Final Exam Math 5365, Second Summer Session 2007 Please give me a hardcopy of your exam and email a copy to lance.drager@ttu.edu Problem 1. Part A. Let f:=x^3-2*x^2-5*x+6;

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Find (numerically) the x and y intercepts, the critical points, local max's, local min's and inflection points (meaning a point where the second derivative changes sign). Provide graphs to illustrate these features. Part B. Let g := (x^3-4*x^2-x+4)/(x^2-9);

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Find (numerically) x and y intercepts Vertical asymptotes Any slant asymptote critical points, local max's and local min'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 inflection points. Give graphs to illustrate these features. Problem 2. 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(a,b). If 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 Part A. Find the critical points of the following function and classify them according to the theorem above. Give graphs to illustrate the behavior near each point. f := 1/3*x^3-x*y+1/2*y^2-2*x;

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

Part B. For each of the critical points above, graph the second order taylor polynomial centered at that point, to see it has the same kind of critical pont as the function. (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is classifying the behaviour of the second order taylor polynomial). Problem 3. In this problem, we'll show how to find the sums LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzY0LUkjbWlHRiQ2OVEhRicvJSdmYW1pbHlHUTBUaW1lc35OZXd+Um9tYW5GJy8lJXNpemVHUSMxMkYnLyUlYm9sZEdRJmZhbHNlRicvJSdpdGFsaWNHUSV0cnVlRicvJSp1bmRlcmxpbmVHRjcvJSpzdWJzY3JpcHRHRjcvJSxzdXBlcnNjcmlwdEdGNy8lK2ZvcmVncm91bmRHUShbMCwwLDBdRicvJStiYWNrZ3JvdW5kR1EuWzI1NSwyNTUsMjU1XUYnLyUnb3BhcXVlR0Y3LyUrZXhlY3V0YWJsZUdGNy8lKXJlYWRvbmx5R0Y3LyUpY29tcG9zZWRHRjcvJSpjb252ZXJ0ZWRHRjcvJStpbXNlbGVjdGVkR0Y3LyUscGxhY2Vob2xkZXJHRjcvJTBmb250X3N0eWxlX25hbWVHUSgyRH5NYXRoRicvJSptYXRoY29sb3JHRkMvJS9tYXRoYmFja2dyb3VuZEdGRi8lK2ZvbnRmYW1pbHlHRjEvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLyUpbWF0aHNpemVHRjQtSSVtc3ViR0YkNiYtRiw2OVEiU0YnRi9GMkY1RjhGO0Y9Rj9GQUZERkdGSUZLRk1GT0ZRRlNGVUZYRlpGZm5GaG5GW28tRiw2OVEicEYnRi9GMkY1RjhGO0Y9Rj9GQUZERkdGSUZLRk1GT0ZRRlNGVUZYRlpGZm5GaG5GW28vJS9zdWJzY3JpcHRzaGlmdEdRIjBGJy8lLHBsYWNlaG9sZGVyR0Y3LUYsNjhGLkYvRjJGNUY4RjtGPUY/RkFGREZHRklGS0ZNRk9GUUZTRlhGWkZmbkZobkZbby1JKG1mZW5jZWRHRiQ2Iy1GIzYjLUYsNjlRIm5GJ0YvRjJGNUY4RjtGPUY/RkFGREZHRklGS0ZNRk9GUUZTRlVGWEZaRmZuRmhuRltvLUkjbW9HRiQ2M1EiPUYnLyUlZm9ybUdRJmluZml4RicvJSZmZW5jZUdGNy8lKnNlcGFyYXRvckdGNy8lJ2xzcGFjZUdRL3RoaWNrbWF0aHNwYWNlRicvJSdyc3BhY2VHRmJxLyUpc3RyZXRjaHlHRjcvJSpzeW1tZXRyaWNHRjcvJShtYXhzaXplR1EpaW5maW5pdHlGJy8lKG1pbnNpemVHUSIxRicvJShsYXJnZW9wR0Y3LyUubW92YWJsZWxpbWl0c0dGNy8lJ2FjY2VudEdGNy8lMGZvbnRfc3R5bGVfbmFtZUdGVy8lJXNpemVHRjQvJStmb3JlZ3JvdW5kR0ZDLyUrYmFja2dyb3VuZEdGRi1GIzYnRltwLUkrbXVuZGVyb3ZlckdGJDYnLUZmcDYyUSYmU3VtO0YnL0ZqcFEncHJlZml4RidGXHFGXnEvRmFxUSQwZW1GJy9GZHFRLnRoaW5tYXRoc3BhY2VGJy9GZnFGOkZncUZpcUZcci9GYHJGOi9GYnJGOkZjckZnckZpckZbcy1GIzYmLUYsNjhRImtGJ0YvRjJGNUY4RjtGPUY/RkFGREZHRklGS0ZNRk9GUUZTRlhGWkZmbkZobkZbby1GZnA2MkZocEZpcEZccUZecUZgcUZjcUZlcUZncUZpcUZcckZfckZhckZjckZnckZpckZbcy1JI21uR0YkNjlGXnJGL0YyRjUvRjlGN0Y7Rj1GP0ZBRkRGR0ZJRktGTUZPRlFGU0ZVRlhGWkZmbi9GaW5RJ25vcm1hbEYnRltvRltwLUYsNjhGZHBGLy9GM1EjMTBGJ0Y1RjhGO0Y9Rj8vRkJRLFsyMDAsMCwyMDBdRidGREZHRklGS0ZNRk9GUS9GVEY6L0ZZRmB1RlpGZm5GaG4vRlxvRl51RmNyLyUsYWNjZW50dW5kZXJHRjdGW3AtSSVtc3VwR0YkNiVGYHRGY28vJTFzdXBlcnNjcmlwdHNoaWZ0R0Zob0ZbcC1GZnA2M1ExJkludmlzaWJsZVRpbWVzO0YnRmlwRlxxRl5xRmBxRmNxRmVxRmdxRmlxRlxyRl9yRmFyRmNyRmVyRmdyRmlyRltzRmVwLUZmcDYzRl12RmlwRlxxRl5xL0ZhcVEwbWVkaXVtbWF0aHNwYWNlRicvRmRxRmF2RmVxRmdxRmlxRlxyRl9yRmFyRmNyRmVyRmdyRmlyRltzLUZndTYlRmV0RmNvRml1LUZmcDYzUScmcGx1cztGJ0ZpcEZccUZecUZgdkZidkZlcUZncUZpcUZcckZfckZhckZjckZlckZnckZpckZbcy1GZ3U2JS1GZnQ2OVEiMkYnRi9GMkY1Rmh0RjtGPUY/RkFGREZHRklGS0ZNRk9GUUZTRlVGWEZaRmZuRml0RltvRmNvRml1RmV2LUZmcDYzUSMuLkYnRmlwRlxxRl5xRmdzL0ZkcUZoc0ZlcUZncUZpcUZcckZfckZhckZjckZlckZnckZpckZbcy1GZnA2M1EiLkYnRmlwRlxxRl5xRmdzRmB3RmVxRmdxRmlxRlxyRl9yRmFyRmNyRmVyRmdyRmlyRltzRmV2LUZndTYlRmJwRmNvRml1Ris= For 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 Here's a maple abbreviation. psum := p -> Sum(k^p, k=1..n);

