Second Midterm (2007) 1.(a) For what values of LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2JVEickYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJw== does the Ratio Test indicate that the series LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2I1EhRictRiM2Ji1JK211bmRlcm92ZXJHRiQ2Jy1JI21vR0YkNi1RJiZTdW07RicvJSxtYXRodmFyaWFudEdRJ25vcm1hbEYnLyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0Y9LyUpc3RyZXRjaHlHUSV0cnVlRicvJSpzeW1tZXRyaWNHRj0vJShsYXJnZW9wR0ZCLyUubW92YWJsZWxpbWl0c0dGQi8lJ2FjY2VudEdGPS8lJ2xzcGFjZUdRJjAuMGVtRicvJSdyc3BhY2VHUSwwLjE2NjY2NjdlbUYnLUYjNiUtRiw2JVEibkYnLyUnaXRhbGljR0ZCL0Y5USdpdGFsaWNGJy1GNTYtUSI9RidGOEY7Rj4vRkFGPUZDL0ZGRj0vRkhGPUZJL0ZMUSwwLjI3Nzc3NzhlbUYnL0ZPRltvLUkjbW5HRiQ2JFEiMUYnRjgtRiw2JVEoJmluZmluO0YnRlZGWC8lJ2FjY2VudEdGPS8lLGFjY2VudHVuZGVyR0Y9RistSSdtc3BhY2VHRiQ2Ji8lJ2hlaWdodEdRJjAuMGV4RicvJSZ3aWR0aEdRJDUuMEYnLyUmZGVwdGhHRl1wLyUqbGluZWJyZWFrR1ElYXV0b0YnLUkmbWZyYWNHRiQ2KC1GIzYoLUklbXN1cEdGJDYlLUkobWZlbmNlZEdGJDYkLUYjNiQtRjU2LVEqJnVtaW51czA7RidGOEY7Rj5GZ25GQ0ZobkZpbkZJL0ZMUSwwLjIyMjIyMjJlbUYnL0ZPRmdxRl1vRjhGUy8lMXN1cGVyc2NyaXB0c2hpZnRHUSIwRictRjU2LVExJkludmlzaWJsZVRpbWVzO0YnRjhGO0Y+RmduRkNGaG5GaW5GSUZLL0ZPRk0tRlxxNiUtRjU2LVEvJkV4cG9uZW50aWFsRTtGJ0Y4RjtGPkZnbkZDRmhuRmluRklGSy9GT1EsMC4xMTExMTExZW1GJy1GIzYlLUYsNiVRInJGJ0ZWRlhGXHJGU0ZpcUZcci1GIzYkRlMtRjU2LVEiIUYnRjhGO0Y+RmduRkNGaG5GaW5GSS9GTEZmckZlckYrLUYjNiMtRlxxNiVGU0ZTRmlxLyUubGluZXRoaWNrbmVzc0dRIjFGJy8lK2Rlbm9tYWxpZ25HUSdjZW50ZXJGJy8lKW51bWFsaWduR0ZbdC8lKWJldmVsbGVkR0Y9Ris= converges? a:= n -> (-1)^n*exp(r*n)*n!/n^n;

NiM+SSJhRzYiZio2I0kibkc2IjYiNiRJKW9wZXJhdG9yRzYiSSZhcnJvd0c2IjYiKiopISIiSSJuRzYiIiIiLUkkZXhwRzYiNiMqJkkickc2IiIiIkkibkc2IiIiIiIiIi1JKmZhY3RvcmlhbEclKnByb3RlY3RlZEc2I0kibkc2IiIiIilJIm5HNiJJIm5HNiIhIiI2IjYiNiI=

ratio:= simplify(a(n+1)/a(n));

NiM+SSZyYXRpb0c2IiwkKigtSSRleHBHNiQlKnByb3RlY3RlZEdJKF9zeXNsaWJHNiI2I0kickc2IiIiIiksJkkibkc2IiIiIiIiIiIiIiwkSSJuRzYiISIiIiIiKUkibkc2Ikkibkc2IiIiIiEiIg==

limit(ratio,n=infinity);

NiMsJComLUkkZXhwRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiNiMhIiIiIiItSSRleHBHNiQlKnByb3RlY3RlZEdJKF9zeXNsaWJHNiI2I0kickc2IiIiIiEiIg==

