Math 1A - Fall 2004 - Borcherds - Final

Math 1A - Fall 2004 - Borcherds - Final - Math 1A...

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Math 1A Final 2004-12-15 12:30-3:30 You arc allowed 1 shcct of notes. Calculators are not allowed. Each question is worth 1 nrark, which will be givcn only for a clcar correct answer and corrcct working. Thcre is no p:rrtial credit for wrong answors. 1. Firxl a forrnula for the inverse of the function tl = 3r3 + 2. 2. Evaluate the limit limr.-.o Glp!. 3. Prove that et = 2 + x has at least one real root. 4. Diflerentiatc f(x) = (tm + b)lQr + tl). 5. FiId the derivative of the function g = tan3(2g). 6. Fitnl dp/dt if 4 cos(r) sin(g) - L 7. Find thc critical numbers of the function [email protected]) = x.2". 8. Show that thc equationxa + 4x + c = O has at most two real roots. 9. Find lim,*o !e{9:-!. 10. Find the points on the ellipse4o2 * y2 = 4 that are fd,rthest away ftom the point (r, 0). 11. Explain why Newton's method does not work for finding the root of the equation s3 - 3z + 6 = 0 if the initial approximation is chosen to be or = 1. 12. Usc onc iteration of Newton's method applied to the initial approxirnation irl : 2 to find 361/5 correct to two dccimal places.
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