1. (13%) Find the derivative of the functions.
(a) (5%) ln (
√
1 − x2−x). (b) (8%) (tan x)sin x.
2. (10%) Evaluate (a) (5%) lim
x→0x (cos x + cos1 x).
(b) (5%) lim
x→1
√x x2+1−
√2 2
x − 1 . 3. (8%) Let f (x) = (x + 1)(x3+x2+1)(x2−x + 1)
ex+1x+2
√
x2+1 . Find f′(−1).
4. (12%) Let
√ 3
2 +xy = sin y.
(a) (6%) Find the equation of the tangent line at (0,π 3).
(b) (6%) Find d2y
dx2 and evaluate at (0,π 3). 5. (12%) Let f (x) = x3+x + cos x, x ∈ R.
(a) (6%) Show that f (x) is a one-to-one function.
(b) (6%) Let g(x) be the inverse function of f (x). Find g′(1).
6. (10%) Show that y = 1 − x and y = cos x intersect at only one point.
7. (25%) Let y = f (x) = x(x − 1) + 2
x + 1 . Find the following (a) the intervals on which y = f (x) increases
the intervals on which y = f (x) decreases (6%)
(b) the intervals on which y = f (x) is concave up
the intervals on which y = f (x) is concave down (6%)
(c) the local maximum(if exists) of y = f (x): (coordinates) the local minimum(if exists) of y = f (x): (coordinates) (6%)
(d) all asymptotes of y = f (x) (4%)
(e) Sketch the graph of y = f (x). (3%)
8. (10%) A boat leaves a dock at 5 ∶ 00 PM and travels due north at a speed of 20 km/h. Another boat has been heading due west at 15 km/h and reaches the same dock at 6 ∶ 00 PM. At what time were the two boats closest together?
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