課程名稱︰微積分 1 課程性質︰生物環境系統工程學系/工程科學及海洋工程學系/地質科學系必修 課程教師︰呂治鴻 開課學院：理學院 開課系所︰數學系 考試日期︰2025/10/13(一)，17:30-18:20 考試時限：50 分鐘 試題 : National Taiwan University - Calculus 1 (Class 09) - Quiz 2 2025/10/13 (Monday) - 50 minutes Name: ＿＿＿＿＿＿＿＿＿＿＿＿＿＿ Student ID Number: ＿＿＿＿＿＿＿＿＿＿＿＿ There are FOUR questions in this quiz. Your work is graded on the quality of your writing as well as the validity of the mathematics. 1. (20 points) Find the derivative of each of the following functions. 1/x (a) (5 points) Let f (x) = (sin x) for 0 < x < π. Find f '(x) 1 1 3 h(2exp(x-2)) (b) (5 points) Let f (x) = x g( ---------------). Find f '(2). 2 x 2 Here functions g and g satisfy ┌────┬───┬────┬───┬────┐ │ x = a │ g(x) │ g'(x) │ h(x) │ h'(x) │ ├────┼───┼────┼───┼────┤ │ a = 2 │ -1 │ 2 │ 6 │ 5 │ ├────┼───┼────┼───┼────┤ │ a = 3 │ 7 │ -4 │ 20 │ -12 │ └────┴───┴────┴───┴────┘ -1 -1 (c) (10 points) Let f (x) = 5x - 2arcsin(x). Find (f )'(0) and (f )''(0). 3 3 2. (8 points) Consider the equation x ln(y) = sin(x-2y). Suppose that y can be expressed implicitly as a function y = y(x) in x near the point (2,1). dy │ (a) (6 points) Find the value of ---- │ dx │(x,y) = (2,1). (b) (2 points) Use the linear approximation to estimate a solution of the equation 2.01ln(y) = sin(2.01-2y). x^3 - 2x^2 + 4x - 2 3 1 3. (17 points) Let f(x) = ─────────── = x + ─── + ──── (x-1)^2 x-1 (x-1)^2 for x ≠ 1. (a) (4 points) Find all asymptotes. (b) (4 points) Find f'(x) and f''(x). (c) (2 points) Find the interval on which f is increasing and decreasing. (d) (1 point) Find the point(s) of inflection of y = f(x). (e) (6 points) Sketch the graph of y = f(x). 4. (5 points) (a) (1 point) State the statement of Rolle's theorem. (b) (1 point) Let f and g be differentiable on |R, and a < b be real numbers. Define h : [a,b] → |R by h(x) = (f(b)-f(a))(g(x)-g(a)) - (g(b)-g(a))(f(x)-f(a)) for x ∈ [a,b]. By using Rolle's theorem to show thatthere exists c ∈ (a,b) such that h'(c) = 0. (c) (3 points) Use the result of (b) to show ln(b) - ln(a) 　　　　　　　　　　──────────── ≧ 2 for 0 < a < b. arctan(b) - arctan(a) --

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