1. (15%) On the curve y

1. (15%) On the curve y³+ xy²+ x²y− 2x³= 1, use implicit differentiation to find (a)^dy_dx and (b) the values of ^dy_dx, ^d_dx^2y₂ at the point x= 1, y = 1.

2. (12%) Find the point on y²= 2x that is closest to the point (1, 4).

3. (21%) Let y= f(x) = _1+x^x32. Answer the following questions. Fill each blank and give your reasons (and computations). Put None in the blank if the item asked does not exist.

(a) The function is increasing on the interval(s) (5%).

The local maximal point(s)(x, y) = .The local minimal point(s)(x, y) = .(2%

total)Reason:

(b) The function is concave upward on the interval(s) and concave downward

on the interval(s) .(5% total)

The inflection point(s)(x, y) = (2%). Reason: (c) The asymptote lines of the function are (3%). Reason:

(d) Sketch the graph of the function. Indicate, if any, where it is increasing/decreasing, where it concave upward/downward, all relative maxima/minima, inflection points and asymptotic line(s) (if any).(4%)

4. (10%) (a) Find the linear approximation of(1 + x)^1/3 at x= 0.

(b) Use (a) to estimate √³ 8.03.

5. (12%) Let f(x) = 3x − sin x.

(a) Prove that f(x) is an increasing function.

(b) Let g(x) be the inverse function of f(x). Find g^′(0).

6. (20%) Evaluate y^′. (a) y= 3^{cos x}

(b) y= tan⁻¹(1 + x²) (c) y= ln(x +√

1+ x²) (d) y=_{1+sin x}^{cos x}

7. (10%) Evaluate the following limits.

(a) lim

x→1⁺ ln x x−1

(b) lim

x→2

√

x³+x²−8−2 x−2

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