question 6

#$&*

course Mth 174

Question: Section 7.5 Problem 3

4th edition 7.5.14 problem 7.5.7 graph concave up and decreasing

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

greatest

**** between which approximations does the actual integral lie?

.

YYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYY

Your solution:

Because the function is decreasing and concave up, RIGHT(n)

Therefore, RIGHT(n)< MID(n)

confidence rating #$&*: 3

^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^

.............................................

Given Solution:

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.

.........................................

23:04:39

&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&

Self-critique (if necessary):

------------------------------------------------

Self-critique Rating: ok

problem 4th edition 7.5.13 was 7.5.10 graph concave DOWN and decreasing (note changes indicated by CAPS) **** list the approximations and their rules in order, from least to greatest

.......!!!!!!!!...................................

11:18:42

YYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYY

Your solution:

Because the function is decreasing and concave down, RIGHT(n)

Therefore,

RIGHT(n)< TRAP(n)< int(f(x)dx

confidence rating #$&*: 3

^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^

.............................................

Given Solution:

Right: if f is decreasing Right < `intf(x) < Left

Trapeziod: if f is concave down Trap < intf(x) < Mid

Exact: if f is concave down Trap < intf(x) < Mid and if f is decreasing Right < `intf(x) < Left, Trap and Mid are closer approximations than right and left

Mid: if f is concave down Trap < intf(x) < Mid

Left: if f is decreasing Right < `intf(x) < Left

..........................................

11:18:43

.......!!!!!!!!...................................

**** between which approximations does the actual integral lie?

.......!!!!!!!!...................................

11:18:54

Trapeziod and midpoint

..........................................

11:18:54

.......!!!!!!!!...................................

**** Explain your reasoning

.......!!!!!!!!...................................

11:20:11

Because those two approximations are closer approximations than right and left, because they have some areas on both sides of the function so the spots that are over and under balance out, whereas right and left approximations do not

..........................................

11:20:12

.......!!!!!!!!...................................

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

.......!!!!!!!!...................................

11:26:14

Because the corners touch the points a and b, and since the curve is pushed upward, there is a gap between the curve and the trapezoid

..........................................

11:26:14

.......!!!!!!!!...................................

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

.......!!!!!!!!...................................

11:30:21

The midpoint error is less clear, but to see that one, you have to draw a line tangent to the curve at the midpoint of the interval, then by shifting the area of the rectangle that is outside this tangent to make a rectangle to form a trapezoid along this tangent, you'll see that on a concave down curve, the tangent is on top of the curve therefore having area outside the curve, producing an overestimate.

......!!!!!!!!...................................

&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&

Self-critique (if necessary):

------------------------------------------------

Self-critique Rating: ok

**** Query temporarily dumbed out problem 7.5.18 graph positive, decreasing, concave upward

over interval 0 < x < h.

The trapezoidal approximation to this graph consists of 'left altitude' L1, 'right altitude' L2 and 'width' h.

The 'left altitude' L1 corresponds to f(0), the value of the function at x = 0.

The 'right altitude' L2 corresponds to f(h), the value of the function at x = h.

The graph is decreasing so L1 > L2.

The graph is concave up so it 'dips below' the trapezoidal approximation.

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

(L1 + L2) / 2 is the 'average altitude', or 'midpoint altitude' of the trapezoidal approximation, and so is an approximation to the average value and midpoint value of the fucttion.

Note that this is only an approximation. Since the graph is curved the 'average altitude' or 'midpoint altitude' of the trapezoid does not correspond to either the average value or the midpoint value of the function on this interval.

The area of a trapezoid is the average altitude multiplied by the width of the corresponding interval.

We have used trapezoidal approximation graphs since the early part of MTH 173.

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

h * f(0) is the area of a rectangle of width h and altitude f(0), which is the 'left altitude' of the graph.

Since the graph is decreasing this is an upper bound for the area beneath the curve.

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

h * f(h) is the area of a rectangle of width h and altitude f(h), which is the 'right altitude' of the graph.

Since the graph is decreasing this is a lower bound for the area beneath the curve.

.........................................

23:09:09

......!!!!!!!!...................................

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

x = 0 is the left end of the interval of the x axis, x = h the right end.

h / 2 is the midpoint of the interval.

f(h/2) is the 'midpoint altitude' of the actual graph.

Since the graph is concave up, this 'midpoint altitude' of the actual graph is less than the 'midpoint altitude' of the trapezoidal approximation.

......!!!!!!!!...................................

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

C is merely the area of the trapezoidal approximation, [ f(0) + f(h) ] / 2 being the average of the altitudes over the interval h and h being the width of the interval.

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

STUDENT SOLUTION:

This is merely the sum of two areas, which makes it look like the previous graph, only it is cut in half at h/2, making it more accurate than the trap approx.

