restart;PROBLEM 1p:=x->x^4-2*x^3-19*x^2+70*x-50;

NiM+SSJwRzYiZio2I0kieEdGJUYlNiRJKW9wZXJhdG9yR0YlSSZhcnJvd0dGJUYlLCwqJDkkIiIlIiIiKiRGLiIiJCEiIyokRi4iIiMhIz5GLiIjcSEjXUYwRiVGJUYl

To write p as a product of polynomials of the smallest degree possible (*that is, to factorize), we do : q:=factor(p(x));

NiM+SSJxRzYiKigsJkkieEdGJSIiIiEiIkYpRiksJkYoRikiIiZGKUYpLCgqJEYoIiIjRilGKCEiJyIjNUYpRik=

Then we find the complex roots. solve(p(x),x);

NiYiIiIhIiZeJCIiJEYjXiRGJiEiIg==

The factorization of the polinomial p(x) is the given by (x-1)*(x+5)*(x^2-6*x+10). The complex roots of p(x) are two: 3+I and 3-I. PROBLEM 2 One first defines a function f:f:=x->2*sin(x)-x^3-5;

NiM+SSJmRzYiZio2I0kieEdGJUYlNiRJKW9wZXJhdG9yR0YlSSZhcnJvd0dGJUYlLCgtSSRzaW5HRiU2IzkkIiIjKiRGMCIiJCEiIiEiJiIiIkYlRiVGJQ==

The following is the plot of the map f(x) plot(f(x),x=-4..4);

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

To check whether the function has more real zeroes than the one we see in the above graph, we plot again, with domain the real line. plot(f(x),x=-infinity..infinity);

