1. (25%) Let y = x2 2 +1
x. Answer the following questions. Fill each blank and give your reasons (and computations). Put None in the blank if the item asked does not exist.
(a) The function is increasing on the interval(s) and decreasing on the
interval(s) (6% total). The local maximal point(s) (x, y) = (2%),
The local minimal point(s) (x, y) = (2%). Reason:
(b) The function is concave upward on the interval(s) and concave downward
on the interval(s) (6% total).
The inflection point(s) (x, y) = (2%). Reason: (c) The asymptote lines of
the function are (3%). Reason:
(d) Sketch the graph of the function. Indicate, if any, where it is increasing/decreasing, where it concaves upward/downward, all relative maxima/minima, inflection points and asymptotic line(s) (if any).(4%)
2. (15%) Suppose y = f (x) satisfy tan−1 y
x + lnx2+ y2
2 = π
4. Find y0 and y00 in terms of x and y.
Find their explicit values for (x, y) = (1, 1).
3. (12%) Find the equation of the line tangent to y = cos x
1 + sin x at x = π 6. 4. (12%) Find lim
x→0
cos x − 1 x sin x .
5. (12%) Find the linear approximation of (128)13 (by a differential).
6. (12%) Let g(x) be the inverse function of f (x) = x5+ 2x3+ x − 2. Find g0(f (1)).
7. (12%) Prove that y = x and y = tan−1x intersect at only one point.
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