Math 1A Final 2004-12-15 12:30-3:30You are allowed 1 sheet of notes. Calculators are not allowed. Each question is worth1 mark, which will be given only for a clear correct answer and correct working. There isno partial credit for wrong answers.1. Find a formula for the inverse of the functiony= 3x3+ 2.2. Evaluate the limit limh→0(2+h)3-8h.3. Prove thatex= 2 +xhas at least one real root.4. Differentiatef(x) = (ax+b)/(cx+d).5. Find the derivative of the functiony= tan3(2x).6. Finddy/dxif 4 cos(x) sin(y) = 1.7. Find the critical numbers of the functionf(x) =xe2x.8. Show that the equationx4+ 4x+c= 0 has at most two real roots.9. Find limx→0cos(x)-1x2.10. Find the points on the ellipse 4x2+y2= 4 that are farthest away from the point(1,0).11. Explain why Newton’s method does not work for finding the root of the equationx3-3x+ 6 = 0 if the initial approximation is chosen to bex1= 1.12. Use one iteration of Newton’s method applied to the initial approximationx1= 2 tofind 361/5correct to two decimal places.
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