LSUlUExPVEc2Ji0lJ0NVUlZFU0c2JDdncDckJCEzSCsrKyF5d2opKiohIz0kIjJTNHpPKioqKioqKioqISM8NyQkITNZKioqKioqZk52cyoqRiwkIjJJdkslXCoqKioqKioqRi83JCQhM3YqKioqKipSLjhmKipGLCQiMndgTiRIKSoqKioqKipGLzckJCEzaSoqKioqXDYyYiUqKkYsJCIycyc9WSZmKioqKioqKkYvNyQkITN5KioqKioqcDFFPSoqRiwkIjJbVyRvTScpKioqKioqRi83JCQhM08rKytJVSwiKikqRiwkIjJ3cSJwam4qKioqKipGLzckJCEzKCoqKioqKlxNQGwkKSpGLCQiMiNvKFJ1MiopKioqKipGLzckJCEzZyoqKioqKmYlRz95KkYsJCIzOy5jVjRUKCoqKioqRiw3JCQhMyUqKioqKioqKm9VSW4qRiwkIjMhUWJEJz1FIioqKioqRiw3JCQhMzArKys8cDBrJipGLCQiMztRW3FoR3oqKioqRiw3JCQhM1ArKytYNVN1JCpGLCQiM1M8M2l5eFEqKioqRiw3JCQhM2ErKytyXnUlPSpGLCQiMzdsTGA7XWspKioqRiw3JCQhMy4rKytVJXAiZSgpRiwkIjMjeiIzSl11PyYqKipGLDckJCEzXioqKioqPjRtKEckKUYsJCIzJ1JMJCopKjMvJCkpKipGLDckJCEzXCsrK0BPUyx6RiwkIjNdKzs8RWB3dyoqRiw3JCQhMzgrKysvUj0wdkYsJCIzPzZVNkJ0a2cqKkYsNyQkITNzKioqKipwTEBcNChGLCQiMyc9I2VdKnlLdiQqKkYsNyQkITNKKysrL3NpcW1GLCQiMyM9cyJcdTYhZSEqKkYsNyQkITNXKysrKHkkcFppRiwkIjMlSGgoUkBaIlwnKSpGLDckJCEzcioqKioqSHlhRSJlRiwkIjM5Nj5PLF9DNikqRiw3JCQhMyopKioqKio0PnMlSGFGLCQiMyp5YE04aSo0XygqRiw3JCQhMyUpKioqKioqXCQqNCkqXEYsJCIzQXpxZHBmM3AnKkYsNyQkITM4KysrXV8mXGMlRiwkIjNkKEg0eUAqUl8mKkYsNyQkITMwKysrXTFhWlRGLCQiMztudWIjR2JFUCpGLDckJCEzMCsrK0ZXLGVSRiwkIjMtRF0iKT1qamAjKkYsNyQkITMxKysrLyMpW29QRiwkIjMpKUc6XjoqKipwNCpGLDckJCEzPisrKyU+QkphJEYsJCIzenl0RCJmK14kKSlGLDckJCEzRCsrKyQ9ZXhKJEYsJCIzbFpMLnBwIkdWKUYsNyQkITMpKSoqKioqelhXbzckRiwkIjNrdEprJVI2YSl5Riw3JCQhMzIrKytMMiRmJEhGLCQiM15beGwjUUYkUXBGLDckJCEzdioqKioqKjM8KlsjR0YsJCIzJT0iZSNSKGZlKSpmRiw3JCQhMyoqKioqKipcb19RciNGLCQiMydbRUcsRnUheldGLDckJCEzQSsrK2hPIkdnI0YsJCIzV1FNdStcNmJHRiw3JCQhMyMqKioqKipwanU8XCNGLCQiM3ImZmQrIyl6OlAiRiw3JCQhMy0rKysnRydbJFIjRiwkIjNueiFRd24oej48ISM+NyQkITM5KysrTno+JkgjRiwkITMnKjNYInovZkdDKkZkdjckJCEzKSoqKioqKlJlNHA+I0YsJCEzKip6PVtgISpcPz5GLDckJCEzMysrK0w3aSk0I0YsJCEzbFRQO2hPLj9HRiw3JCQhMyUqKioqKipcVlhIKT1GLCQhMytqVkkjKT0/c1dGLDckJCEzMysrK1AncHNtIkYsJCEzWG0wJFFjVFppJkYsNyQkITMiKioqKioqUkYmKj1ZIkYsJCEzci9kNHJkSCU9J0YsNyQkITM1KysrNzRfYzdGLCQhM2RoUSc9Rj5XWidGLDckJCEzJSoqKioqKjRoeCRcNkYsJCEzJFEkSG5RJlImZmxGLDckJCEzMCsrKzVWQlU1RiwkITNkQ2Z3ZlUjNGgnRiw3JCQhM0grKyshNDU0TipGZHYkITMmeSJ5bnkqSFNqJ0YsNyQkITMwKysrITN4JXojKUZkdiQhM1MnSCYpMzFRRWonRiw3JCQhMzwrKytTcmw2akZkdiQhM3lkRWpMY2h1bEYsNyQkITNIKysrK3MkUU0lRmR2JCEzSyg9QzdvaUxYJ0YsNyQkITM1KysrYnZGQEFGZHYkITNxKipbNUk/QmdpRiw3JCQhM2MqKioqKioqKjR6cikqISNAJCEzNXQ6a3piZjdnRiw3JCQiM0MrKysrIW8ySiVGZHYkITNJUHQuT1UoelQmRiw3JCQiMyQpKioqKioqKj4uK0InRmR2JCEzMSkpNCc0W2g3PyZGLDckJCIzNisrKyslUSNcIilGZHYkITNsMHpzJnolSHldRiw3JCQiMyoqKioqKioqXF1XbicpRmR2JCEzV1JtV15EInoxJkYsNyQkIjMnKSoqKioqKipwXmM9KkZkdiQhM3M7WDNPbCgpb11GLDckJCIzdSoqKioqKlwkZVFxKkZkdiQhM1J2V2J3MiM9MyZGLDckJCIzJyoqKioqKioqXDFBLSJGLCQhM3EkXF8iPk0+Ml5GLDckJCIzJSoqKioqKipIeSVlNyJGLCQhM3c/dmFLJltmPiZGLDckJCIzMCsrK2ciKltIN0YsJCEzKCo9MG4iRypHTWBGLDckJCIzLSsrK0lDalY5RiwkITMhR3E/IVtXYGFkRiw3JCQiMyYpKioqKioqKnB2eGwiRiwkITNbYSE+IilSNDRIJ0YsNyQkIjMsKysrSTB4dz9GLCQhM2ZkOTFDKD1rTyhGLDckJCIzKikqKioqKipmJnBAWyNGLCQhMy9SJ1JfPno8PSlGLDckJCIzSSsrKyEzJ0hLSEYsJCEzdCdIWSVvYFUneSlGLDckJCIzKSoqKioqKipwWnZPTEYsJCEzck1JJyozIVJZOCpGLDckJCIzPCsrK10yZ29QRiwkITNdN3Ncb2UkelAqRiw3JCQiMyoqKioqKioqKlI8KmZURiwkITMiZnhKaUF6al8qRiw3JCQiM0ArKytdKUh4ZSVGLCQhMykpeWNqOVpIUicqRiw3JCQiMz8rKytJIW8tKlxGLCQhM0FBb2JycCJbcipGLDckJCIzcCoqKioqKjRrLjZhRiwkITNMP0tXXlokcHgqRiw3JCQiM1UqKioqKio+V1RBZUYsJCEzQD5NPEE+bEcpKkYsNyQkIjMpKioqKioqKmYhKjNgaUYsJCEzWTJFPVttKFIoKSpGLDckJCIzQisrK0kqenltJ0YsJCEzOy5jMHpbIyk0KipGLDckJCIzYyoqKioqKjROMSM0KEYsJCEzaz51NSQqKlwjUioqRiw3JCQiM3YqKioqKipIWXQ3dkYsJCEzIVIieUheWXZoKipGLDckJCIzYysrKyt4RyoqeUYsJCEzS2pBWj5EJ3AoKipGLDckJCIzcCoqKioqKlI2S1UkKUYsJCEzKEcoXDUnb10nKSkqKkYsNyQkIjM0KysrSWJkUSgpRiwkITN3eEAjW3QpKVwqKipGLDckJCIzRisrK2dgMWgiKkYsJCEzc3FHakNZXykqKipGLDckJCIzWCsrKytQRGokKkYsJCEzQzgvLSpvYSQqKioqRiw3JCQiM2EqKioqKipSP1dsJipGLCQhM2FZT0deW3oqKioqRiw3JCQiM1srKytJOjN1JypGLCQhMydcYllFWDgqKioqKkYsNyQkIjNMKysrPzVzI3kqRiwkITNzdzt5YlYoKioqKipGLDckJCIzQysrK2wyL1ApKkYsJCEyZ0cmRyI9KikqKioqKkYvNyQkIjM7KysrNTBPIiopKkYsJCEyWzVaV3onKioqKioqRi83JCQiMyIqKioqKioqei5fPSoqRiwkITIvJ3BsWicpKioqKioqRi83JCQiMzMrKytiLW9YKipGLCQhMiFSZEkqZioqKioqKipGLzckJCIzJyoqKioqKioqPWcjZioqRiwkITJzMmQ0JCkqKioqKioqRi83JCQiM0QrKytJLCVHKCoqRiwkITJfQTgqXCoqKioqKioqRi83JCQiMzgrKytsK1UnKSoqRiwkITJHOlJQKioqKioqKioqRi83JEkqdW5kZWZpbmVkR0kqcHJvdGVjdGVkR0ZbaGxGamdsLSUmQ09MT1JHNiYlJFJHQkckIiM1ISIiJCIiIUZiaGxGY2hsLSUrQVhFU0xBQkVMU0c2JFEieDYiUSFGaWhsLSUqQVhFU1RJQ0tTRzYlNyUvJCEjNUZiaGxRKi1pbmZpbml0eUZpaGwvRmNobFEiMEZpaGwvRmBobFEpaW5maW5pdHlGaWhsRl5pbC0lJUZPTlRHNiMlKERFRkFVTFRHLSUlVklFV0c2JDtGYGlsRmBobDskITI7dlskKioqKioqUjUhIzskIjJldVUkKioqKioqUjVGYmps

