96年 - 96 國立政治大學轉學生招生考試_應用數學系：微積分(一)#122393

1. Prove that if θ is measured in radians, then = 1. (10%)

2. Use the definition of derivative to find f'(x) for the function f(x) = 2/x. (10%)

(a) x + tan(xy) = 0.

(b) y = 2^sin3t

(c) y = (θ+sinθ)/

(d) y = (5% each)

(a) Find the equations of the asymptotes.

(b) Determine rise and fall (increasing, decreasing).

(c) Determine the concavity.

(d) Sketch the graph. (5% each)

5. Find all local maxima, local minima, and saddle points of the function f(x,y) = 6x²-2x³+3y²+6xy. (10%)

6. Find the points on the hyperbolic cylinder x²-z²-1=0 closest to the origin. (10%) ( Use the Method of Lagrange Multipliers)

7. Water runs into a conical tank at the rate of 9 ft³/min. The tank stands point down and has a height of 10 ft and a base radius of 5 ft. How fast is the water level rising when the water is 6 ft deep? (see the figure below) (10%)

the conical tank in problem 7.

8. Show that f(x,y) =

is continuous at every point except the origin. (10%)

96年 - 96 國立政治大學轉學生招生考試_應用數學系：微積分(一)#122393

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