course Mth 174 y˻ assignment #006 V˪˵}Ӿ Physics II 06-23-2008
......!!!!!!!!...................................
17:25:09 Query problem 7.5.13 (3d edition #10) graph concave DOWN and decreasing (note changes indicated by CAPS)
......!!!!!!!!...................................
RESPONSE --> My order, from least to most, Exact, Right, Trapezoid, Mid, Left I checked in the back of the book and found out I was wrong. I messed up in figuring the exact value because when I used the Fundamental Theorem I did not create a the correct graph for int( f(x))
.................................................
......!!!!!!!!...................................
17:27:30 list the approximations and their rules in order, from least to greatest
......!!!!!!!!...................................
RESPONSE --> Right: f(1.5) * 1.5 + f(3) * 1.5 Trapezoid: (Left + Right) / 2 Exact: int[0,3] f'(x) dx = f(3) - f(0) Mid: f(.75) * 1.5 + f(1.25) * 1.5 Left; f(0) * 1.5 + f(1.5) * 1.5
.................................................
......!!!!!!!!...................................
17:27:52 between which approximations does the actual integral lie?
......!!!!!!!!...................................
RESPONSE --> Mid and Trapezoid
.................................................
......!!!!!!!!...................................
17:28:48 Explain your reasoning
......!!!!!!!!...................................
RESPONSE --> Well, my reasoning is that T < exact < M That was kind of a joke answer. I do not really know how to explain why.
.................................................
......!!!!!!!!...................................
17:30:02 if you have not done so explain why when a function is concave down the trapezoidal rule UNDERestimates the integral
......!!!!!!!!...................................
RESPONSE --> Because the trapezoids lie below the graph.
.................................................
......!!!!!!!!...................................
17:32:08 if you have not done so explain why when a function is concave down the midpoint rule OVERrestimates the integral
......!!!!!!!!...................................
RESPONSE --> If the graph is concave down, the top of the trapezoid (being the tangent line to the curve at the midpoint) lies above the graph, so it is overestimating
.................................................
......!!!!!!!!...................................
17:32:28 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
......!!!!!!!!...................................
RESPONSE -->
.................................................
......!!!!!!!!...................................
17:32:34 why is the area of the trapezoid h (L1 + L2) / 2?
......!!!!!!!!...................................
RESPONSE -->
.................................................
......!!!!!!!!...................................
17:32:37 Describe how you sketched the area E = h * f(0)
......!!!!!!!!...................................
RESPONSE -->
.................................................
......!!!!!!!!...................................
17:32:39 Describe how you sketched the area F = h * f(h)
......!!!!!!!!...................................
RESPONSE -->
.................................................
......!!!!!!!!...................................
17:32:41 Describe how you sketched the area R = h*f(h/2)
......!!!!!!!!...................................
RESPONSE -->
.................................................
......!!!!!!!!...................................
17:32:45 Describe how you sketched the area C = h * [ f(0) + f(h) ] / 2
......!!!!!!!!...................................
RESPONSE -->
.................................................
......!!!!!!!!...................................
17:32:48 Describe how you sketched the area N = h/2 * [ f(0) + f(h/2) ] / 2 + h/2 * [ f(h/2) } f(h) ] / 2
......!!!!!!!!...................................
RESPONSE -->
.................................................
......!!!!!!!!...................................
17:32:58 why is C = ( E + F ) / 2?
......!!!!!!!!...................................
RESPONSE -->
.................................................
......!!!!!!!!...................................
17:33:00 Why is N = ( R + C ) / 2?
......!!!!!!!!...................................
RESPONSE -->
.................................................
......!!!!!!!!...................................
17:33:02 Is E or F the better approximation to the area?
......!!!!!!!!...................................
RESPONSE -->
.................................................
......!!!!!!!!...................................
17:33:04 Is R or C the better approximation to the area?
......!!!!!!!!...................................
RESPONSE -->
.................................................
......!!!!!!!!...................................
18:03:28 query problem 7.5.24 show trap(n) = left(n) + 1/2 ( f(b) - f(a) ) `dx
......!!!!!!!!...................................
RESPONSE --> I am not sure if this is correct, but this is my attempt. (T = TRAP, L = Left, E = Exact, R = Right) f(b) - f(a) = Exact T(n) = L(n) + (1/2)(E(n)) T(n) = (L(n) + R(n)) / 2 so (L(n) + R(n)) / 2 = L(n) + (1/2)(E(n)) L(n) + R(n) = 2L(n) + E(n) R(n) = L(n) + E(n) Exact = Right - Left This is as far as I could get.
.................................................
......!!!!!!!!...................................
18:03:30 Explain why the equation must hold.
......!!!!!!!!...................................
RESPONSE -->
.................................................