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

psum(1); psum(2);

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

We know psum(0)=n;

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

It's well known that psum(1)=n*(n+1)/2;

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

Let's record these formulas := [psum(0)=n, psum(1)=n*(n+1)/2];

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

Now, let's find 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 with(student): #to get the right version of Sum Consider the sum 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 On the one hand, this is a telescoping sum, so Sum((k+1)^3-k^3, k=1..n)= (n+1)^3-1;

NiQtSSVtcm93RzYjL0krbW9kdWxlbmFtZUc2IkksVHlwZXNldHRpbmdHSShfc3lzbGliR0YoNiUtRiQ2JS1JK211bmRlcm92ZXJHRiU2Jy1JI21vR0YlNjRRJiZTdW07RigvJSVmb3JtR1EncHJlZml4RigvJSZmZW5jZUdRJmZhbHNlRigvJSpzZXBhcmF0b3JHRjovJSdsc3BhY2VHUSQwZW1GKC8lJ3JzcGFjZUdRLnRoaW5tYXRoc3BhY2VGKC8lKXN0cmV0Y2h5R1EldHJ1ZUYoLyUqc3ltbWV0cmljR0Y6LyUobWF4c2l6ZUdRKWluZmluaXR5RigvJShtaW5zaXplR1EiMUYoLyUobGFyZ2VvcEdGRS8lLm1vdmFibGVsaW1pdHNHRkUvJSdhY2NlbnRHRjovJTBmb250X3N0eWxlX25hbWVHUSoyRH5PdXRwdXRGKC8lJXNpemVHUSMxMkYoLyUrZm9yZWdyb3VuZEdRLlsxNDQsMTQ0LDE0NF1GKC8lK2JhY2tncm91bmRHUS5bMjU1LDI1NSwyNTVdRigvSSttc2VtYW50aWNzR0YlUSZpbmVydEYoLUYkNiUtSSNtaUdGJTY5USJrRigvJSdmYW1pbHlHUTBUaW1lc35OZXd+Um9tYW5GKC8lJXNpemVHRlkvJSVib2xkR0Y6LyUnaXRhbGljR0ZFLyUqdW5kZXJsaW5lR0Y6LyUqc3Vic2NyaXB0R0Y6LyUsc3VwZXJzY3JpcHRHRjovJStmb3JlZ3JvdW5kR1EqWzAsMCwyNTVdRigvJStiYWNrZ3JvdW5kR0Zpbi8lJ29wYXF1ZUdGOi8lK2V4ZWN1dGFibGVHRjovJSlyZWFkb25seUdGRS8lKWNvbXBvc2VkR0Y6LyUqY29udmVydGVkR0Y6LyUraW1zZWxlY3RlZEdGOi8lLHBsYWNlaG9sZGVyR0Y6LyUwZm9udF9zdHlsZV9uYW1lR0ZWLyUqbWF0aGNvbG9yR0ZkcC8lL21hdGhiYWNrZ3JvdW5kR0Zpbi8lK2ZvbnRmYW1pbHlHRmVvLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGKC8lKW1hdGhzaXplR0ZZLUYyNjNRIj1GKC9GNlEmaW5maXhGKEY4RjsvRj5RL3RoaWNrbWF0aHNwYWNlRigvRkFGaHIvRkRGOkZGRkhGSy9GT0Y6L0ZRRjpGUkZURlcvRmVuRmRwRmduLUkjbW5HRiU2OUZNRmNvRmZvRmhvL0ZbcEY6RlxwRl5wRmBwRmJwRmVwRmdwRmlwRltxRl1xRl9xRmFxRmNxRmVxRmdxRmlxRltyL0ZeclEnbm9ybWFsRihGYHItRmBvNjlRIm5GKEZjb0Zmb0Zob0Zqb0ZccEZecEZgcEZicEZlcEZncEZpcEZbcUZdcUZfcUZhcUZjcUZlcUZncUZpcUZbckZdckZgckZSLyUsYWNjZW50dW5kZXJHRjotSSdtc3BhY2VHRiU2Ji8lJ2hlaWdodEdRJzAuMH5leEYoLyUmd2lkdGhHUSU1LjB+RigvJSZkZXB0aEdGXnQvJSpsaW5lYnJlYWtHUSVhdXRvRigtRiQ2JS1GMjYzUSIoRihGNS9GOUZFRjsvRj5GQkZARkNGRkZIRktGW3NGXHNGUkZURldGXXNGZ24tRiQ2JS1GJDYjLUklbXN1cEdGJTYlLUYkNiVGaXQtRiQ2JUZfby1GMjYzUSIrRihGZXJGOEY7L0Y+UTBtZWRpdW1tYXRoc3BhY2VGKC9GQUZddkZqckZGRkhGS0Zbc0Zcc0ZSRlRGV0Zdc0ZnbkZecy1GMjYzUSIpRigvRjZRKHBvc3RmaXhGKEZcdUY7Rl11L0ZBUTJ2ZXJ5dGhpbm1hdGhzcGFjZUYoRkNGRkZIRktGW3NGXHNGUkZURldGXXNGZ24tRl9zNjlRIjNGKEZjb0Zmb0Zob0Zhc0ZccEZecEZgcEZicEZlcEZncEZpcEZbcUZdcUZfcUZhcUZjcUZlcUZncUZpcUZbckZic0Zgci8lMXN1cGVyc2NyaXB0c2hpZnRHUSIwRigtRjI2M1EoJm1pbnVzO0YoRmVyRjhGO0ZcdkZedkZqckZGRkhGS0Zbc0Zcc0ZSRlRGV0Zdc0Znbi1GJDYjLUZjdTYlRl9vRmZ2Rml2Rl92RmJyLUYkNiUtRiQ2Iy1GY3U2JS1GJDYlRml0LUYkNiVGZHNGaXVGXnNGX3ZGZnZGaXZGXHdGXnM3Iy8tSSRTdW1HRio2JCwmKiQpLCZJImtHRigiIiJGZ3hGZ3giIiRGZ3hGZ3gqJClGZnhGaHhGZ3ghIiIvRmZ4O0ZneEkibkdGKCwmKiQpLCZGXnlGZ3hGZ3hGZ3hGaHhGZ3hGZ3hGZ3hGW3k=

On the other hand, we can expand the left-hand side. expand(%);

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

Of course, we know two of these sums, so let's substitue our formulas for them. subs(formulas, %);

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

We can solve this for what we want. solve(%, psum(2));

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

factor(%);

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

We can add this to our formulas formulas := [op(formulas), psum(2)=%];

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

Now, write a program to extend this list up to 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 Problem 4. Let 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 be the unit circle. The center of a wheel of radius 1/4 lies on 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 the center of the wheel travels one around 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, the wheel revolves 3 times. Given an animation showing the moving wheel and the path traced out by a point on the rim of the wheel (choose your own starting positions).