MATH 156 MIDTERM OCTOBER 2006In this exam you will work with the integral NiMtSSRpbnRHNiI2JCwmKiYiIiUiIiItSSVzcXJ0RzYkSSpwcm90ZWN0ZWRHRi5JKF9zeXNsaWJHRiU2IywmIiIjRioqJEkieEdGJUYyISIiRipGKkYyRjUvRjQ7IiIhRio= . It ends up that it is equal to NiMlI1BpRw==. So this exam has as purpose to find approximations of NiMlI1BpRw==.Answers all the questions. When an explanation is required, your answer should not just be: "I saw it on the graph" or "The numbers show that A<B". Explain your answers using the properties of the function involved. Do not forget the student and plots packages. If you are graphing two sums on the same graph, the order you are displaying them plays a role: you may need to interchange the order of display to see both of them e.g. display(A, B) will not produce the same result as display(B, A).1. Graph the function NiMtJSJmRzYjJSJ4Rw===4*sqrt(2-x^2)-2 in the range [0,1]. f:=x->4*sqrt(2-x^2)-2;NiM+SSJmRzYiZio2I0kieEdGJUYlNiRJKW9wZXJhdG9yR0YlSSZhcnJvd0dGJUYlLCYtSSVzcXJ0R0YlNiMsJiIiIyIiIiokOSRGMSEiIiIiJSEiI0YyRiVGJUYlplot(f(x), x=0..1);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2. Use Maple to decide whether the function NiMtJSJmRzYjJSJ4Rw== is increasing or decreasing.From the graph we see that it is a decreasing function. However, we check that the derivative is negative:g:=D(f);NiM+SSJnRzYiZio2I0kieEdGJUYlNiRJKW9wZXJhdG9yR0YlSSZhcnJvd0dGJUYlLCQqJjkkIiIiLUklc3FydEc2JEkqcHJvdGVjdGVkR0YzSShfc3lzbGliR0YlNiMsJiIiI0YvKiRGLkY3ISIiRjkhIiVGJUYlRiU=plot(g(x), x=0..1);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3. Use Maple to decide whether the function NiMtJSJmRzYjJSJ4Rw== is concave upwards or concave downwards.From the graph we see that f is concave downwards. We check that the second derivative is negative:h:=D(g);NiM+SSJoRzYiZio2I0kieEdGJUYlNiRJKW9wZXJhdG9yR0YlSSZhcnJvd0dGJUYlLCYqJC1JJXNxcnRHNiRJKnByb3RlY3RlZEdGMUkoX3N5c2xpYkdGJTYjLCYiIiMiIiIqJDkkRjUhIiJGOSEiJSomRjhGNUYuISIkRjpGJUYlRiU=plot(h(x), x=0..1);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4. Plot on the same graph the function and the left-hand sums and right-hand sums with the same number of subintervals for NiMlIm5H=2, 4, 8, e.g. LHS(8) and RHS(8). Which are larger and why?5. Plot on the same graph the function and the left-hand sums and midpoint sums with the same number of subintervals NiMlIm5H=2, 4, 8, e.g. LHS(8) and MID(8). Which are larger and why?6. Plot on the same graph the function and the