From the plot it follows that the function f has one real zero.Finally, one finds the real root of the equation. u:=evalf(fsolve(f(x),x),21);

NiM+SSJ1RzYiJCE2ai8hUnYmeXdBSCE+ISM/

PROBLEM 3curve:=x^2 *y^3 + y^2 + y -2 *exp(x);

NiM+SSZjdXJ2ZUc2IiwqKiZJInhHRiUiIiNJInlHRiUiIiQiIiIqJEYqRilGLEYqRiwtSSRleHBHNiRJKnByb3RlY3RlZEdGMUkoX3N5c2xpYkdGJTYjRighIiM=

We find the slope of the tangent line at (0,1)slope:=eval(subs({x=0,y=1},implicitdiff(curve,y,x)));

NiM+SSZzbG9wZUc2IiMiIiMiIiQ=

The tangent line has slope 2/3 and passes through the points (0,1). Therefore, it has equation y=x*2/3+1We do the required graph.with(plots):

Warning, the name changecoords has been redefined

plots[display](plot(x*slope+1,x=-4..4,thickness=1), implicitplot(curve,x=-4..4,y=-4..4,color=blue,thickness=1));

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

PROBLEM 4One first defines a function g that gives the sum of the first k primesg:=k->sum(ithprime(j),j=1..k);

NiM+SSJnRzYiZio2I0kia0dGJUYlNiRJKW9wZXJhdG9yR0YlSSZhcnJvd0dGJUYlLUkkc3VtR0YlNiQtSSlpdGhwcmltZUdGJTYjSSJqR0YlL0YyOyIiIjkkRiVGJUYl

By definition, g is an increasing funtion. One computes the value of g at certain values to find an interval of integers where the solution lies.seq([i*100+1,g(i*100+1)],i=0..10);

Ni03JCIiIiIiIzckIiQsIiImIW9DNyQiJCwjIic7RzY3JCIkLCQiJ2FJRjckIiQsJSIndTBeNyQiJCwmIid1I0cpNyQiJCwnIig7dkEiNyQiJCwoIihxRHIiNyQiJCwpIigtVEcjNyQiJCwqIihzQyVINyQiJSw1IihTM3Ak

Since g is increasing the solution is in the interval [101,200] (because g(101)=24680 and g(200)=112816) Now we look for a smaller interval.seq([100+i*10,g(100+i*10)],i=1..10);

Niw3JCIkNSIiJigqKUg3JCIkPyIiJkZpJDckIiRJIiImLEslNyQiJFMiIiYoKTMmNyQiJF0iIiZwI2Y3JCIkZyIiJlQkbzckIiRxIiImXCJ5NyQiJCE9IiYmZSkpNyQiJCE+IiYmbyoqNyQiJCsjIicoZTYi

Hence, the value we are looking for is between 180 and 190.To find the exact value we compute all the values of g between 180 and 190seq([180+i,g(180+i)],i=0..10);

Ni03JCIkIT0iJiZlKSk3JCIkIj0iJnMnKik3JCIkIz0iJmoyKjckIiQkPSImYz0qNyQiJCU9IiZgSCo3JCIkJj0iJmNTKjckIiQnPSImbF4qNyQiJCg9IiYjRycqNyQiJCk9IiYwdSo3JCIkKj0iJk0mKSo3JCIkIT4iJiZvKio=

Therefore, the value we are looking for is 181.PROBLEM 5Since maple ussually works with less digits, first one makes it use as many digits as we need. Digits:=105;

NiM+SSdEaWdpdHNHNiIiJDAi

Now, we can define a function that gives the kth decimal digit of pidc:=k->evalf(modp(trunc(10^k*Pi),10));

NiM+SSNkY0c2ImYqNiNJImtHRiVGJTYkSSlvcGVyYXRvckdGJUkmYXJyb3dHRiVGJS1JJmV2YWxmR0kqcHJvdGVjdGVkR0YuNiMtSSVtb2RwR0YuNiQtSSZ0cnVuY0dGLjYjKiYpIiM1OSQiIiJJI1BpR0YuRjpGOEYlRiVGJQ==

The above function dc does the following: Takes an integer k, and then "moves" the period of pi k places (that is, multiplies by 10^k). Then removes the decimal part of the obtained number (this is what the command "trunc" does). Hence, after doing trunc we obtain an integer number. Then, by considering this number mod 10, takes the last digit of this integer number. This is the kth decimal digit of pi.We list the sequence.PI:=[seq([i,dc(i)],i=1..100)];

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

We do the required plot (since we do not ask style=point, it automatic produces the required polyline.plot(PI);

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