The series converges when the limiting ratio has absolute value < 1, i.e. for LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2I1EhRictRiM2JS1GLDYlUSJyRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUkjbW9HRiQ2LVEiPEYnL0Y4USdub3JtYWxGJy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGQi8lKXN0cmV0Y2h5R0ZCLyUqc3ltbWV0cmljR0ZCLyUobGFyZ2VvcEdGQi8lLm1vdmFibGVsaW1pdHNHRkIvJSdhY2NlbnRHRkIvJSdsc3BhY2VHUSwwLjI3Nzc3NzhlbUYnLyUncnNwYWNlR0ZRLUkjbW5HRiQ2JFEiMUYnRj5GKw==. (b) Plot one graph showing 5 different partial sums of this series as functions of LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2JVEickYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJw==, using an interval for LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2JVEickYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJw== that includes some values for which the series converges and some for which it diverges. Choose intervals for LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2JVEickYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJw== and for the LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2JVEieUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJw== axis so that places where the curves cross are clearly visible. psum:= N -> add(a(n),n=1..N);

NiM+SSVwc3VtRzYiZio2I0kiTkc2IjYiNiRJKW9wZXJhdG9yRzYiSSZhcnJvd0c2IjYiLUkkYWRkRyUqcHJvdGVjdGVkRzYkLUkiYUc2IjYjSSJuRzYiL0kibkc2IjsiIiJJIk5HNiI2IjYiNiI=

Some experimenting may be needed to get good bounds. plot([seq(psum(N),N=1..5)],r=0..2,y=-3..1);

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2. (a) Find the Simpson's Rule approximations LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUklbXN1YkdGJDYlLUkjbWlHRiQ2JVEiU0YnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1GIzYjLUkjbW5HRiQ2JFEjMTBGJy9GNlEnbm9ybWFsRicvJS9zdWJzY3JpcHRzaGlmdEdRIjBGJw== and LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUklbXN1YkdGJDYlLUkjbWlHRiQ2JVEiU0YnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1GIzYjLUkjbW5HRiQ2JFEjMjBGJy9GNlEnbm9ybWFsRicvJS9zdWJzY3JpcHRzaGlmdEdRIjBGJw== to 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. If you use ApproximateInt from the Student[Calculus1] package, note that S(n) in our notation corresponds to partition=n/2. with(Student[Calculus1]): S10 := evalf(ApproximateInt(exp(x^2), x=1..2, method=simpson, partition=5));

NiM+SSRTMTBHNiIkIisrRkQqXCIhIik=

S20 := evalf(ApproximateInt(exp(x^2), x=1..2, method=simpson, partition=10));

NiM+SSRTMjBHNiIkIitiUiwqXCIhIik=

Alternatively, you could have used the definitions in Lesson 19: a:= 1: b:= 2: f:= x -> exp(x^2); h := n -> (b-a)/n; X := (k,n) -> a + k*h(n); S := n -> add((1/3*f(X(2*k-2,n)) + 4/3*f(X(2*k-1,n)) + 1/3*f(X(2*k,n))) * h(n), k=1..n/2);

NiM+SSJmRzYiZio2I0kieEc2IjYiNiRJKW9wZXJhdG9yRzYiSSZhcnJvd0c2IjYiLUkkZXhwRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiNiMqJClJInhHNiIiIiMiIiI2IjYiNiI= NiM+SSJoRzYiZio2I0kibkc2IjYiNiRJKW9wZXJhdG9yRzYiSSZhcnJvd0c2IjYiKiYsJkkiYkc2IiIiIkkiYUc2IiEiIiIiIkkibkc2IiEiIjYiNiI2Ig== NiM+SSJYRzYiZio2JEkia0c2Ikkibkc2IjYiNiRJKW9wZXJhdG9yRzYiSSZhcnJvd0c2IjYiLCZJImFHNiIiIiIqJkkia0c2IiIiIi1JImhHNiI2I0kibkc2IiIiIiIiIjYiNiI2Ig== NiM+SSJTRzYiZio2I0kibkc2IjYiNiRJKW9wZXJhdG9yRzYiSSZhcnJvd0c2IjYiLUkkYWRkRyUqcHJvdGVjdGVkRzYkKiYsKComIyIiIiIiJCIiIi1JImZHNiI2Iy1JIlhHNiI2JCwmKiYiIiMiIiJJImtHNiIiIiIiIiIiIiMhIiJJIm5HNiIiIiIiIiIqJiMiIiUiIiQiIiItSSJmRzYiNiMtSSJYRzYiNiQsJiomIiIjIiIiSSJrRzYiIiIiIiIiIiIiISIiSSJuRzYiIiIiIiIiKiYjIiIiIiIkIiIiLUkiZkc2IjYjLUkiWEc2IjYkLCQqJiIiIyIiIkkia0c2IiIiIiIiIkkibkc2IiIiIiIiIiIiIi1JImhHNiI2I0kibkc2IiIiIi9JImtHNiI7IiIiLCQqJiMiIiIiIiMiIiJJIm5HNiIiIiIiIiI2IjYiNiI=

evalf(S(10)), evalf(S(20));

