課程名稱︰微積分甲上 課程性質︰物理系必帶 課程教師︰張志中 開課學院：理學院 開課系所︰數學系 考試日期（年月日）︰2017/11/9 考試時限（分鐘）：110分鐘 是否需發放獎勵金：是 （如未明確表示，則不予發放） 試題 : 1.(12%) Let f(x) be a function defined R\{0} such that lim(x→0)f(x)=L exists. Now a new function g is defined on (-1,∞) by (cos(x)-1)ln(1+x^2)f(x) ------------------------, x>-1,x≠0 g(x)={ x^3 a, x=0 (a)(6%) Determine a so that g(x) is continuous at x=0. (b)(6%) Find L if g(x) is differentiable at x=0 and g'(0)=2. 2.(24%) Evaluate each limit. (a)(8%) lim (3-x)^tan(πx/4), x→2 (b)(8%) x^(3/2)(cos(1/x)-1)cos(x) lim -------------------------, x→∞ e^cos(x) (c)(8%) First find the value of k∈R such that the limit 2xcos(x)+sin(x)+tan^-1(kx) lim -------------------------- =L∈R exists. Next find L. x→0 x^3 3.(18%) Let f(x)=ln((1+x)/(1-x))-2x, g(x)=x^3, -1<x<1. (a)(8%) For -1<a<b<1, what does Cauchy mean value theorem tell about the quotient (f(b)-f(a))/(g(b)-g(a))? (b)(6%) Apply the above result to find an upper bound and a lower bound of ln2 . (c)(4%) Use the mid-point to find an approximation of ln2, and then estimate the error. 4.(18%) Jane is 2km offshore in a boat and wishes to reach a coastal village 6 km down a straight shoreline from the point nearest the boat. She can row 3km/h and can walk 5km/h. Where should she land her boat to reach the village in the least amount of time? 5.(8%) Suppose that f''(x)>0 on an open interval I. Show that the graph of y=f(x) lies above all of its tangents on I. (Hint. Investigate the monotonicity of certain function.) 6.(30%) Answer the following questions and sketch the graph of x^3 y=f(x)=---------,x≠1. (x-1)^2 (a) Evaluate f'(x)=_______________________________________________________ (b) Evaluate f''(x)=______________________________________________________ (c) f(x) is increasing on the interval(s):________________________________ (d) f(x) is concave upward on the interval(s):____________________________ (e) f(x) has (a) local maximum/maxima at (x,y)= (if any)__________________ and (a) local minimum/mimima at (x,y)= (if any)_______________________ (f) f(x) has inflection point(s) at (x,y)= (if any)_______________________ (g) f(x) has asymptote(s) (if any)________________________________________ (h) Sketch the graph of y=f(x). --

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