right-hand sums and midpoint sums with the same number of subintervals NiMlIm5H=2, 4, 8, e.g. RHS(8) and MID(8). Which are larger and why?lhs2:=leftbox(f(x), x=0..1, 2):lhs4:=leftbox(f(x), x=0..1, 4):lhs8:=leftbox(f(x), x=0..1, 8):rhs2:=rightbox(f(x), x=0..1, 2):rhs4:=rightbox(f(x), x=0..1, 4):rhs8:=rightbox(f(x), x=0..1, 8):display(rhs2, lhs2);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display(rhs4, lhs4);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(rhs8, lhs8);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The right-hand sums are smaller than the left-hand sums since the function is decreasing.mid2:=middlebox(f(x), x=0..1, 2):mid4:=middlebox(f(x), x=0..1, 4):mid8:=middlebox(f(x), x=0..1, 8):display(mid2, lhs2);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display(mid4, lhs4);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display(mid8, lhs8);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The left-hand sums are larger than the midpoint sums, since the function is decreasing.display(rhs2, 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midpoint sums are larger than the right-hand sums, since the function is decreasing.7. Compute the left-hand sums, right-hand sums, midpoint sums and trapezoid rule with NiMlIm5H=2, 4, 8, 16, 32, 64, 126, 256, 512, 1024 subintervals. Which ones are overestimates and which ones are underestimates for NiMlI1BpRw==? Why? Put these numbers in increasing order, e.g. LHS(8)<NiMlI1BpRw==<TRAP(8)<MID(8)<RHS(8). This order is not correct, your work should show the correct order. Explain your ordering, i.e. explain why LHS(8)<TRAP(8) or the other way around. For this part you should use a loop. If you do not know how to use a loop, stop at NiMlIm5H =126.8. How many decimal digits of NiMlI1BpRw== can you get using left-hand sums and right-hand sums? How many digits of NiMlI1BpRw== can you get using the trapezoid rule and midpoint sums?for j from 1 to 10 do: n:=2^j;left:=evalf(leftsum(f(x), x=0..1, n)); right:=evalf(rightsum(f(x), x=0..1, n)); mid:=evalf(middlesum(f(x), x=0..1, n)); trap:=(left+right)/2; od;NiM+SSJuRzYiIiIjNiM+SSVsZWZ0RzYiJCIrTyV5VFokISIqNiM+SSZyaWdodEc2IiQiKzY4dlhFISIqNiM+SSRtaWRHNiIkIitWenoiPSQhIio=NiM+SSV0cmFwRzYiJCIrdVsnKmZJISIqNiM+SSJuRzYiIiIlNiM+SSVsZWZ0RzYiJCIrIT4pKXpLJCEiKg==NiM+SSZyaWdodEc2IiQiK0dZeDhIISIqNiM+SSRtaWRHNiIkIis/SyE+OiQhIio=NiM+SSV0cmFwRzYiJCIrNDkpMzckISIqNiM+SSJuRzYiIiIpNiM+SSVsZWZ0RzYiJCIrMGQlKlJLISIqNiM+SSZyaWdodEc2IiQiK0MqUUcuJCEiKg==NiM+SSRtaWRHNiIkIitSKSo9V0ohIio=NiM+SSV0cmFwRzYiJCIrOUJST0ohIio=NiM+SSJuRzYiIiM7NiM+SSVsZWZ0RzYiJCIrc3gxIz4kISIqNiM+SSZyaWdodEc2IiQiKyJROSYpMyQhIio=NiM+SSRtaWRHNiIkIitfS0NVSiEiKg==NiM+SSV0cmFwRzYiJCIrdzVIU0ohIio=NiM+SSJuRzYiIiNLNiM+SSVsZWZ0RzYiJCIrN2I6bkohIio=NiM+SSZyaWdodEc2IiQiKzwpeWA2JCEiKg==NiM+SSRtaWRHNiIkIispUWI8OSQhIio=NiM+SSV0cmFwRzYiJCIra3JFVEohIio=NiM+SSJuRzYiIiNrNiM+SSVsZWZ0RzYiJCIrW2FYYUohIio=NiM+SSZyaWdodEc2IiQiKy1yY0dKISIqNiM+SSRtaWRHNiIkIitVTGpUSiEiKg==NiM+SSV0cmFwRzYiJCIrdjdeVEohIio=NiM+SSJuRzYiIiRHIg==NiM+SSVsZWZ0RzYiJCIrJlJXIVtKISIqNiM+SSZyaWdodEc2IiQiK0EtNU5KISIqNiM+SSRtaWRHNiIkIitFR2dUSiEiKg==NiM+SSV0cmFwRzYiJCIrNEJkVEohIio=NiM+SSJuRzYiIiRjIw==NiM+SSVsZWZ0RzYiJCIrNk8jWzkkISIqNiM+SSZyaWdodEc2IiQiK0M6TlFKISIqNiM+SSRtaWRHNiIkIisoPiZmVEohIio=NiM+SSV0cmFwRzYiJCIrb3ZlVEohIio=NiM+SSJuRzYiIiQ3Jg==NiM+SSVsZWZ0RzYiJCIrLyU0SzkkISIqNiM+SSZyaWdodEc2IiQiK2ZMKCpSSiEiKg==NiM+SSRtaWRHNiIkIisqRyRmVEohIio=NiM+SSV0cmFwRzYiJCIrI1EiZlRKISIqNiM+SSJuRzYiIiVDNQ==NiM+SSVsZWZ0RzYiJCIrWjhTVUohIio=NiM+SSZyaWdodEc2IiQiK0RMeVNKISIqNiM+SSRtaWRHNiIkIis3R2ZUSiEiKg==NiM+SSV0cmFwRzYiJCIrT0JmVEohIio=The correct order is: LHS(n)> MID(n)> NiMlI1BpRw== >TRAP(n) >RHS(n). Here is the explanation. Since the function is concave downwards, the midpoint sum is an overestimate (it is a midpoint-tangent sum, with the tangent above the graph), while the trapezoid sum is an underestimate, since the chord joining the end-points is below the graph. Clearly, the trapezoid is between the left-hand sum and the right-hand-sum. We explained the midpoint and the left-hand-sums and right-hand sums above. This gives this ordering. We get only two decimal digits correct with the left-hand sum and the right hand sum with 1024 subintervals.We get 6 decimal digits of NiMlI1BpRw== with the midpoint and trapezoid rules and 1024 subintervals. We answers we got are 3.141592812 and 3.141592336, which bound above and below the correct value. These answers agree to the first 6 decimals, so we are certain of the first 6 decimal digits of NiMlI1BpRw==.9. Plot on the same graph the function and the trapezoid rule with NiMvJSJuRyIiIg== subintervals. Explain why it is an underestimate or an overestimate.10. Plot on the same graph the function and the trapezoid rule with NiMvJSJuRyIiIw== subintervals. Which is larger? TRAP(1) or TRAP(2)? Why?with(student): with(plots):Warning, the name changecoords