NiQkIisrRkQqXCIhIikkIitiUiwqXCJGJQ==

(b) Estimate the error in LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUklbXN1YkdGJDYlLUkjbWlHRiQ2JVEiU0YnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1GIzYjLUkjbW5HRiQ2JFEjMjBGJy9GNlEnbm9ybWFsRicvJS9zdWJzY3JpcHRzaGlmdEdRIjBGJw== using Richardson extrapolation. According to this result, approximately what n would be needed in order for LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUklbXN1YkdGJDYlLUkjbWlHRiQ2JVEiU0YnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1GIzYjLUYvNiVRIm5GJ0YyRjUvJS9zdWJzY3JpcHRzaGlmdEdRIjBGJw== to be accurate to within LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUklbXN1cEdGJDYlLUkjbW5HRiQ2JFEjMTBGJy8lLG1hdGh2YXJpYW50R1Enbm9ybWFsRictRiM2JC1JI21vR0YkNi1RKiZ1bWludXMwO0YnRjIvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRj0vJSlzdHJldGNoeUdGPS8lKnN5bW1ldHJpY0dGPS8lKGxhcmdlb3BHRj0vJS5tb3ZhYmxlbGltaXRzR0Y9LyUnYWNjZW50R0Y9LyUnbHNwYWNlR1EsMC4yMjIyMjIyZW1GJy8lJ3JzcGFjZUdGTEYuLyUxc3VwZXJzY3JpcHRzaGlmdEdRIjBGJw==? The error in S(n) is approximately LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkmbWZyYWNHRiQ2KC1GIzYjLUkjbWlHRiQ2JVEiQUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1GIzYlLUYxNiNRIUYnLUYjNiMtSSVtc3VwR0YkNiUtRjE2JVEibkYnRjRGNy1JI21uR0YkNiRRIjRGJy9GOFEnbm9ybWFsRicvJTFzdXBlcnNjcmlwdHNoaWZ0R1EiMEYnRjwvJS5saW5ldGhpY2tuZXNzR1EiMUYnLyUrZGVub21hbGlnbkdRJ2NlbnRlckYnLyUpbnVtYWxpZ25HRlUvJSliZXZlbGxlZEdRJmZhbHNlRic=, where (with LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2I1EhRictRiM2JS1GLDYlUSJwRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUkjbW9HRiQ2LVEiPUYnL0Y4USdub3JtYWxGJy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGQi8lKXN0cmV0Y2h5R0ZCLyUqc3ltbWV0cmljR0ZCLyUobGFyZ2VvcEdGQi8lLm1vdmFibGVsaW1pdHNHRkIvJSdhY2NlbnRHRkIvJSdsc3BhY2VHUSwwLjI3Nzc3NzhlbUYnLyUncnNwYWNlR0ZRLUkjbW5HRiQ2JFEiNEYnRj5GKw==) the Richardson approximation for LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2JVEiQUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJw== is 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 AR := (S20 - S10)*20^4/(2^4-1);

NiM+SSNBUkc2IiQhJ2hZRCEiJQ==

The error in S(20) is thus approximately AR/20^4;

NiMkIStdN2oiZiIhIzg=

The error will have absolute value approximately LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUklbXN1cEdGJDYlLUkjbW5HRiQ2JFEjMTBGJy8lLG1hdGh2YXJpYW50R1Enbm9ybWFsRictRiM2JC1JI21vR0YkNi1RKiZ1bWludXMwO0YnRjIvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRj0vJSlzdHJldGNoeUdGPS8lKnN5bW1ldHJpY0dGPS8lKGxhcmdlb3BHRj0vJS5tb3ZhYmxlbGltaXRzR0Y9LyUnYWNjZW50R0Y9LyUnbHNwYWNlR1EsMC4yMjIyMjIyZW1GJy8lJ3JzcGFjZUdGTEYuLyUxc3VwZXJzY3JpcHRzaGlmdEdRIjBGJw== when abs(AR)/n^4 = 10^(-10);

