課程名稱︰微積分上 課程性質︰工科海洋大一必修 課程教師︰黃維信 開課學院：工學院 開課系所︰工科海洋系 考試日期（年月日）︰98.11.12 考試時限（分鐘）：110min 是否需發放獎勵金：是 （如未明確表示，則不予發放） 試題 : 1. Give an ε δ proof for lim √x=1 (10%) x→1 2. True or False?(10%)Justify your answers.A function f is defined on the interval[a,b] (a)If f(a)>0 and f(b)<0 then there must exist at least one number c in (a,b),for which f(c)=0 (b)If f is continuous on [a,b]with f(c)=0 for some number c in (a,b), then f(a) and f(b) have opposite signs. 3. Let f be a differentiable function. Use the chain rule to show that:if f is odd,then f' is even.(10%) 4. Find equations for the lines tangent and normal to the curve ysin2x-xsiny=π/4 at the points (π/4,π/2).(10%) 5. Find a formula for all order of derivative of y=(a-bx)^n.(10%) 6. Let G(x)={xsin(1/x) x≠0 Sketch a figure that shows the general nature { 0 x=0 of the graph of G.Determine whether G is continuous or differentiable at x=0?(20%) x 7. Let F(x)=∫t(t-3)^2 dt. Find the critical points,the points of inflection, 0 if any,determine the concavity,and sketch the graph of F.(20%) x t 8. Let f be everywhere continuous and set F(x)=∫[t∫f(u)du]dt.Find (a)F'(x) 0 1 (b)F'(1) (c)F''(x) (d) F''(1) (10%) --

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