(You can not use l’Hospital’s Rule.) (a) (4%) lim x

1. (22%) Compute the following limits. (You can not use l’Hospital’s Rule.)

(a) (4%) lim

x→−∞

e^−x+2

3e^x−e^−x. (b) (6%) lim

x→∞

√

x²−x − x + tan⁻¹x.

(c) (6%) lim

x→0

cos x − 1

x² . (d) (6%) lim

x→1

ln x x³−1. 2. (20%) (a) (6%) Let f (x) = x

3x²+4. Calculate f^′(x).

(b) (7%) Let f (x) = tan⁻¹(e^x) +sin⁻¹(cos(2x)). Calculate f^′(x).

(c) (7%) Let f (x) = x^{sin x}. Calculate f^′(π).

3. (10%) Consider the plane curve given by the equation x³+xy +^y₈³ =4.

(a) (6%) Compute ^dy_dx at the point (1, 2).

(b) (4%) The curve near the point (1, 2) can be described as y = f (x). Use the linear approximation to estimate f (1.01).

4. (16%) Let f (x) = tan x, x ∈ [0,^π₂).

(a) (6%) Show that tan x > x for x ∈ (0,^π₂).

(b) (4%) Using the result from (a), explain that f^′(x) = 1 + f²(x) > 1 + x², for x ∈ (0,^π₂). (c) (6%) Show that tan x > x +¹₃x³ for x ∈ (0,^π₂).

5. (20%) Sketch the graph of the function f (x) = ln ∣x + 1∣ + 5x + x². (a) (2%) Write down the domain of f (x). Find lim

x→−1⁻f (x) and lim

x→−1⁺f (x).

(b) (7%) Compute f^′(x). Find the interval(s) of increase and interval(s) of decrease of f (x). Find local extreme values of f (x).

(c) (7%) Compute f^′′(x). Determine the concavity of y = f (x) and find the inflection point(s).

(d) (4%) Sketch the graph of f (x).

6. (12%) A lamp is hung over the center of a circular table with radius 2m. Find the height h of the lamp over the table such that the illumination I at the perimeter of the table is maximum. We know that I = ^{k sin α}_s2 , where k > 0 is a constant, s is the slant length, and α is the angle at which the light strikes the perimeter. (You need to explain that the extreme value you find is the absolute maximum.)

s

2 h

α

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