课程名称︰微积分 1 课程性质︰生物环境系统工程学系/工程科学及海洋工程学系/地质科学系必修 课程教师︰吕治鸿 开课学院：理学院 开课系所︰数学系 考试日期︰2025/10/13(一)，17:30-18:20 考试时限：50 分钟 试题 : 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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