has been redefined a:=0;b:=1;n:=1;Dx:=(b-a)/n;NiM+SSJhRzYiIiIhNiM+SSJiRzYiIiIiNiM+SSJuRzYiIiIiNiM+SSNEeEc2IiIiIg==xpoints:=[seq( a+Dx*i, i=0..n)];NiM+SSh4cG9pbnRzRzYiNyQiIiEiIiI=valuesoflist:=map(f, xpoints);NiM+SS12YWx1ZXNvZmxpc3RHNiI3JCwmKiQiIiMjIiIiRikiIiUhIiNGK0Yppair:=(x,y)->[x,y];NiM+SSVwYWlyRzYiZio2JEkieEdGJUkieUdGJUYlNiRJKW9wZXJhdG9yR0YlSSZhcnJvd0dGJUYlNyQ5JDklRiVGJUYldatapoints:=zip(pair, xpoints,valuesoflist);NiM+SStkYXRhcG9pbnRzRzYiNyQ3JCIiISwmKiQiIiMjIiIiRisiIiUhIiNGLTckRi1GKw==trap1:=plot(datapoints, style=line, color=blue):display(trap1, plot(f(x), x=0..1));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The trapezoid rule is an underestimate of the area since the blue line lies below the graph. This is so because the function is concave downwards.If you want to shade the region, here is the command. The range of y has been adjusted to 2..4. for i from 1 to n do trapezoid[i]:=inequal({y<f(xpoints[i])+(f(xpoints[i+1])-f(xpoints[i]))/Dx *(x-xpoints[i])}, x=xpoints[i]..xpoints[i+1], y=2..4, optionsexcluded=(color=white)): od;NiM+JkkqdHJhcGV6b2lkRzYiNiMiIiItSS9JTlRFUkZBQ0VfUExPVEdGJjYmLUkpUE9MWUdPTlNHNiRJKnByb3RlY3RlZEdGL0koX3N5c2xpYkdGJjYkNyY3JCQiIiFGNSQiMFMjXFxVJm9sJCEjOTckRjQkIiIlRjU3JCRGKEY1Rjo3JEY9JCIiI0Y1LUknQ09MT1VSR0YuNiZJJFJHQkdGLkYoRihGKC1GLTYkNyc3JEY0Rj9GOUY8Rj5GSC1JJkNPTE9SR0YuNiZGRCQiI3EhIiNGTCQiIzUhIiItSSdDVVJWRVNHRi42JDckRj5GMy1JKkxJTkVTVFlMRUdGJjYjRkAtSSZTVFlMRUdGLjYjSSxQQVRDSE5PR1JJREdGLg==display(plot(f(x), x=0..1), trapezoid[1]);LSUlUExPVEc2Jy0lJ0NVUlZFU0c2JDdTNyQkIiIhRiskIjNlIVEjXFxVJm9sJCEjPDckJCIzZG1tbTthcnpAISM+JCIzVl84PiVIIz1jT0YuNyQkIjNbTEwkZTl1aTIlRjIkIjNlNl5VLVJdYU9GLjckJCIzbm1tbSJ6XyI0aUYyJCIzYSpRVStLKlJeT0YuNyQkIjNbbW1tVCZwaE4pRjIkIjMuV2J1KXlxcGskRi43JCQiM0JMTGUqPSlIXDUhIz0kIjMybyopZSIqPUVUT0YuNyQkIjNmbW0iei8zdUMiRkckIjNZTTwzUWQhW2okRi43JCQiMyUpKioqXDdMUkRYIkZHJCIzTyRceE8xUHBpJEYuNyQkIjNdbW0ielInb2s7RkckIjNHNSxOSnJfPE9GLjckJCIzdioqKlxpNWBoKD1GRyQiM04hUUlqbGBvZyRGLjckJCIzV0xMTDNFbiQ0I0ZHJCIzUSVweW1DPlhmJEYuNyQkIjNwbW07L1JFJkcjRkckIjNfPTooUlE0RGUkRi43JCQiMyIpKioqKipcS100XSNGRyQiMyIqRzkyd2dwbk5GLjckJCIzJCoqKioqKlxQQXZyI0ZHJCIzISk9XEIoR0w5YiRGLjckJCIzKSoqKioqKlxuSGkjSEZHJCIzUko6aThJVk1ORi43JCQiM2ptbSJ6KmV2OkpGRyQiMyN6JD5rK2MmeV4kRi43JCQiMz9MTEwzNDdUTEZHJCIzdVhpO0Z4cidcJEYuNyQkIjMrTExMTFkuS05GRyQiM3k8RmgoeSdleE1GLjckJCIzdioqKlw3bzdUdiRGRyQiMy8xUiEpekwhUlgkRi43JCQiMyZHTExMUSpvXVJGRyQiMy1tV2RBOWtKTUYuNyQkIjNAKytEIj1sajslRkckIjM4Jm9WZ1UmejBNRi43JCQiMzErK3ZWJlI