the triangle is added to that of the rectangle to get the area of the trapezoid. If the slope is negative the area is subtracted.

The area of the triangle is 1/2 * | f(b) - f(a) | * `dx.

1/2 ( f(b) - f(a) ) * `dx is equal to the area if the slope is positive so that f(b) > f(a), and to the negative of the area if the slope is negative so that f(b) < f(a).

It follows that left(n) * `dx + 1/2 ( f(b) - f(a) ) * `dx = area of the trapezoid.

If you have n trapezoids then if a = x0, x1, x2, ..., xn = b represents the partition of the interval [ a, b ] then the contributions of the small triangles at the tops of the trapezoids are 1/2 ( f(x1) - f(x0) ), 1/2 ( f(x2) - f(x1) ), ..., 1/2 ( f(xn) - f(x(n-1)) ). These contributions add up to

1/2 ( (f (x1) - f(x0)) + (f (x2) - f(x1)) + ... + (f (xn) - f(x(n-1))) ). Rearranging we find that all terms except f(x0) and f(xn) cancel out:

1/2 ( -f(x0) + f(x1) - f(x1) + f(x2) - f(x2) + ... + f(x(n-1)) - f(x(n-1) + f(xn) ) =

1/2 (f(xn) - f(x0) ) = 1/2 ( f(b) - f(a) ).

Left(n) is the sum of all the areas of the left-hand rectangles, so left(n) + 1/2 ( f(b) - f(a) ) * `dx represents the areas of all the rectangles plus the contributions of all the small triangles, which gives us the trapezoidal approximation.**

SYMBOLIC SOLUTION: Assuming a < b and partition a = x0, x1, x2, ..., xn = b we have

left(n)  = [ f(x0) + f(x1) + ... + f(x(n-1) ] * `dx

and

trap(n) = 1/2 [ f(x0) + 2 f(x1) + 2 f(x2) + ... + 2 f(x(n-1)) + f(xn) ] * `dx.

So

trap(n) - left(n) = [-1/2 f(x0) + 1/2 f(xn) ] * `dx. Since x0 = a and xn = b this gives us

trap(n) = left(n) + 1/2 ( f(b) - f(a) ) * `dx.

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20:50:33

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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You figure out left(n) by dividing the graph into sections, multiplying the width of each section by its left corner, or f(a) and then adding them all together by a Riemann sum. You figure out trap(n) by going through this same procedure to find left(n), then finding right(n) by multiplying the right corners, or f(b) to the widths and adding them together. Then you take the average of the left(n) and right(n).

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&#This looks good. See my notes. Let me know if you have any questions. &#