1. (14%) Find the following limits.
(a) (7%) lim
x→0
x sin x
1 − cos x. (b) (7%) lim
x→∞(1 + 2 x)
3x
. (Hint: lim
x→∞(1 +1 x)x=e) 2. (14%) Find the first derivative of the following functions.
(a) (7%) f (x) = tan(2x). (b) (7%) f (x) = (sin x)x.
3. (12%) Given tan−1y
x =2xy − 2y2+ π
4, find the first derivative and the second derivative, dy dx, d2y
dx2 at (1, 1).
4. (8%) Estimate 4
√
10004 by a linear approximation.
5. (8%) Show that − ln(1 − x) > x +1
2x2for 0 < x < 1.
6. (14%) As in the picture, consider a trapezoid inscribed in the unit circle such that one base is the diameter.
(a) (4%) Describe the area of the trapezoid as a function of θ. Denote this function by f (θ).
(b) (7%) Find critical numbers of f (θ) for 0 < θ <π 2.
(c) (3%) Find the absolute maximum value of f (θ) for 0 ≤ θ ≤ π 2. 7. (14%) Let y = f (x) =
√
4x2+x, for x ≤ −1
4 or x ≥ 0. Find slant asymptotes of y = f (x).
8. (16%) Consider f (x) = 3 ln(x2−1) − 4x.
(a) The domain of f (x) is .
(b) f′(x) = . f (x) is increasing on (intervals).
f (x) is decreasing on (intervals).
(c) f′′(x) = .
f (x) is concave upward on (intervals, if any).
f (x) is concave downward on (intervals, if any).
(d) At x = , f (x) has local maximum value . (If there is any
local maximum value.)
At x = , f (x) has local minimum value . (If there is any
local minimum value.)
(e) Find vertical asymptotes of y = f (x).
(f) Draw the graph of y = f (x).
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