MATH 115 Final: MATH 115 UPenn 115FinalFall 11 Exams
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Student no: answer all questions from section a, you may use a handwritten sheet of notes. Calculators are not permitted: read all questions carefully, unless explicitly told otherwise, you should explain all your answers fully, do not seperate the pages of your exam. In each case, nd dy dx . (a) y = xe x. Using the quotient-rule dy dx (1 + x2) d dx (xe x) xe x d (1 + x2)2 dx (cid:0)1 + x2(cid:1) (1 + x2)(e x xe x) xe x(2x) (1 + x2)2. = e x 1 x x2 x3 (1 + x2)2 (b) y = (sin x)x. Use logarithmic di erentiation: so hence (c) xy3 = 4. Use implicit di erentiation: so ln y = x ln(sin x) = ln(sin x) + x cos x sin x dy dx. = (sin x)x (ln(sin x) + x cot x) . y3 + 3xy2 dy dx. Problem a2: [16 pts] find the following integrals. (a) z.
