**Unformatted text preview: **10/03/2001 WED 17:33 FAX 6434330 MOFFITT LIBRARY I001
George M. Bergman Fall 1998, Math 1AW 6 November, 1998 2050 VLSB Second Midterm 3: 10—4:00 PM 1. (50 points, 10 points apiece) Find the following. 2x” ex+1 (a) llmxal 16—1 1— cos 2x 03) llmeO 1 u-cos 3x' (c) An antiderivative of the function x + x_1 (d) A function satisfying the differential equation f ’(x) = ~—9 f(x), and such that
f (1) - f(0) = 1 7 i cosh x
(6) dx 1 + x 2. (25 points) (a) (10 points) State the principle of mathematical induction. (b) (15 points) Suppose f is an inﬁnitely differentiable function (i.e., a function such that
f ’, f ”, , fol), all exist). Prove that for all positive integers 11, one has
D" (x f(x)) : xf(n)(x) + nf("_1)-(x). Here fan means f. Suggestion: Use mathematical induction. x2+1 5 3. (25 points) (a) (15 points) Give the information asked for below about the curve y = in If any of the items asked for does not exist, write “None”. (For limits, write “None”
only if the function does not approach either a real number or i co.) (b) (10 points) Sketch the curve._ Your sketch does not need to reﬂect accurate
numerical values of the coordinates of the various transition points, as long as it correctly shows the order in which these occur. x—intercept(s): y—intercept: increasing on interval(s): decreasing on interval(s): concave up on interval(s): concave down on interval(s): vertical asymptote(s): 11m hmx __>_oo y = _ extrema: 36—)on = ...

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