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4 Pages

### oz83

Course: MA 114, Fall 2008
School: Kentucky
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Word Count: 691

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Homework # from Ostobee Zorn # # 8.3 Arclength # # #2 Verify that the arclength formula gives the correct # answer for the length of the graph of y = mx +b from # x =0 to x = 1. # # Solution: The correct answer is the distance from (0,b) # to (1,m+b), or sqrt(1+m^2). The arclength formula # gives &gt; Int(sqrt(1+diff(m*x+b,x)^2),x=0..1) &gt; =int(sqrt(1+diff(m*x+b,x)^2),x=0..1);...

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Homework # from Ostobee Zorn # # 8.3 Arclength # # #2 Verify that the arclength formula gives the correct # answer for the length of the graph of y = mx +b from # x =0 to x = 1. # # Solution: The correct answer is the distance from (0,b) # to (1,m+b), or sqrt(1+m^2). The arclength formula # gives > Int(sqrt(1+diff(m*x+b,x)^2),x=0..1) > =int(sqrt(1+diff(m*x+b,x)^2),x=0..1); 1 / | 2 1/2 2 1/2 | (1 + m ) dx = (1 + m ) | / 0 # # 6 Find the length of the curve y = x^2 fro 1 to 2 to # within .005 # # # Solution: The length is the integral, which can # be evaluated easily by Maple. # > Int(sqrt(1+diff(x^2,x)^2),x=1..2)=int(sqrt(1+diff(x^2,x)^2),x=1..2); > evalf("); 2 / | 2 1/2 | (1 + 4 x ) dx = | / 1 1/2 1/2 1/2 1/2 17 - 1/4 ln(-4 + 17 ) - 1/2 5 - 1/4 ln(2 + 5 ) 3.168 = 3.168 # We want to see how many partitions we would need # to approximate this number to within .005 using # the right rule Rn. # # We can use theorem 2 to bound the error, # taking k1 to be the maximum of the absolute value # of the derivative, which we can get # # > restart; > f := x->sqrt(1+4*x^2); 2 f := x -> sqrt(1 + 4 x ) > fp := D(f); x fp := x -> 4 -------------- 2 sqrt(1 + 4 x ) > plot(fp, 1..2); > # So a suitable value for k1 is 2 # We want n large enough that k1(b-a)^2/(2n) = 1/n < .005 # # solving the inequality for n, we see that any n > 200 # will work by theorem 2. Using Maple, > bound := 2*(2-1)^2/(2*n); bound := 1/n > solve(bound = .005,n); 200. # checking this > R200 := (2-1) /(200)* sum(f(1+i/200.) ,i=1..200) ; R200 := 3.172558813 > evalf(R200-int(f(x),x=1..2) ); .0047179088 # The actual error using R200 is just slightly under the maximum error # predicted by theorem 2. # # # 8.4 Work # # #1 Suppose that the spring on a bathroom scale is stretched by .06 # inches when a 115 lb person stands # on the scale. # # a) how much work is done on the spring when this person steps on the # scale? # # Solution: Get the spring constant k using Hooke's law # # 115 = k * .06, so > k := 115/.06; k := 1917. # The work is the integral of the force k*x with respect to distance dx # as x goes from 0 to .06 > work := int(k*x,x=0..(.06)); work 3.451 # := So around 3 and 1/2 inch-pounds of work is done on the # spring. # # Note: The statement of the problem is subject to interpretation. We have # calculated the amount of work needed to stretch the spring out .06 # inches. This means that as the person steps on the scale, the spring # begins to stretch out before his full weight gets on the scale. # # b) How much work is done when a 175 lb person steps on # the scale? # # Solution: How far will the spring stretch? # Then integrate the force from 0 to that distance > eqn := 175 = k*x0; eqn := 175 = 1917. x0 > x0 := solve(eqn,x0); x0 := .09129 > work := int(k*x,x=0..x0); work := 7.988 # Note that increasing the weight by 50 lbs (less than 50 # percent) increases the work by more than 100 percent. # # We can see that more generally by letting the weight # increase by p percent and asking to calculate the percentage increase # in work > eqn := 115 + p*115 = k*xp; eqn := 115 + 115 p = 1917. xp > xp := solve(eqn,xp); xp := .05999 + .05999 p > work := int(k*x,x=0..xp); 2 work := 3.449 + 6.899 p + 3.449 p > wp := (work - 3.45)/3.45*100; 2 wp := -.01 + 200.0 p + 99.99 p > plot(wp,p=0..20); # We can conclude that the percentage increase # in work wp is a quadratic function of p, a percentage increase in # the weight on the spring. # # #4 A cylindrical gasoline tank with radius 4 and # length 15 is buried on its side under a service # station. The top of th...

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Kentucky - MA - 114
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# Ostobee Zorn Chapter 9# # 9.1 #7 The hint is not needed.# &gt; with(student);Warning, new definition for D[D, Diff, Doubleint, Int, Limit, Lineint, Product, Sum, Tripleint, changevar, combine, completesquare, distance, equate, extrema
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Kentucky - MA - 114
%!PS-Adobe-2.0 %Creator: dvipsk 5.58f Copyright 1986, 1994 Radical Eye Software %Title: ws2.dvi %Pages: 1 %PageOrder: Ascend %BoundingBox: 0 0 612 792 %EndComments %DVIPSCommandLine: dvips -p1 -l1 ws2 %DVIPSParameters: dpi=300, comments removed %DVIP
Kentucky - MA - 114
%!PS-Adobe-2.0 %Creator: dvipsk 5.58f Copyright 1986, 1994 Radical Eye Software %Title: ws3.dvi %Pages: 2 %PageOrder: Ascend %BoundingBox: 0 0 612 792 %EndComments %DVIPSCommandLine: dvips ws3 %DVIPSParameters: dpi=300, comments removed %DVIPSSource:
Kentucky - MA - 114
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Kentucky - MA - 114
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Kentucky - MA - 114
%!PS-Adobe-2.0 %Creator: dvipsk 5.58f Copyright 1986, 1994 Radical Eye Software %Title: hw3.dvi %Pages: 6 %PageOrder: Ascend %BoundingBox: 0 0 612 792 %EndComments %DVIPSCommandLine: dvips hw3 %DVIPSParameters: dpi=300, comments removed %DVIPSSource:
Kentucky - MA - 114
# 7.1 &gt; with(student);[D, Diff, Doubleint, Int, Limit, Lineint, Product, Sum, Tripleint, changevar, combine, completesquare, distance, equate, extrema, integrand, intercept, intparts, isolate, leftbox, leftsum, makeproc, maximiz
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%!PS-Adobe-2.0 %Creator: dvipsk 5.58f Copyright 1986, 1994 Radical Eye Software %Title: skills1.dvi %Pages: 4 %PageOrder: Ascend %BoundingBox: 0 0 612 792 %EndComments %DVIPSCommandLine: dvips skills1 %DVIPSParameters: dpi=300, comments removed %DVIP
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