1. (20%) Find the following limits.
(a) (5%) lim
x→1
√
2x2+3x − 4 − x
x − 1 . (b) (5%) lim
x→0−x ⋅
√ 1 + 9
x2. (c) (5%) lim
x→0
cos 3x − 1
x2 . (d) (5%) lim
x→∞(1 + e x)
2x
. (Hint: lim
x→∞(1 + 1 x)
x
=e.)
2. (15%) Find the derivative of the function.
(a) (5%) f (x) = sin 2x
1 − cos 2x. (b) (5%) f (x) = tan−1(
√ ex−1).
(c) (5%) f (x) = (sec x)x, −π
2 <x < π 2. 3. (14%) f (x) = x5+2x − 3.
(a) (4%) Explain that f (x) is 1-1.
(b) (5%) We know that f (x) has inverse function from (a). Find f−1(−3) and (f−1)′(−3).
(c) (5%) Write down the linear approximation of f−1(x) at x = −3. Estimate f−1(−3.01) by the linear approximation.
4. (9%) Find the equation of the tangent line to the curve ln(x2−3y) = x − y − 1 at the point (2, 1).
5. (6%) By the Mean Value Theorem, explain that a
√ 1 − a2
<sin−1(2a) − sin−1(a) < a
√
1 − 4a2, where 0 < a <1 2.
6. (12%) For a right cone with base radius r and height h, r, h > 0, find the inscribed right cylinder (as figure) of maximal volume.
h
r
7. (24%) Follow the steps to sketch the graph of the function f (x) = 1 x−
1 3x3. (a) (2%) Discuss the symmetry of y = f (x).
(b) (6%) Compute f′(x). Find interval(s) of increase and interval(s) of decrease of f (x).
(c) (2%) Classify (local) extreme values.
(d) (6%) Compute f′′(x). Discuss concavities of y = f (x).
(e) (2%) Find inflection point(s) of y = f (x).
(f) (2%) Find asymptotes of y = f (x).
(g) (4%) Sketch the graph of f (x).
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