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MATHEMATICS 120 PROBLEM SET 7 Due October 30, 2002 For full credit, please show all work. 1. (5 marks) Find all values of a for which the equation ln x = ax2 has exactly one solution. 2. (5 marks) A radioactive substance decays at a rate proportional to the amount present. If 10% of the substance decays in 2 years, how long will it take for 80% of it to decay? x 3. (5 marks) Show that arctan x = arcsin √ for all x. 1 + x2 4. (5 marks) Prove that cosh x > 1 + x2 for all x 6= 0. 2 Please read Sections 3.3–3.6 of the textbook (you may skip “interest on investments” and ”logistic growth”, Section 3.4). The recommended practice problems are: Section 3.3, 1–50 and 57–68; Section 3.4, 9–15; Section 3.5, 1–34 and 45–48; Section 3.6, 1,2,5,7. 1