NiMvLCQqJiQiJ2hZRCEiJSIiIilJIm5HNiIiIiUhIiIiIiIjIiIiIiwrKysrKyI=

solve(%);

NiYkISsxIil6LnIhIigsJComJCIrMSIpei5yRiUiIiJeI0YqRiohIiJGJ0Yo

So approximately n=710 would be needed. (c) Transform the integral LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2I1EhRictRiM2KC1JKG1zdWJzdXBHRiQ2Jy1JI21vR0YkNi1RJiZpbnQ7RicvJSxtYXRodmFyaWFudEdRJ25vcm1hbEYnLyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0Y9LyUpc3RyZXRjaHlHRj0vJSpzeW1tZXRyaWNHRj0vJShsYXJnZW9wR0Y9LyUubW92YWJsZWxpbWl0c0dGPS8lJ2FjY2VudEdGPS8lJ2xzcGFjZUdRJjAuMGVtRicvJSdyc3BhY2VHRkwtSSNtbkdGJDYkUSIwRidGOC1GUDYkUSIxRidGOC8lMXN1cGVyc2NyaXB0c2hpZnRHUSIyRicvJS9zdWJzY3JpcHRzaGlmdEdRIjBGJy1JJm1mcmFjR0YkNigtRiM2Iy1JJW1zdXBHRiQ2JS1GLDYlUSJ4RicvJSdpdGFsaWNHUSV0cnVlRicvRjlRJ2l0YWxpY0YnLUZnbjYoLUYjNiNGUy1GIzYjLUZQNiRRIjRGJ0Y4LyUubGluZXRoaWNrbmVzc0dRIjFGJy8lK2Rlbm9tYWxpZ25HUSdjZW50ZXJGJy8lKW51bWFsaWduR0ZkcC8lKWJldmVsbGVkR0Y9L0ZXRmVuLUYjNiVGKy1GIzYmRistRiM2Iy1GXG82JUZeby1GUDYkUSIyRidGOEZpcC1GNTYtUSIrRidGOEY7Rj5GQEZCRkRGRkZIL0ZLUSwwLjIyMjIyMjJlbUYnL0ZORmlxRlNGK0ZfcEZicEZlcEZncEYrLUknbXNwYWNlR0YkNiYvJSdoZWlnaHRHUSYwLjBleEYnLyUmd2lkdGhHUSYwLjNlbUYnLyUmZGVwdGhHRmByLyUqbGluZWJyZWFrR1ElYXV0b0YnLUY1Ni1RMCZEaWZmZXJlbnRpYWxEO0YnRjhGO0Y+RkBGQkZERkZGSEZKRk1GXm9GKw== into something for which Simpson's Rule will produce good approximations. The problem is the LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUklbXN1cEdGJDYlLUkjbWlHRiQ2JVEieEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JJm1mcmFjR0YkNigtRiM2Iy1JI21uR0YkNiRRIjFGJy9GNlEnbm9ybWFsRictRiM2Iy1GPjYkUSI0RidGQS8lLmxpbmV0aGlja25lc3NHUSIxRicvJStkZW5vbWFsaWduR1EnY2VudGVyRicvJSludW1hbGlnbkdGTS8lKWJldmVsbGVkR1EmZmFsc2VGJy8lMXN1cGVyc2NyaXB0c2hpZnRHUSIwRic=, which is not differentiable at LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2I1EhRictRiM2JS1GLDYlUSJ4RicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUkjbW9HRiQ2LVEiPUYnL0Y4USdub3JtYWxGJy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGQi8lKXN0cmV0Y2h5R0ZCLyUqc3ltbWV0cmljR0ZCLyUobGFyZ2VvcEdGQi8lLm1vdmFibGVsaW1pdHNHRkIvJSdhY2NlbnRHRkIvJSdsc3BhY2VHUSwwLjI3Nzc3NzhlbUYnLyUncnNwYWNlR0ZRLUkjbW5HRiQ2JFEiMEYnRj5GKw==. Rule[change,x=t^4](Int(x^(1/4)/(x^2+1),x=0..1));

NiMvLUkkSW50RzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiNiQqJilJInhHNiIjIiIiIiIlIiIiLCYqJClJInhHNiIiIiMiIiIiIiIiIiIiIiIhIiIvSSJ4RzYiOyIiISIiIi1JJEludEc2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkc2IjYkLCQqKCIiJSIiIilJInRHNiIiIiUiIiIsJiokKUkidEc2IiIiKSIiIiIiIiIiIiIiIiEiIiIiIi9JInRHNiI7IiIhIiIi