1. Let f (x) = x
√ x2+3.
(a) (5%) Show that f (x) is a one-to-one function.
(b) (5%) Let g(x) be the inverse function of f (x). Find g′(−2).
2. (a) (6%) Set a, b ∈ R. If limx→0
√
ax + b − 2
x =2, then a, b =?
(b) (6%) Find lim
n→∞( n + 2 n − 2)2n.
3. (a) (5%) Let y = cot(cos2(3x)). Find dy dx. (b) (5%) Let y = xln x, x > 0. Find dy
dx. (c) (5%) Find the 52th derivative of sin 3x.
4. (10%) Prove the inequality ∣ tan a − tan b∣ ≥ ∣a − b∣ for a, b ∈ (−π 2,π
2), a ≠ b.
5. (10%) Use linear approximation to estimate the value of esin−1(−0.0002).
6. (10%) Find an equation of the tangent line to the curve x cos2y = sin y at the point (0, π).
7. (23%) Let y = f (x) = (x + 1)3
x2+2x. Find the following
(a) the intervals on which y = f (x) increases (3%)
the intervals on which y = f (x) decreases (3%)
(b) the intervals on which y = f (x) is concave up (3%)
the intervals on which y = f (x) is concave down (3%)
(c) the local maximum(if exists) of y = f (x): (coordinates) (2%) the local minimum(if exists) of y = f (x): (coordinates) (2%)
(d) all asymptotes of y = f (x) (4%)
(e) Sketch the graph of y = f (x). (3%)
8. (10%) A truck gets 500/x kilometers per liter when driven at a constant speed of x kph (between 60 and 120 kph).
If the price of fuel is$20/liter and the driver is paid $400/hour, at what speed between 60 and 120 kph is it most economical(minimal sum of the fuel cost and driver’s pay) to drive for 400 kilometers?
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