INSTRUCTOR COMMENT:

Right. If you take the original graph and approximate it by two trapezoids, one running from x = 0 to x = h/2 and the other from x = h/2 to x = h, the three 'graph altitudes' are f(0), f(h/2) and f(h). You get trapezoids with areas h/2 * [ f(0) + f(h/2) ] / 2 and h/2 * [ f(h/2) } f(h) ] / 2, with total area h/2 * [ f(0) + f(h/2) ] / 2 + h/2 * [ f(h/2) + f(h) ] / 2.

The line segments from (0, f(0)) to (h/2, f(h/2)) to (h, f(h)) lie closer to the curve than the line segment from (0, f(0)) to (h, f(h)), so the area approximation h/2 * [ f(0) + f(h/2) ] / 2 + h/2 * [ f(h/2) } f(h) ] / 2 is closer to the actual area beneath the curve than the original area approximation h * [ f(0) + f(h) ] / 2.

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

Geometrically, E is the area of the 'left rectangle' and F the area of the 'right rectangle'. The trapezoid 'splits the difference' between these areas, so its area lies halfway between those of the two rectangles. (E + F) / 2 is halfway between E and F.

Symbolically, E = h * f(0) and F = h * f(h) so (E + F) / 2 = (h * f(0) + h ( f(h) ) / 2 = h * ( f(0) + f(h) ) / 2, which is identical to C.

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

N is an avg of the graphs of R and C. This can be observed by looking at the

corresponding graphs

**** Is E or F the better approximation to the area?

** F is closer. The area 'wrongly included' under the graph of E is greater than the area 'wrongly excluded' by the graph of F because the average thickness of the former region is greater than that of the latter. The upward concavity of f means that it's closer to the right-hand approximation than to the left-hand approximation for most of the interval. **

**** Is R or C the better approximation to the area?

** This is the midpoint vs. trapezoidal approximation situation again. Sketch a picture similar to that described on the preceding problem and identify the regions corresponding to 'wrongly included' and 'wrongly excluded' areas under each of the approximations. Compare the two and see if you can figure out whether the underestimate of the midpoint graph is greater or less than the overestimate of the trapezoidal graph. **

......!!!!!!!!...................................

&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&

Self-critique (if necessary):

------------------------------------------------

Self-critique Rating:

**** query problem 7.5.22 show trap(n) = left(n) + 1/2 ( f(b) - f(a) ) ] `dx **** Explain why the equation must hold.

......!!!!!!!!...................................

YYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYY

Your solution:

LEFT(n) = (b-a)/n f(a) + f(x1) + ... + f(xn) + f(b)]

TRAP(n) = (b-a)/n (f(a) + f(x1))/2 + (b-a)/n (f(x1) + f(x2))/2 + .... + (b-a)/n (f(xn) + f(b))/2 = (b-a)/n [f(a)/2 + f(b)/2 + f(x1) + ... + f(xn)]

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

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

confidence rating #$&*: 3

^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^

.............................................

Given Solution:

I understand and see that the equation does indeed work, but I am having trouble explaining why. I can picture on a graph the fact that by taking f(b) - f(a) we are taking the difference in the value of the function at points a and b. We then multiply this value by 'dx, which is determined by the number of subdivisions chosen. After doing this, we multiply by 1/2 and arrive at the value representing the area of the trapezoid. This is the best I can come up with, but I'm almost sure there is a more rigorous explanation.

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.

.........................................

23:17:22

......!!!!!!!!...................................

.........................................

&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&

Self-critique (if necessary):

------------------------------------------------

Self-critique Rating: ok

**** Query Add comments on any surprises or insights you experienced

as a result of this assignment.

......!!!!!!!!...................................

23:19:34

This was a very informative exercise, and it shows how important learning

about that trapezoidal approximation graph was in early Calculus I. I am

interested in seeing how f'' affects the accuracy of these techniques, or

how it shows something more about them. I think I am supposed to find that

out in the next assingment, according to class notes. I find this part of

the course to be very interesting.

I don’t know why a couple of the question came with the answer right after it, including this one.

"

Self-critique (if necessary):

------------------------------------------------

Self-critique rating:

**** Query Add comments on any surprises or insights you experienced

as a result of this assignment.

......!!!!!!!!...................................

23:19:34

This was a very informative exercise, and it shows how important learning

about that trapezoidal approximation graph was in early Calculus I. I am

interested in seeing how f'' affects the accuracy of these techniques, or

how it shows something more about them. I think I am supposed to find that

out in the next assingment, according to class notes. I find this part of

the course to be very interesting.

I don’t know why a couple of the question came with the answer right after it, including this one.

"

@& If the second derivative was zero, the graph would be linear and the trapezoid would be a perfect approximation.

You will find it interesting to think about how the curvature of the graph affects the trapezoidal and midpoint approximations in subtly different ways. As you expect, that is coming up very soon.

The editing of these documents is in pretty good shape, but there are some glitches.*@

Self-critique (if necessary):

------------------------------------------------

Self-critique rating:

#*&!

&#Very good responses. Let me know if you have questions. &#