8UCVGRyQiMyEpZjtgJ3ojeXpMRi43JCQiM1dMTCRlOUVnZSVGRyQiM0stNiIqR0I7XkxGLjckJCIzR0xMZVIiM0d5JUZHJCIzZXFUPFwpR05LJEYuNyQkIjNjbW07L1QxJipcRkckIjMhKXA3YVElW0FIJEYuNyQkIjMlZW07elJRYkAmRkckIjN5OHFSWCYzImVLRi43JCQiM1sqKipcKD0+WTJhRkckIjMlcG56KkdwKnBBJEYuNyQkIjM5bW07elh1OWNGRyQiM0twKUckKUg2Pj4kRi43JCQiM2sqKioqKipceSkpR2VGRyQiMzk2JkgxMThTOiRGLjckJCIzJyopKioqXGlfUVFnRkckIjN3XTI0anZGOkpGLjckJCIzQCoqKlw3eSUzVGlGRyQiM3ZEJT02NixpMiRGLjckJCIzNSoqKipcUCFbaFknRkckIjNWNCMpZkJzIzQuJEYuNyQkIjNqS0xMJFF4JG9tRkckIjMnNCh5dSlvNSYpKUhGLjckJCIzISkqKioqKlxQK1Ypb0ZHJCIzRToiKXlfJ2U4JUhGLjckJCIzP21tInpwZSp6cUZHJCIzQzZWYCxYI3AqR0YuNyQkIjMlKSoqKioqXCNcJ1FIKEZHJCIzd1tKOjRWVllHRi43JCQiM0dLTGU5UzgmXChGRyQiM3NPPGtGJltxeiNGLjckJCIzUioqKlxpPz1icShGRyQiMyczYF93STtNdSNGLjckJCIzIkhMTCQzcz82ekZHJCIzYl5iUlcjRyopbyNGLjckJCIzYSoqKlw3YFdsNylGRyQiMzVDb0BwNWpIRUYuNyQkIjMjcG1tbScqUlJMKUZHJCIzdyI9bnNUZi1kI0YuNyQkIjNRbW07YTwuWSYpRkckIjNdIm9NRSRmOjJERi43JCQiMzxMTGU5dE9jKClGRyQiM3EnejhbOyE0VUNGLjckJCIzdCoqKioqKlxRa1wqKUZHJCIzIUgrPkw2RCtRI0YuNyQkIjNDTEwkM2RnNjwqRkckIjNwVUYjSEQmMzFCRi43JCQiM0htbW1teEdwJCpGRyQiM3p0KVtCZSFIUEFGLjckJCIzQSsrRCJvSzBlKkZHJCIzJypvU0FzbS1oQEYuNyQkIjNBKyt2PTVzI3kqRkckIjN3PSkpcEVIMSYzI0YuNyQkIiIiRiskIiIjRistJSZDT0xPUkc2JiUkUkdCRyQiIzUhIiIkRitGYVtsRmJbbC0lKVBPTFlHT05TRzYmNyY3JEYqJCIwUyNcXFUmb2wkISM5NyRGKiQiIiVGKzckRmd6RlxcbEZmejcnNyRGKkZpekZbXGxGXlxsRmZ6RmBcbC1GXFtsNilGXltsRl9bbEZfW2xGX1tsJCIxc0N5PyEzJz5xISM7RmNcbEZfW2wtJSZTVFlMRUc2IyUsUEFUQ0hOT0dSSURHLUYmNiU3JEZmekZnW2wtJSpMSU5FU1RZTEVHNiNGanpGZlxsLSUrQVhFU0xBQkVMU0c2JFEieDYiUSFGZF1sLSUlVklFV0c2JDtGYltsRl9bbDskIiQnPiEiIyQiJC8lRl1ebA==a:=0;b:=1;n:=2;Dx:=(b-a)/n;NiM+SSJhRzYiIiIhNiM+SSJiRzYiIiIiNiM+SSJuRzYiIiIjNiM+SSNEeEc2IiMiIiIiIiM=xpoints:=[seq( a+Dx*i, i=0..n)];NiM+SSh4cG9pbnRzRzYiNyUiIiEjIiIiIiIjRik=valuesoflist:=map(f, xpoints);NiM+SS12YWx1ZXNvZmxpc3RHNiI3JSwmKiQiIiMjIiIiRikiIiUhIiNGKywmKiQiIihGKkYpRi1GK0Yppair:=(x,y)->[x,y];NiM+SSVwYWlyRzYiZio2JEkieEdGJUkieUdGJUYlNiRJKW9wZXJhdG9yR0YlSSZhcnJvd0dGJUYlNyQ5JDklRiVGJUYldatapoints:=zip(pair, xpoints,valuesoflist);NiM+SStkYXRhcG9pbnRzRzYiNyU3JCIiISwmKiQiIiMjIiIiRisiIiUhIiNGLTckRiwsJiokIiIoRixGK0YvRi03JEYtRis=trap2:=plot(datapoints, style=line, color=blue):display(plot(f(x), x=0..1), trap2, trap1);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We see that TRAP(2) is larger than TRAP(1). This is so TRAP(2) graphs two chords on the graph. The function is concave downwards. The two chords lie above the one chord used by TRAP(1).